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# Isospectral graphs via spectral bracketing

John Stewart Fabila-Carrasco  School of Engineering, Institute for Digital
Communications, University of Edinburgh, Edinburgh EH9 3FB, U.K.
<EMAIL_ADDRESS>,  Fernando Lledó  Department of Mathematics, University
Carlos III de Madrid, Avda. de la Universidad 30, 28911 Leganés (Madrid),
Spain and Instituto de Ciencias Matemáticas (CSIC-UAM-UC3M-UCM), Madrid
<EMAIL_ADDRESS>and  Olaf Post  Fachbereich 4 – Mathematik, Universität
Trier, 54286 Trier, Germany<EMAIL_ADDRESS>
###### Abstract.

In this article, we develop a perturbative technique to construct families of
non-isomorphic discrete graphs which are isospectral for the standard (also
called normalised) Laplacian and its signless version. We use vertex
contractions as a graph perturbation and spectral bracketing with auxiliary
graphs which have certain eigenvalues with high multiplicity. There is no need
to know explicitly the eigenvalues or eigenfunctions of the corresponding
graphs. We illustrate the method by presenting several families of examples of
isospectral graphs including fuzzy complete bipartite graphs and subdivision
graphs obtained from the previous examples. All the examples constructed turn
out to be also isospectral for the standard (Kirchhoff) Laplacian on the
associated equilateral metric graph.

###### Key words and phrases:

discrete normalised Laplacian, isospectral graphs, spectral bracketing

###### 2010 Mathematics Subject Classification:

05C50, 05C76, 47B39, 47A10

JSFC was supported by the Leverhulme Trust via a Research Project Grant
(RPG-2020-158). FLl was supported by the Severo Ochoa Programme for Centres of
Excellence in R&D (SEV-2015-0554) and from the Spanish National Research
Council, through the Ayuda extraordinaria a Centros de Excelencia Severo Ochoa
(20205CEX001) and by the Madrid Government under the Agreement with UC3M in
the line of _Research Funds for Beatriz Galindo Fellowships_ (C&QIG-BG-CM-
UC3M), and in the context of the V PRICIT

##  1\. Introduction

Perturbative techniques in the study of linear Hilbert space operators (or
matrices) have a long and fertile history (see, e.g. [K95, B85]). From a
general point of view, they provide a method of investigation based on the
knowledge of the spectrum of a free (non-perturbed) operator (typically a
Laplacian on some domain of $\mathbb{R}^{n}$ or on a graph) and the analysis
of its stability or variation under a controlled perturbation expressed in
terms of a potential or a geometrical manipulation of the underlying spaces
(see, e.g. [So99, LP07]). In [FLP22a] we presented a systematic study of the
spectral effects of graph perturbations (like edge deletion, vertex
contraction, vertex virtualisation etc.) for arbitrary weights and including
arbitrary magnetic potentials. These effects can be quantified in terms of a
natural spectral preorder. In this article, we present a perturbative method
leading to the construction of isospectral graphs which are interpreted as
perturbations (via vertex merging) of auxiliary graphs which have certain
eigenvalues with high enough multiplicity.

Spectral graph theory studies the relationship between the structure of a
graph and the spectrum of operators naturally associated with the graph which
are usually represented by matrices with respect to a suitable orthonormal
basis. In this article, we consider only simple finite graphs, i.e., there are
no multiple edges nor loops. For a graph $G=(V,E)$, one can define different
natural operators associated with it. Typically, one considers the _adjacency
matrix_ $A_{G}$ with respect to a numbering of the vertices
$V=\\{v_{1},\dots,v_{n}\\}$ having entries $1$ and $0$ depending on whether
there is an edge between two vertices or not. The _combinatorial Laplacian_ is
given by $D_{G}-A_{G}$, where $D_{G}$ is the diagonal matrix with the degree
of each vertex on the diagonal. The _signless combinatorial Laplacian_ is
defined by $D_{G}+A_{G}$. Another well known choice is the _standard
Laplacian_ (also called _geometric_ or _normalised_ in the literature, cf.
Remark 2.3 (a)), represented by

$\Delta_{{G}}=D_{G}^{-1/2}(D_{G}-A_{G})D_{G}^{-1/2}=I_{n}-D_{G}^{-1/2}A_{G}D_{G}^{-1/2},$

where $I_{n}$ is the identity $(n\times n)$-matrix (see also [Ch97] for
additional references and motivation). The _signless standard Laplacian_ is
defined as $\Delta_{{G^{+}}}=I_{n}+D_{G}^{-1/2}A_{G}D_{G}^{-1/2}$ (see also
Subsection 2.2). Both standard Laplacians have their spectrum contained in the
interval $[0,2]$.

An important question with many applications is whether the spectrum of a
certain operator on a graph determines the graph up to isomorphism or not. Two
non-isomorphic graphs having the same spectrum for a certain operator are
called _isospectral graphs_ with respect to that operator. Two non-isomorphic
graphs such that their adjacency matrices have the same spectrum are often
referred to as _cospectral_ cf. e.g. [CDGT88], i.e., isospectral with respect
to the adjacency matrix in our notation. Often, classical problems related to
isospectrality are considered for the _combinatorial_ Laplacian, see e.g.
[CDS95, BrH12, DH03]. Much less is known for the standard Laplacian, as most
previous works concentrate on the adjacency matrix or the combinatorial
Laplacian. We refer in this article to _isospectral graphs_ as _isospectral_
with respect to the _standard Laplacian_ (or its signless version). In recent
time, though, results and constructions of isospectral graphs for the standard
Laplacian have been considered by several authors. In [BuG11], the authors
construct from given isospectral bipartite graphs (e.g. unions of complete
bipartite graphs) new isospectral graphs. The proof is based on a concrete
construction of eigenfunctions of the new graphs from the old ones and
checking that all eigenvalues are exhausted. Examples where isospectral graphs
with _different_ number of edges are present are constructed in [Bu15], based
on so-called _twin vertices_ and simultaneously scaling the edge and vertex
weights. Other recent references describing isospectral graphs for the
standard Laplacian are given in [DP16, L22, Bu10].

In this article we present a new perturbative technique to construct
isospectral graphs of the standard Laplacian (including its signless version).
This new philosophy is based on the elementary perturbation of a graph and the
control of the effect of this perturbation on some of its eigenvalues. From
known multiplicity of certain eigenvalues of some auxiliary graphs (and _no_
knowledge of the corresponding eigenfunctions), we construct isospectral
graphs using spectral bracketing, i.e., enclosing unknown eigenvalues between
the corresponding eigenvalues of the auxiliary graphs. Concretely, a spectral
bracketing of a graph $G$ with another graph $G^{\prime}$ means that the
eigenvalues interlace, i.e., that for some $t\in\mathbb{N}$ the relations

$\lambda_{1}(G^{\prime})\leq\lambda_{1}(G)\leq\lambda_{1+t}(G^{\prime}),\quad\lambda_{2}(G^{\prime})\leq\lambda_{2}(G)\leq\lambda_{2+t}(G^{\prime}),\quad\dots,\quad\lambda_{n}(G^{\prime})\leq\lambda_{n}(G)\leq\lambda_{n+t}(G^{\prime})$
(1.1)

hold, where $n=\lvert G\rvert$ and $n+t=\lvert G^{\prime}\rvert$ are the
orders of $G$ and $G^{\prime}$, respectively. Symbolically, we denote this as
$G^{\prime}\preccurlyeq G\stackrel{{\scriptstyle
t}}{{\preccurlyeq}}G^{\prime}$ for this eigenvalue interlacing, where
$t\in\mathbb{N}_{0}$ denotes the _spectral shift_ in the discrete label
counting the eigenvalues that fixes the bracketing, i.e.,
$\lambda_{k}(G)\in[\lambda_{k}(G^{\prime}),\lambda_{k+t}(G^{\prime})]$,
$k=1,\dots,n$. If $G$ now has an eigenvalue of multiplicity $\mu>t$, then
$G^{\prime}$ also has this eigenvalue with multiplicity at least $\mu-t>0$. In
Remark 3.3 we also give a diagrammatic visualisation of the spectral
bracketing method. See also [CDHLPS04] for an interlacing result under edge
deletion.

This technique is more efficient the smaller the spectral shift $t$ is, since
then, the bracketing becomes tighter. Ideally one looks for elementary
perturbations of the graph that have $t=1$ in only one of the spectral
relations leading to the classical interlacing of eigenvalues. In this sense,
the spectral shift quantifies the _spectral cost_ of the perturbation. This
cost depends much on which weights are used on the graph. For example, an edge
deletion has a minimal spectral cost for combinatorial weights but not for
standard weights while a single vertex contraction is a convenient
perturbation of the graph with standard weights but not for combinatorial
weights (see [FLP22a, Corollaries 4.2 and 4.7]). For this reason we will focus
in this article on perturbations where the graph $\widetilde{G}$ is obtained
from $G$ by contracting vertices of $G$ such that the difference of the number
of vertices is given by $t=\lvert G\rvert-\lvert\widetilde{G}\rvert$ and the
spectral bracketing of this operation is given by

$G\preccurlyeq\widetilde{G}\stackrel{{\scriptstyle t}}{{\preccurlyeq}}G.$

The task is now to find suitable auxiliary graphs $G$ having eigenvalues with
high multiplicity such that one can exhaustively determine eigenvalues
$\widetilde{G}$ by the respective bracketings. This can be done, for example,
if the different eigenvalue sets of the auxiliary graphs are disjoint (and
hence the corresponding eigenspaces must be orthogonal). If two non-isomorphic
graphs $\widetilde{G}_{1}$ or $\widetilde{G}_{2}$ can be sandwiched by the
same auxiliary graphs as above and if all eigenvalues are exhausted, then they
are isospectral by Theorem 3.7. In the final section we apply this method to
different classes of examples where one knows the spectrum of the
corresponding auxiliary graphs and, hence, one can also determine the spectrum
of the corresponding isospectral graphs. But the construction technique of
isospectral graphs proposed in this article _only requires to know the
multiplicity of certain eigenvalues_ of the auxiliary graphs.

This article is structured as follows: Section 2 contains a brief introduction
of the notation and results on graphs with standard weights and their
Laplacians. In the following section the main tools for the analysis are
presented, the spectral bracketing of graphs and vertex contraction as the
main geometrical perturbation of the graph. It is proved that vertex
contraction has small spectral cost for the standard weights. Theorem 3.7 is
the main result in this section and shows how the multiplicity of certain
eigenvalues of the auxiliary graphs may lead to the determination of the
spectrum of the sandwiched graph. This idea is exploited in Section 4 to
present different families of isospectral and non-isomorphic graphs. These
examples include fuzzy ball and fuzzy complete graph families. Finally, new
examples obtained as subdivision graphs are shown to be isospectral using
directly the diagrammatic picture associated to different spectral
bracketings. All the examples of isospectral discrete graphs can be naturally
turned into equilateral metric graphs which, again, are isospectral for the
corresponding (unbounded) self-adjoint standard Laplacian (cf. [BK13]).

##  2\. Discrete graphs and their standard Laplacians

In this section, we introduce briefly the discrete structures and operations
needed for the construction for families of isospectral graphs. We will only
consider here discrete, simple and unoriented finite graphs together with its
Laplacian and signless Laplacian with standard weights. We refer to [FLP22a]
for a general analysis in the context of directed multigraphs with an
additional magnetic potential described as a $S^{1}$-valued function on the
edges.

###  2.1. Discrete graphs

A _(simple) graph_ $G$ consists of a finite _vertex_ set $V$ and an adjacency
relation on $V$ (i.e., a symmetric and non-reflexive relation on $V$); two
vertices being in this relation are called _adjacent_ and $\\{u,v\\}$ is
called an _edge_ joining $u$ and $v$. The _order_ of a finite graph $G$ is the
cardinality of its vertex set which we denote by $\lvert G\rvert:=\lvert
V\rvert$. We define the neighbourhood of a vertex $v$ as

$N_{v}:=\\{\,u\in V\,|\,\text{$u$ and $v$ are adjacent}\,\\}$

and the _degree_ of $v$ is the number of vertices adjacent with $v$ which we
denote as

$\deg v:=\lvert N_{v}\rvert.$

Since we focus in this article on Laplacians with standard weights we exclude
_isolated_ vertices (i.e., vertices of degree $0$ or, alternatively, we assume
$\deg_{G}v>0$ for all $v\in V$). A vertex of degree $1$ is called _pendant_.
Let $G$ be a graph with $n$ vertices labelled as $v_{1},v_{2},\dots,v_{n}$ and
such that $\deg v_{1}\leq\deg v_{2}\leq\dots\leq\deg v_{n}$. The list

$\deg G:=\bigl{(}\deg v_{1},\deg v_{2},\dots,\deg v_{n}\bigr{)}$

is called the _degree list_ of $G$ and provides a graph invariant, i.e., if
the degree sequence or the degree list of two graphs differ, then the graphs
cannot be isomorphic. A graph $G$ is _connected_ if for any two vertices $u,v$
there exists a path from $u$ to $v$, i.e., there is a sequence of distinct
vertices $v_{0},v_{1},\dots,v_{n}$ such that $v_{i-1}$ and $v_{i}$ are
adjacent for $1\leq i\leq n$ and such that $u=v_{0}$ and $v=v_{n}$. If the
explicit dependence of the graph $G$ is needed, we write $V=V(G)$ or
$\deg^{G}v$ etc.

A central operation needed in this article is the _contraction of vertices_
(also the name _gluing_ or _merging_ is used in the literature). Recall that
the choice of a subset of vertices $V_{0}\subset V$ naturally specifies an
equivalence relation of vertices. In this article the result of contracting
vertices must give, again, a simple graph. We refer to Definition 2.1 and
Remark 2.2 [FLP22a] for a general description of contractions in the context
of multiple directed graphs.

######  Definition 2.1 (Contracting vertices).

Let $G$ be a graph with vertex set $V$, consider an equivalence relation
$\sim$ on the vertex set $V$ and denote by $[v]$ the class of vertices related
with $v$. The quotient graph $G/{\sim}$ is the graph with vertex set
$V/{\sim}=\\{\,[v]\,|\,v\in V\,\\}$ and keeping all edges.

We say that $G/{\sim}$ is obtained from $G$ by _contracting the vertices_
(with respect to $\sim$). Two vertices $[u]$ and $[v]$ in $G/{\sim}$ are
adjacent if $[u]\not=[v]$ and if some vertex in the class of $[v]$ is adjacent
in $G$ to some vertex in the class of $[u]$.

The _shrinking number_ $r$ of the equivalence relation $\sim$ is defined by
$r:=\lvert G\rvert-\lvert G/{\sim}\rvert=\lvert V\rvert-\lvert V/{\sim}\rvert$
and quantifies the reduction of vertices in the quotient graph.

In this article we will only allow contractions that respect the category of
simple graphs. In particular, the quotient graph must be simple as well. For
example, adjacent vertices in $G$ can not be contracted since the quotient
graph is not allowed to have loops in this article.

###  2.2. Standard Laplacians

For a finite graph $G$, we associate the following natural weighted Hilbert
space

$\displaystyle\ell_{2}({V,\deg}):=\Bigl{\\{}\,f\colon
V\longrightarrow\mathbb{R}\,\Bigl{|}\Bigr{.}\,\|{f}\|^{2}_{\ell_{2}({V,\deg})}=\sum_{v\in
V}\lvert{f(v)}\rvert^{2}\deg v\,\Bigr{\\}}.$

The standard Laplacian is defined as follows (for an approach using discrete
exterior derivatives, more general weights and magnetic potentials we refer to
[FLP22a, Sec. 3] and references therein):

######  Definition 2.2 (standard Laplacian and standard signless Laplacian).

The _standard (discrete) Laplacian_ of a graph $G$ is the self-adjoint
operator
$\Delta_{{G}}\colon\ell_{2}({V,\deg})\longrightarrow\ell_{2}({V,\deg})$
defined for $v\in V$ and $f\in\ell_{2}({V,\deg})$ by
$\bigl{(}\Delta_{{G}}f\bigr{)}(v)=\frac{1}{\deg v}\sum_{u\in
N_{v}}\bigl{(}f(v)-f(u)\bigr{)}=f(v)-\frac{1}{\deg v}\sum_{u\in N_{v}}f(u).$
(2.1a) If $G$ has order $n$, then we write the spectrum of the corresponding
Laplacian $\Delta_{{G}}$ as the list
$\sigma_{\mathrm{}}(G)=\bigl{(}\lambda_{1}(G),\lambda_{2}(G),\dots,\lambda_{n}(G)\bigr{)},$
where the eigenvalues $\lambda_{1}(G),\dots,\lambda_{n}(G)$ are written in
ascending order and repeated according to their multiplicities.

Similarly, we define the _signless standard Laplacian_ $\Delta_{{G^{+}}}$ for
$v\in V$ and $f\in\ell_{2}({V,\deg})$ by111Note that in [FLP18] and [FLP22a,
Sec. 5.3] we reserved the superscript + to denote Dirichlet Laplacians. Since
in this article we do not need the operation of vertex virtualisation we use
the superscript + in a different way: the results in the last section can be
written in a much more convenient way using - for the Laplacian and - for the
signless Laplacian.

$\bigl{(}\Delta_{{G^{+}}}f\bigr{)}(v)=\frac{1}{\deg v}\sum_{u\in
N_{v}}\bigl{(}f(v)+f(u)\bigr{)}=f(v)+\frac{1}{\deg v}\sum_{u\in N_{v}}f(u)$
(2.1b)

Its eigenvalues are denoted by
$\sigma_{\mathrm{}}(G^{+})=\bigl{(}\lambda_{1}(G^{+}),\lambda_{2}(G^{+}),\dots,\lambda_{n}(G^{+})\bigr{)}$.

We refer to [LP08a] for a description of the unoriented homology related to
the signless Laplacian and to [FLP22a, Sec. 2.4 and Sec. 3.1] for the use of
the magnetic potential as an interpolation parameter connecting both
Laplacians. More applications of the magnetic potential to analyse
combinatorial properties of the graph are given in [FLP22b].

######  Remark 2.3.

  1. (a)

Some authors call the Laplacian with degree weight _geometric_ , as some
results referring to this Laplacian are closer to the Laplacian on a manifold
or a metric graph. Sometimes, the standard Laplacian is also called
_normalised_ , but we find that this name should better be reserved for edge
weights $w_{e}$ with edge $e=\\{u,v\\}$ and associated vertex weights
$\deg^{w}v:=\sum_{u\in N_{v}}w_{\\{u,v\\}}$.

  2. (b)

Recall that the spectra of the previous Laplacians is contained in the
interval $[0,2]$, i.e., $\sigma_{\mathrm{}}(G^{\pm})\subset[0,2]$. Moreover,
we always have $0\in\sigma_{\mathrm{}}(G)$, and $0$ is simple if and only if
$G$ is connected. Similarly, we have $2\in\sigma_{\mathrm{}}(G^{+})$. The
eigenfunction of both eigenvalues is a constant function on $V$ as one easily
sees from (2.1a) and (2.1b).

  3. (c)

With respect to the orthonormal basis $\\{\,\delta_{v}\,|\,v\in V\,\\}$ with
$\delta_{v}(u)=(\deg v)^{-1/2}$ if $u=v$ and $\delta_{v}=0$ if $u\neq v$, the
standard Laplacian denoted with $(-)$ as $\Delta_{{G}}=\Delta_{{G^{-}}}$ and
the standard signless Laplacian denoted with $(+)$ as $\Delta_{{G^{+}}}$ are
represented by the matrices with entries

$(\Delta_{{G^{\mp}}})_{u,v}=\begin{cases}1,&u=v,\\\ \mp\bigl{(}(\deg u)(\deg
v)\bigr{)}^{-1/2},&\text{$u$ and $v$ are adjacent,}\\\
0,&\text{otherwise\;.}\end{cases}$ (2.2)

##  3\. Spectral bracketing and graph perturbations

We present in this section our original idea of how to construct isospectral
graphs based on certain graph perturbations whose spectral effect we can
control. We will sandwich the original graphs between two auxiliary graphs
with high symmetry which will specify the spectral bracketing. Our method here
is based on the concept of “spectral preorder” in relation to vertex
contraction (see also [FLP22a, Section 4] for a general spectral analysis of
elementary perturbations of multidigraphs).

###  3.1. Spectral preorder

In this section we introduce a spectral preorder in the class of simple graphs
with standard weights based on the order relation of consecutive lists of
eigenvalues of the corresponding Laplacians written in increasing order and
repeated according to their multiplicities. From an algebraic viewpoint the
spectral preorder is a quite flexible generalisation of the eigenvalue
interlacing known for matrices (see e.g. [HoJ13, Theorem 4.3.28] and [BrH12,
Section 2.5 and 3.2]). This relation can be generalised to include a shift
$t\in\mathbb{N}_{0}$ in the list of eigenvalues. We refer to [FLP18, FLP22a]
for proofs and additional motivation. We will use here this preorder to
control the spectral effect of vertex contraction.

######  Definition 3.1 (Spectral preorder $\preccurlyeq$).

Let $G$ and $G^{\prime}$ be two graphs and denote the corresponding increasing
lists of the eigenvalues of the respective standard Laplacians, repeated
according to their multiplicity, by

$\bigl{(}\lambda_{1}(G),\lambda_{2}(G),\dots,\lambda_{\lvert
G\rvert}(G)\bigr{)}\qquad\text{and}\qquad\bigl{(}\lambda_{1}(G^{\prime}),\lambda_{2}(G^{\prime}),\dots,\lambda_{\lvert
G^{\prime}\rvert}(G^{\prime})\bigr{)}.$

We say that _$G$ is spectrally smaller than $G^{\prime}$ with shift
$t\in\mathbb{N}_{0}$_, and we denote this by

$G\stackrel{{\scriptstyle t}}{{\preccurlyeq}}G^{\prime},$

if the following two conditions hold:

$\lvert G\rvert\geq\lvert
G^{\prime}\rvert-t\quad\text{and}\quad\lambda_{k}(G)\leq\lambda_{k+t}(G^{\prime})\qquad\text{for
all $1\leq k\leq\lvert G^{\prime}\rvert-t$.}$

If $t=0$ we write simply $G\preccurlyeq G^{\prime}$.

The same definition applies to the signless Laplacian, i.e., we write
$G^{+}\stackrel{{\scriptstyle t}}{{\preccurlyeq}}(G^{\prime})^{+}$ if the
eigenvalues of the signless Laplacians fulfil
$\lambda_{k}(G^{+})\leq\lambda_{k+t}((G^{\prime})^{+})$ for all $1\leq
k\leq\lvert G^{\prime}\rvert-t$.

######  Remark 3.2 (spectral preorder visualised).

We present here the following useful diagrammatic representation of the
relation $G\stackrel{{\scriptstyle t}}{{\preccurlyeq}}G^{\prime}$.

$G$ |  |  |  |  $\lambda_{1}$ |  $\lambda_{2}$ |  … |  $\lambda_{n-1}$ |  $\lambda_{n}$
---|---|---|---|---|---|---|---|---
$G^{\prime}$ |  $\lambda_{1}^{\prime}$ |  … |  $\lambda_{t}^{\prime}$ |  $\lambda_{1+t}^{\prime}$ |  $\lambda_{2+t}^{\prime}$ |  … |  $\lambda_{n-1+t}^{\prime}$ |

We list the eigenvalues $\lambda_{k}=\lambda_{k}(G)$, $n=\lvert G\rvert$ and
$\lambda_{k}^{\prime}=\lambda_{k}(G^{\prime})$ of $G$ and $G^{\prime}$ in the
first and second row, respectively. The condition $\lvert G\rvert\geq\lvert
G^{\prime}\rvert-t$ just means that the last box of the second row can not
exceed the last box of the first row. Moreover, eigenvalues in the same column
increase when going down the column and one can easily visualise the spectral
relation just looking at the successive columns.

Let $G$, $G^{\prime}$ and $G^{\prime\prime}$ be finite graphs. We describe
next some useful conventions and easy consequences of the preceding
definition. We write $G\stackrel{{\scriptstyle
t^{\prime}}}{{\preccurlyeq}}G^{\prime}$ and $G^{\prime}\stackrel{{\scriptstyle
t^{\prime\prime}}}{{\preccurlyeq}}G^{\prime\prime}$ simply by
$G\stackrel{{\scriptstyle
t^{\prime}}}{{\preccurlyeq}}G^{\prime}\stackrel{{\scriptstyle
t^{\prime\prime}}}{{\preccurlyeq}}G^{\prime\prime}$, and call such estimates
_spectral bracketing_ or _eigenvalue interlacing_.

######  Remark 3.3 (spectral bracketing visualised).

Later we will often consider the graph sandwiching $G^{\prime}\preccurlyeq
G\stackrel{{\scriptstyle t}}{{\preccurlyeq}}G^{\prime}$ which implies that
$\lvert G\rvert+t=\lvert G^{\prime}\rvert\geq\lvert G\rvert$. If $\lvert
G\rvert=n$, then $G^{\prime}\preccurlyeq G\stackrel{{\scriptstyle
t}}{{\preccurlyeq}}G^{\prime}$ is equivalent with the _interlacing_ of the
spectra corresponding to the Laplacians of $G$ and $G^{\prime}$ similarly as
in [BrH12, Section 2.5], i.e., (1.1) holds. Using the diagrammatic
representation of Remark 3.2, we write

$G^{\prime}$ |  |  |  |  $\lambda^{\prime}_{1}$ |  $\lambda^{\prime}_{2}$ |  $\lambda^{\prime}_{3}$ |  … |  $\lambda^{\prime}_{n-1}$ |  $\lambda^{\prime}_{n}$ |  $\lambda^{\prime}_{n+1}$ |  … |  $\lambda^{\prime}_{n+t}$
---|---|---|---|---|---|---|---|---|---|---|---|---
$G$ |  |  |  |  $\lambda_{1}$ |  $\lambda_{2}$ |  $\lambda_{3}$ |  … |  $\lambda_{n-1}$ |  $\lambda_{n}$ |  |  |
$G^{\prime}$ |  $\lambda_{1}$ |  … |  $\lambda^{\prime}_{t}$ |  $\lambda^{\prime}_{1+t}$ |  $\lambda^{\prime}_{2+t}$ |  $\lambda^{\prime}_{3+t}$ |  … |  $\lambda^{\prime}_{n-1+t}$ |  $\lambda^{\prime}_{n+t}$ |  |  |

The condition $\lvert G\rvert=\lvert G^{\prime}\rvert-t$ means here that the
second and third columns are aligned on the right hand side. In particular, if
$t=1$ it becomes the usual eigenvalue interlacing (justifying also the name)

$\lambda_{1}(G^{\prime})\leq\lambda_{1}(G)\leq\lambda_{2}(G^{\prime})\leq\lambda_{2}(G)\leq\dots\leq\lambda_{n}(G^{\prime})\leq\lambda_{n}(G)\leq\lambda_{n+1}(G^{\prime}).$

The following result is a direct consequence of the definition of spectral
preorder and quantifies the stability of eigenvalues with high multiplicity
under vertex contraction perturbation.

######  Proposition 3.4.

Consider two graphs $G,G^{\prime}$ satisfying the spectral relation
$G^{\prime}\stackrel{{\scriptstyle
t^{\prime}}}{{\preccurlyeq}}G\stackrel{{\scriptstyle
t^{\prime\prime}}}{{\preccurlyeq}}G^{\prime}$. If $\lambda$ is an eigenvalue
of the Laplacian on $G^{\prime}$ with multiplicity
$\mu>t^{\prime}+t^{\prime\prime}$, then $\lambda$ is an eigenvalue of the
Laplacian on $G$ with multiplicity at least
$\mu-(t^{\prime}+t^{\prime\prime})$.

###  3.2. Spectral bracketing and vertex contraction

The following result will be central to the class of examples considered in
the next section. An important observation for our method is that multiple
eigenvalues remain eigenvalues with lower multiplicity in the quotient graph
provided the shrinking number of the quotient is small enough (cf. Definition
2.1). We present next vertex contraction as a perturbation with low spectral
cost for standard weights; see Theorem 3.14 in [FLP22a] for a proof.

######  Proposition 3.5 (contracting vertices and spectral bracketing).

Let $G$ be a finite discrete graph. If $\widetilde{G}=G/{\sim}$ is a graph
obtained from $G$ by contracting vertices according to an equivalence relation
$\sim$ on $V(G)$, then we have the following spectral relation between the
corresponding standard Laplacians

$G\preccurlyeq\widetilde{G}\stackrel{{\scriptstyle t}}{{\preccurlyeq}}G,$

where $t=\lvert G\rvert-\lvert\widetilde{G}\rvert\geq 0$ is the shrinking
number of $\sim$ (cf. Definition 2.1). The same result is true for the
signless Laplacians, i.e., we have
$G^{+}\preccurlyeq\widetilde{G}^{+}\stackrel{{\scriptstyle
t}}{{\preccurlyeq}}G^{+}$.

Note that Proposition 3.4 can be immediately applied to the graph perturbation
introduced before. In fact, the total spectral shift $t$ is the shrinking
number of the relation $\sim$ (cf. Proposition 3.5).

###  3.3. Spectral bracketing and determination of the spectrum

We tacitly understand subsets $\Lambda$ of a spectrum $\sigma_{\mathrm{}}(G)$
as _multisets_ without formally introducing them. A simple workaround is to
think of $\Lambda$ as an ordered list
$\Lambda=(\lambda_{1},\dots,\lambda_{n})$ with
$\lambda_{1}\leq\lambda_{2}\leq\dots\leq\lambda_{n}$, and multiple members are
repeated according to their multiplicity. The notation $\lambda\in\Lambda$
then means that there is $j\in\\{1,\dots,n\\}$ such that
$\lambda=\lambda_{j}$. Similarly, we understand
$\Lambda^{\prime}\subset\Lambda$, i.e., each $\lambda\in\Lambda^{\prime}$
appearing $\mu$ times appears at least $\mu$ times also in $\Lambda$.

######  Definition 3.6 (multiplicity and subsets).

We say that a subset of non-negative numbers $\Lambda$ has _multiplicity $\mu$
in $\sigma_{\mathrm{}}(G)$_ if each element $\lambda\in\Lambda$ having
multiplicity $s$ in $\Lambda$ has at least multiplicity $s\mu$ in
$\sigma_{\mathrm{}}(G)$. We denote this relation by
$\Lambda^{(\mu)}\subset\sigma_{\mathrm{}}(G)$.

Next, we present the key result on which we base our construction of
isospectral graphs in Section 4. Namely, we specify conditions that guarantee
that the spectrum of a graph $\widetilde{G}$ is obtained by spectrally
bracketing the original graph with the auxiliary graphs $G_{j}$ having
eigenvalues of high multiplicity.

######  Theorem 3.7.

Let $J$ be a finite index set and consider for each $j\in J$ the finite set of
non-negative numbers $\Lambda_{j}\subset[0,2]$. Assume that the graphs $G$ and
$G_{j}$ ($j\in J$) with standard Laplacians satisfy the following conditions:

  1. (a)

$G_{j}\stackrel{{\scriptstyle
t^{\prime}_{j}}}{{\preccurlyeq}}G\stackrel{{\scriptstyle
t^{\prime\prime}_{j}}}{{\preccurlyeq}}G_{j}$, $j\in J$ and denote the total
spectral shift by $t_{j}:=t_{j}^{\prime}+t_{j}^{\prime\prime}$;

  2. (b)

$\Lambda_{j}$ has multiplicity $\mu_{j}$ in $\sigma_{\mathrm{}}(G_{j})$, $j\in
J$, and $\mu_{j}>t_{j}=t_{j}^{\prime}+t_{j}^{\prime\prime}$;

  3. (c)

the sets ${\Lambda_{j}}$, ${j\in J}$ are pairwise disjoint;

  4. (d)

the order of $G$ is

$\lvert G\rvert=\sum_{j\in J}(\mu_{j}-t_{j})\lvert\Lambda_{j}\rvert.$ (3.1)

Then the spectrum of $G$ is determined by the subsets $\Lambda_{j}$, namely we
have $\sigma_{\mathrm{}}(G)=\biguplus_{j\in J}\Lambda_{j}^{(\mu_{j}-t_{j})}$,
where $\biguplus$ means the disjoint union of multisets.

###### Proof.

From Proposition 3.4 and conditions (a)–(b) above we conclude that each
$\Lambda_{j}$ has multiplicity at least $(\mu_{j}-t_{j})$ in
$\sigma_{\mathrm{}}(G)$, where the total shift is
$t_{j}:=t_{j}^{\prime}+t_{j}^{\prime\prime}$. We denote this relation as
$\Lambda_{j}^{(\mu_{j}-t_{j})}\subset\sigma_{\mathrm{}}(G)$ (as multiset).
Moreover, since $\Lambda_{j}\cap\Lambda_{j^{\prime}}=\emptyset$ for all
$j,j^{\prime}\in J$ with $j\neq j^{\prime}$ condition (c) implies that an
eigenvalue of $G$ is contained in only _one_ of the sets $\Lambda_{j}$.
Finally, condition (d) guarantees that all eigenvalues of $G$ are necessarily
in one of the sets $\Lambda_{j}$ and, therefore, the spectrum is exhausted. ∎

######  Remark 3.8.

  1. (a)

The disjointness condition (c) of Theorem 3.7 is necessary to guarantee that
one obtains all _different_ eigenvalues, i.e., that eigenvalues from different
sets $\Lambda_{j}$ and $\Lambda_{j^{\prime}}$ for $j,j^{\prime}\in J$ with
$j\neq j^{\prime}$ correspond to _different_ eigenfunctions.

  2. (b)

Sometimes the condition (d) can be relaxed. For example, recall that $0$
(resp. $2$) is always an eigenvalue of the Laplacian (resp. signless
Laplacian) with constant eigenfunction; also, some eigenvalues may possibly be
known because the graph is bipartite etc.

     * •

It is enough that (3.1) adds up to $n-1$, where we put $n:=\lvert G\rvert$. In
fact, the missing eigenvalue can be recovered from the remaining ones by the
following trace relation for the the standard or signless Laplacians

$\sum_{k=1}^{n}\lambda_{k}(G)=\operatorname{tr}\Delta_{{G}}=n\;.$

The latter equality follows from the fact that the matrix representation of
$\Delta_{{G^{\mp}}}$ has only $1$ on its diagonal, cf. (2.2).

     * •

Any known eigenvalues of $G$ can be put in a (multi-)set $\Lambda_{j_{0}}$ for
some index $j_{0}\in J$. Choosing $G_{j_{0}}=G$,
$t_{j_{0}}^{\prime}=t_{j_{0}}^{\prime\prime}=0$ and $\mu_{j_{0}}=1$, Theorem
3.7 can formally applied and only
$\lvert\widetilde{G}\rvert-\lvert\Lambda_{j_{0}}\rvert$ eigenvalues need to be
recovered. We apply this idea sometimes to the eigenvalue $0$ or to
$\\{0,2\\}$ if the graph is bipartite. Recall finally, that if $G$ is
bipartite, one can also use the fact that $\lambda\in\sigma_{\mathrm{}}(G)$ if
and only if $2-\lambda\in\sigma_{\mathrm{}}(G)$ (cf. [Ch97, Lemma 1.8] or
[LP08a, Proposition 2.3]).

##  4\. A perturbative construction of isospectral graphs

We base our discrete perturbative approach (spectral bracketing technique) on
the result presented in Theorem 3.7 and using auxiliary graphs having suitable
symmetry. Note that in the following classes of examples of isospectral graphs
(see e.g. Theorem 4.2) we can determine the spectrum explicitly due to the
fact that we know the spectra of the auxiliary graphs. But our technique only
requires to know that the multiplicity of certain eigenvalues of the auxiliary
graphs is high enough to guarantee isospectrality. Our bracketing method is,
in this sense, based on _multiplicities_ of certain eigenvalues and not on
explicit computation of the spectrum or eigenfunctions of the auxiliary
graphs. In [FLP22c] we develop a new geometrical construction of families of
isospectral magnetic graphs. This method starts with several copies of an
arbitrary building block graph $G$ which are glued following a specific
pattern. To prove isospectrality in the mentioned reference we lift
eigenfunctions on $G$ and, also, eigenfunctions with certain Dirichlet
conditions on the merged vertices to the constructed graph. We mention that
the perturbative method developed in this article can also be applied to
alternatively prove isospectrality of some families of graphs constructed in
[FLP22c].

Our construction of non-isomorphic graphs in some examples is based on the
combinatorial notion of partition of length $s\in\mathbb{N}$ of a given
natural number $r\in\mathbb{N}$ (which we call an $s$-partition of $r$ for
short). Recall that an $s$ partition of $r$ is given by a multiset
$A=\\{a_{1},a_{2},\dots,a_{s}\\}$ (i.e. taking multiplicities into account)
with $a_{i}\in\mathbb{N}$ and $a_{1}+\dots+a_{s}=r$. We will show that
different $s$-partitions of $r$ lead to isospectral non-isomorphic graphs. As
an example, the smallest natural number having two different partitions is
$r=4$: the length is $s=2$ and the partitions are $\\{1,3\\}$ and $\\{2,2\\}$
(since $4=2+2=1+3$).

###  4.1. The fuzzy ball construction

We revisit and generalise the fuzzy ball construction of [BuG11, Section 4],
proof isospectrality using spectral bracketing technique and _without_
computing eigenfunctions. Let $K_{r}$ be the complete graph with $r$ vertices
and denote by $\widehat{K}_{r}$ the graph with $2r$ vertices obtained from
$K_{r}$ by attaching a pendant edge to each vertex of $K_{r}$.

We first calculate the eigenvalues with high multiplicity since we only need
this information to prove isospectrality of the family of graphs in Theorem
4.2.

######  Lemma 4.1 (Spectrum of auxiliary graphs).

  1. (a)

If $K_{r}$ is the complete graph with $r$ vertices, then

$1\pm\frac{1}{r-1}\in\sigma_{\mathrm{}}(K_{r}^{\mp})$

has multiplicity $r-1$ in the spectrum of the (signless) standard Laplacian
(here and in the sequel, the upper sign is for the standard Laplacian, the
lower sign for the signless Laplacian).

  2. (b)

Let $\widehat{K}_{r}$ be the complete graph with a pendant edge decoration at
each vertex of $K_{r}$. Then

$\frac{2r\pm 1-\sqrt{4r+1}}{2r},\frac{2r\pm
1+\sqrt{4r+1}}{2r}\in\sigma_{\mathrm{}}(\widehat{K}_{r}^{\mp})$

with multiplicity $r-1$.

###### Proof.

Part (a) is standard. To show part (b) we use a factorisation result of
characteristic polynomials in [Hey19]. In fact, the decorated complete graph
$\widehat{K}_{r}$ can be seen as a rooted product of ${K}_{r}$ with edges
$K_{2}$ on each vertex. Since ${K}_{r}$ a is simple $(r-1)$-regular graph and
since ${K}_{2}$ has degree $1$ we obtain from Corollary 1 of [Hey19] after
some straightforward manipulations that the spectrum is as claimed in (b). ∎

$\widehat{K}_{6}$

$\widehat{K}_{6}(A)$

$\widehat{K}_{6}(B)$

$K_{7}$

Figure 1. The figures above correspond to the auxiliary graph
$\widehat{K}_{6}$, the isospectral graphs $\widehat{K}_{6}(A)$ and
$\widehat{K}_{6}(B)$ for $A=(2,2,2)$ and $B=(1,2,3)$, and finally the
auxiliary graph $K_{7}$. Note that $\widehat{K}_{6}(A)$ and
$\widehat{K}_{6}(B)$ are obtained from $\widehat{K}_{6}$ by contracting the
pendant vertices according to $A$ and $B$, and that $K_{7}$ is obtained from
either $\widehat{K}_{6}(A)$ or $\widehat{K}_{6}(B)$ by contracting all
formerly pendant vertices into one vertex.

We now construct the first example class (the so-called _fuzzy balls_). Note
that we can also treat the signless version without any modification of the
proof and that the signless spectrum cannot be calculated from the unsigned
one as the graphs are not regular. We start with the decorated complete graph
$\widehat{K}_{r}$. Let $A=\\{a_{1},\dots,a_{s}\\}$ be an $s$-partition of
$r\in\mathbb{N}$ and label the pendant vertices as $\\{v_{1},\dots,v_{r}\\}$.
Consider the equivalence relation $\sim_{A}$ which contracts the set of
vertices $\\{v_{1},\dots,v_{a_{1}}\\}$,
$\\{v_{a_{1}+1},\dots,v_{a_{1}+a_{2}}\\}$, $\dots$,
$\\{v_{a_{1}+\dots+a_{n-1}+1},\dots,a_{r}\\}$ into $s$ vertices and denote the
corresponding quotient graph by
$\widehat{K}_{r}(A)=\widehat{GrK}_{r}/{\sim_{A}}$. In order to have two
different $s$-partitions of $r$ we necessarily need $r\geq 4$ and
$s\in\\{2,\dots,r-1\\}$.

######  Theorem 4.2 (“The fuzzy ball theorem”).

Assume that $A$ and $B$ are two different $s$-partitions of $r$. Then the
graphs $\widehat{K}_{r}(A)$ and $\widehat{K}_{r}(B)$ constructed above are
non-isomorphic and isospectral for the standard and signless standard
Laplacian.

###### Proof.

We apply Theorem 3.7 and show that the spectrum of $G:=\widehat{K}_{r}(A)$
depends _only_ on $r$ and the length $s$ of the partition. We choose as first
bracketing graph the decorated complete graph $G_{1}=\widehat{K}_{r}$ and as
spectral set

$\Lambda_{1}=\Bigl{\\{}\frac{(2r\pm 1-\sqrt{4r+1})}{2r},\frac{(2r\pm
1+\sqrt{4r+1})}{2r}\Bigr{\\}}.$

From Lemma 4.1 (b) we obtain
$\Lambda_{1}^{(r-1)}\subset\sigma_{\mathrm{}}(G_{1}^{\mp})$ and hence,
$\mu_{1}=r-1$. Since $\widehat{K}_{r}(A)=\widehat{K}_{r}/{\sim}_{A}$ has
shrinking number $t_{1}:=r-s$ we obtain from Proposition 3.5 that

$\widehat{K}_{r}^{\mp}\preccurlyeq\widehat{K}_{r}(A)^{\mp}\stackrel{{\scriptstyle
r-s}}{{\preccurlyeq}}\widehat{K}_{r}^{\mp}.$

For the second bracketing graph we consider $G_{2}=K_{r+1}$ and the spectral
set $\Lambda_{2}:=\\{1\pm\nicefrac{{1}}{{r}}\\}$ which has multiplicity
$\mu_{2}:=r$ in $\sigma_{\mathrm{}}(G_{2}^{\mp})$. Note that $K_{r+1}$ can be
obtained from $\widehat{K}_{r}(A)$ by contracting the $s$ vertices obtained by
the contraction of the pendant vertices of $\widehat{K}_{r}$ given by the
relation $\sim_{A}$. The shrinking number of this process is $t_{2}=s-1$ and
hence $\widehat{K}_{r}(A)^{\mp}\preccurlyeq
K_{r+1}^{\mp}\stackrel{{\scriptstyle
s-1}}{{\preccurlyeq}}\widehat{GrK}_{r}(A)^{\mp}$, or, equivalently,

$K_{r+1}^{\mp}\stackrel{{\scriptstyle
s-1}}{{\preccurlyeq}}\widehat{K}_{r}(A)^{\mp}\preccurlyeq K_{r+1}^{\mp}.$

It is clear that the spectral subsets $\Lambda_{1}$ and $\Lambda_{2}$ are
disjoint. Moreover, we have already detected

$(\mu_{1}-t_{1})\lvert{\Lambda_{1}}\rvert+(\mu_{2}-t_{2})\lvert{\Lambda_{2}}\rvert=\underbrace{((r-1)-(r-s))}_{=s-1}\cdot
2+\underbrace{(r-(s-1))}_{r-s+1}\cdot
1=r+s-1=\lvert{\widehat{K}_{r}(A)}\rvert-1$

eigenvalues, hence the spectrum of the Laplacian
$\Delta_{{\widetilde{K}_{r}(A)}}$ is determined by Theorem 3.7 and Remark 3.8
(b).

Since the values in $\Lambda_{1}$ and $\Lambda_{2}$ as well as the
multiplicities depend only on $r$ and the length $s$ and not on the concrete
partition $A$ or $B$ we obtain
$\sigma_{\mathrm{}}(\widehat{K}_{r}(A)^{\mp})=\sigma_{\mathrm{}}(\widehat{K}_{r}(B)^{\mp})$
as claimed.

Finally, the degree lists of $\widehat{K}_{r}(A)$ and $\widehat{K}_{r}(B)$ are
given respectively by

$\\{r^{(r)}\\}\uplus A\quad\text{and}\quad\\{r^{(r)}\\}\uplus B,$

where the union of multisets takes multiplicities into account. Since the
partitions $A$ and $B$ are different we conclude that the corresponding graphs
cannot be isomorphic. ∎

######  Corollary 4.3.

The spectrum of $\widehat{K}_{r}(A)^{\mp}$ and $\widehat{K}_{r}(B)^{\mp}$ is
given by

$\sigma_{\mathrm{}}(\widehat{K}_{r}(A)^{\mp})=\sigma_{\mathrm{}}(\widehat{K}_{r}(B)^{\mp})=\Bigl{\\{}\frac{2r\pm
1-\sqrt{4r+1}}{2r},\frac{2r\pm
1+\sqrt{4r+1}}{2r}\Bigr{\\}}^{(s-1)}\cup\Bigl{\\{}1\pm\frac{1}{r}\Bigr{\\}}^{(r-s+1)}\cup\\{\lambda_{*}^{\pm}\\},$

where powers $(\cdot)^{(\mu)}$ indicate multiplicity. The remaining eigenvalue
is $\lambda_{*}^{\pm}=1\pm 1$, i.e., $0$ for the standard and $2$ for the
signless Laplacian.

For the next result we need to mention the notion of metric graph. Recall that
any discrete graph $G$ has an associated equilateral metric graph which we
denote here by $\overline{G}$ (see [LP08a, BK13, KM21] and references cited
therein). Recall that the standard Laplacian (sometimes also called
“Kirchhoff”) is an unbounded second order differential operator on the metric
graph with standard conditions on the vertices (the sum of the in-derivatives
equals the sum of the out-derivatives) which guarantee that the operator is
self-adjoint.

We observe that the construction of isospectral graphs as in Theorem 4.2
produces also isospectral equilateral metric graphs for the corresponding
Kirchhoff Laplacians. The proof is based on a beautiful relation between the
spectra of the standard Laplacian and the Kirchhoff Laplacian (see, e.g.
[BK13, LP08a] and references therein).

######  Corollary 4.4.

Let $A$ and $B$ are two different $s$-partitions of $r$ and denote by
$\overline{K}_{r}(A)$ and $\overline{K}_{r}(B)$ the equilateral metric graph
associated with the discrete graphs $\widehat{K}_{r}(A)$ and
$\widehat{K}_{r}(B)$. Then the graphs $\overline{K}_{r}(A)$ and
$\overline{K}_{r}(B)$ are non-isomorphic and isospectral for the standard
metric Laplacian.

###### Sketch of the proof.

We prove in [FLP22c] that $\overline{G}_{1}$ and $\overline{G}_{2}$ are
isospectral with respect to the standard metric Laplacian if and only if
$G_{1}$ and $G_{2}$ are isospectral with respect to the standard discrete
Laplacian and if $G_{1}$ and $G_{2}$ have the same number of edges. A similar
statement is stated in [KM21, Proposition 1]. (Note that isospectrality
guarantees that the number of vertices is the same and, therefore,
preservation of the number of edges guarantees that the corresponding Betti
numbers are also the same.) Note that this follows for eigenvalues
$\lambda\in(0,2)$ by e.g. [LP08a, Proposition 4.1, 4.7 and 5.2]. Recall that
the bipartiteness of a graph can be seen from the spectrum of the standard
discrete Laplacian (a connected graph is bipartite if and only if $2$ is in
its spectrum).

The claim on isospectral metric graphs follows from the fact that in all our
constructions of isospectral graphs using vertex contraction, the number of
edges is the same. ∎

######  Remark 4.5.

Note that the splitted complete graphs $K_{5}$ in Figure 1 of [KM21] fall into
the class of fuzzy balls. In fact, they correspond to $\widehat{K}_{4}(A)$ and
$\widehat{K}_{4}(B)$ with partitions of $r=4$ given by $A=(2,2)$ and
$B=(1,3)$. Therefore the corresponding equilateral metric graphs
$\overline{K}_{4}(A)$ and $\overline{K}_{4}(B)$ will also be isospectral for
the corresponding Kirchhoff Laplacian.

######  Example 4.6.

We illustrate here in a concrete example with $r=6$ and $s=3$ (cf. Figure 1)
how to apply directly and with the help of diagrams the spectral bracketing
technique developed in the proof Theorem 4.2. Recall that the lowest
eigenvalue $0$ is always included in all spectra.

The first bracketing here arises from

$\widehat{K}_{6}\preccurlyeq\widehat{K}_{6}(A)\stackrel{{\scriptstyle
3}}{{\preccurlyeq}}\widehat{K}_{6}.$

with shrinking number $t_{1}:=r-s=3$. Hence, for the eigenvalues we have

$\widehat{K}_{6}$ |  |  |  |  0 |  $\nicefrac{{2}}{{3}}$ |  $\nicefrac{{2}}{{3}}$ |  $\nicefrac{{2}}{{3}}$ |  $\nicefrac{{2}}{{3}}$ |  $\nicefrac{{2}}{{3}}$ |  $*$ |  $\nicefrac{{3}}{{2}}$ |  $\nicefrac{{3}}{{2}}$ |  $\nicefrac{{3}}{{2}}$ |  $\nicefrac{{3}}{{2}}$ |  $\nicefrac{{3}}{{2}}$
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---
$\widehat{K}_{6}(A)$ |  |  |  |  0 |  $\nicefrac{{2}}{{3}}$ |  $\nicefrac{{2}}{{3}}$ |  ? |  ? |  ? |  ? |  $\nicefrac{{3}}{{2}}$ |  $\nicefrac{{3}}{{2}}$ |  |  |
$\widehat{K}_{6}$ |  0 |  $\nicefrac{{2}}{{3}}$ |  $\nicefrac{{2}}{{3}}$ |  $\nicefrac{{2}}{{3}}$ |  $\nicefrac{{2}}{{3}}$ |  $\nicefrac{{2}}{{3}}$ |  $*$ |  $\nicefrac{{3}}{{2}}$ |  $\nicefrac{{3}}{{2}}$ |  $\nicefrac{{3}}{{2}}$ |  $\nicefrac{{3}}{{2}}$ |  $\nicefrac{{3}}{{2}}$ |  |  |

where the simple eigenvalue $*=\nicefrac{{7}}{{6}}$ of $\widehat{K}_{6}$ is
useless for this bracketing and the question marks are the four eigenvalues
that still need to be determined. Note that $\nicefrac{{2}}{{3}}$ and
$\nicefrac{{3}}{{2}}$ are eigenvalues of $\widehat{K}_{6}$ both with
multiplicity $\mu_{1}-(r-s)=5-(6-3)=2$ and determine through enclosure four
eigenvalues of $\widehat{K}_{6}(A)$ (light gray in the diagram).

To determine the remaining eigenvalues we consider a second bracketing by
choosing $G_{2}=K_{7}$ and the spectral set
$\Lambda_{2}:=\\{\nicefrac{{7}}{{6}}\\}$ which has multiplicity $\mu_{2}:=6$
in $\sigma_{\mathrm{}}(G_{2})$. The shrinking number is $t_{2}=2$ and hence

$K_{7}\stackrel{{\scriptstyle
2}}{{\preccurlyeq}}\widehat{K}_{6}(A)\preccurlyeq K_{7}.$

These relations can be represented diagrammatically for the eigenvalues as

$K_{7}$ |  |  |  $0$ |  $\nicefrac{{7}}{{6}}$ |  $\nicefrac{{7}}{{6}}$ |  $\nicefrac{{7}}{{6}}$ |  $\nicefrac{{7}}{{6}}$ |  $\nicefrac{{7}}{{6}}$ |  $\nicefrac{{7}}{{6}}$
---|---|---|---|---|---|---|---|---|---
$\widehat{K}_{6}(A)$ |  0 |  $\nicefrac{{2}}{{3}}$ |  $\nicefrac{{2}}{{3}}$ |  $\nicefrac{{7}}{{6}}$ |  $\nicefrac{{7}}{{6}}$ |  $\nicefrac{{7}}{{6}}$ |  $\nicefrac{{7}}{{6}}$ |  $\nicefrac{{3}}{{2}}$ |  $\nicefrac{{3}}{{2}}$
$K_{7}$ |  $0$ |  $\nicefrac{{7}}{{6}}$ |  $\nicefrac{{7}}{{6}}$ |  $\nicefrac{{7}}{{6}}$ |  $\nicefrac{{7}}{{6}}$ |  $\nicefrac{{7}}{{6}}$ |  $\nicefrac{{7}}{{6}}$ |  |

Therefore the remaining four eigenvalues are equal to $\nicefrac{{7}}{{6}}$
since multiplicity is given by $\mu_{2}-(s-1)=6-(3-1)=4$ (dark grey).
Altogether we have that both bracketings determine the complete spectrum of
$\widehat{K}_{6}(A)$:

$0$ |  $\nicefrac{{2}}{{3}}$ |  $\nicefrac{{2}}{{3}}$ |  $\nicefrac{{7}}{{6}}$ |  $\nicefrac{{7}}{{6}}$ |  $\nicefrac{{7}}{{6}}$ |  $\nicefrac{{7}}{{6}}$ |  $\nicefrac{{3}}{{2}}$ |  $\nicefrac{{3}}{{2}}$
---|---|---|---|---|---|---|---|---

Note that in the reasoning only $r=6$ and the length of the partition $s=3$
matter and so the two different partitions $A=\\{2,2,2\\}$ and $B=\\{1,2,3\\}$
lead to isospectral graphs $\widehat{K}_{6}(A)$ and $\widehat{GrK}_{6}(B)$.
Moreover, the corresponding degree lists

$\deg\widehat{K}_{6}(A)=(\mathbf{2},\mathbf{2},\mathbf{2},6,6,6,6,6,6)\qquad\text{and}\qquad\deg\widehat{K}_{6}(B)=(\mathbf{1},\mathbf{2},\mathbf{3},6,6,6,6,6,6)$

are different and, hence, the graphs are not isomorphic.

Finally, for the signless Laplacian one can reason similarly choosing the same
auxiliary graphs and replacing the eigenvalues $0$ by $2$ (constant
eigenfunction), $\nicefrac{{2}}{{3}}$ by $\nicefrac{{1}}{{2}}$,
$\nicefrac{{3}}{{2}}$ by $\nicefrac{{4}}{{3}}$ and $\nicefrac{{7}}{{6}}$ by
$\nicefrac{{5}}{{6}}$, respectively.

###  4.2. The fuzzy complete bipartite construction

In this section we will show a different class of examples leading to
isospectral graphs. Instead of following the lemma-theorem structure of the
preceding subsection we will directly reason using spectral diagrams in
concrete examples that can be generalised in an obvious way.

For $p,r\in\mathbb{N}$, let $K_{p,r}$ be the complete bipartite graph with
$p+r$ vertices and denote by $\widehat{K}_{p,r}$ the graph with $p+2r$
vertices obtained from $K_{p,r}$ by attaching a pendant edge to each of the
$r$ vertices in the partition of $K_{p,r}$. Note since the resulting classes
of graphs in this subsection will be bipartite there is no need to distinguish
between Laplacian and signless Laplacian as they have the same spectrum. This
is due to the fact that on a bipartite graph, the signatures $1$ and $-1$ on
each edge are gauge-equivalent (see e.g. [FLP18, Sec. 3.1]. We now construct
the isospectral graphs from an $s$-partition $A$ of the natural number $r$,
i.e., $A=(a_{1},\dots,a_{r})$. We start with the decorated complete bipartite
graph $\widehat{K}_{p,r}$ (decorated with $r$ pendant edges numbered by
$\\{v_{1},\dots,v_{r}\\}$). Let $A=\\{a_{1},\dots,a_{s}\\}$ be an
$s$-partition of $r\in\mathbb{N}$. Consider the equivalence $\sim_{A}$ which
contracts the set of vertices $\\{v_{1},\dots,v_{a_{1}}\\}$,
$\\{v_{a_{1}+1},\dots,v_{a_{1}+a_{2}}\\}$, $\dots$,
$\\{v_{a_{1}+\dots+a_{n-1}+1},\dots,v_{r}\\}$ into $s$ vertices and denote the
corresponding quotient graph by
$\widehat{K}_{p,r}(A)=\widehat{K}_{p,r}/{\sim_{A}}$.

$\hat{K}_{2,5}$

$\hat{K}_{2,5}(A)$

$\hat{K}_{2,5}(B)$

${K}_{3,5}$

Figure 2. The figures above represent the auxiliary graph $\widehat{K}_{2,5}$,
the isospectral graphs $\widehat{K}_{2,5}(A)$ and $\widehat{K}_{2,5}(B)$ for
$A=(4,1)$ and $B=(2,3)$, and, finally, the auxiliary graph ${K}_{3,5}$. Note
that $\widehat{K}_{2,5}(A)$ and $\widehat{K}_{2,5}(B)$ are obtained from
$\widehat{K}_{2,5}$ by contracting the pendant vertices according to $A$ and
$B$, and that ${K}_{3,5}$ is obtained from either $\widehat{K}_{2,5}(A)$ or
$\widehat{K}_{2,5}(B)$ by contracting all former pendant vertices once again
into one vertex.

To illustrate the bracketing technique directly in terms of diagrams we will
consider a concrete example of this class by choosing the case $p=2$, $r=5$
and $s=2$ (see Figure 2). The method works similarly for other cases of in
this class.

For the first bracketing we use $G_{1}=\widehat{K}_{2,5}$ as auxiliary graph
which, by construction, give the relations

$\widehat{K}_{2,5}\preccurlyeq\widehat{K}_{2,5}(A)\stackrel{{\scriptstyle
3}}{{\preccurlyeq}}\widehat{K}_{2,5}.$

with shrinking number $t_{1}:=r-s=3$. Diagrammatically, the eigenvalues
satisfy

$\widehat{K}_{2,5}$ |  |  |  |  $0$ |  $\nicefrac{{1}}{{2}}$ |  $\nicefrac{{1}}{{2}}$ |  $\nicefrac{{1}}{{2}}$ |  $\nicefrac{{1}}{{2}}$ |  $*$ |  $*$ |  $\nicefrac{{3}}{{2}}$ |  $\nicefrac{{3}}{{2}}$ |  $\nicefrac{{3}}{{2}}$ |  $\nicefrac{{3}}{{2}}$ |  $2$
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---
$\widehat{K}_{2,5}(A)$ |  |  |  |  0 |  $\nicefrac{{1}}{{2}}$ |  ? |  ? |  ? |  ? |  ? |  $\nicefrac{{3}}{{2}}$ |  2 |  |  |
$\widehat{K}_{2,5}$ |  $0$ |  $\nicefrac{{1}}{{2}}$ |  $\nicefrac{{1}}{{2}}$ |  $\nicefrac{{1}}{{2}}$ |  $\nicefrac{{1}}{{2}}$ |  $*$ |  $*$ |  $\nicefrac{{3}}{{2}}$ |  $\nicefrac{{3}}{{2}}$ |  $\nicefrac{{3}}{{2}}$ |  $\nicefrac{{3}}{{2}}$ |  $2$ |  |  |

where now the double eigenvalue $*=1$ is irrelevant for this bracketing. Note
that we are able to enclose the two eigenvalues $\nicefrac{{1}}{{3}}$ and
$\nicefrac{{3}}{{2}}$, both with multiplicity $\mu_{1}-(r-s)=4-(6-3)=1$ (light
grey in the diagram).

To determine the remaining eigenvalues we consider a second bracketing with
the auxiliary graph $G_{2}=K_{3,5}$. Note that the eigenvalue $1$ has
multiplicity $\mu_{2}:=(p+1)+r-2=6$ in $\sigma_{\mathrm{}}(G_{2})$ and the
shrinking number is $t_{2}=s-1=1$. Therefore we obtain the spectral relations

$K_{3,5}\stackrel{{\scriptstyle
1}}{{\preccurlyeq}}\widehat{K}_{2,5}(A)\preccurlyeq K_{3,5}.$

which diagrammatically can be represented by

$K_{3,5}$ |  |  $0$ |  $1$ |  $1$ |  $1$ |  $1$ |  $1$ |  $1$ |  $2$
---|---|---|---|---|---|---|---|---|---
$\widehat{K}_{2,5}(A)$ |  0 |  $\nicefrac{{1}}{{2}}$ |  $1$ |  $1$ |  $1$ |  $1$ |  $1$ |  $\nicefrac{{3}}{{2}}$ |  2
$K_{3,5}$ |  $0$ |  $1$ |  $1$ |  $1$ |  $1$ |  $1$ |  $1$ |  $2$ |

We have now fixed the eigenvalue $1$ with multiplicity $\mu_{2}-t_{2}=6-1=5$
(dark grey in the diagram).

Altogether, the spectrum of $\widehat{K}_{2,5}(A)$ is given by

$0$ |  $\nicefrac{{1}}{{2}}$ |  $1$ |  $1$ |  $1$ |  $1$ |  $1$ |  $\nicefrac{{3}}{{2}}$ |  $2$
---|---|---|---|---|---|---|---|---

Again, the reasoning only depends on the values $p=2$, $r=6$ and the length of
the partition $s=2$ and so the two different partitions $A=\\{2,3\\}$ and
$B=\\{1,4\\}$ lead to isospectral graphs $\widehat{K}_{2,5}(A)$ and
$\widehat{K}_{2,5}(B)$. Since their degree lists

$\deg\widehat{K}_{2,5}(A)=(\mathbf{2},\mathbf{3},3,3,3,3,3,5,5)\qquad\text{and}\qquad\deg\widehat{K}_{2,5}(B)=(\mathbf{1},3,3,3,3,3,\mathbf{4},5,5)$

are different the graphs are not isomorphic. To make contact with the main
result in Theorem 3.7 recall that in the two bracketings used in this class of
examples we have for the first auxiliary graph $G_{1}=\widehat{K}_{2,5}$ the
spectral set
$\Lambda_{1}=\left\\{\nicefrac{{1}}{{2}},\nicefrac{{3}}{{2}}\right\\}$ with
multiplicity $\mu_{1}=5-1=4$ and for the second auxiliary graph
$G_{2}={K}_{3,5}$ the spectral set is $\Lambda_{2}=\left\\{1\right\\}$ with
$\mu_{2}=7-1=6$.

As in Corollary 4.4 we can extend isospectrality to the metric graph scenario.
Let $A$ and $B$ are two different $s$-partitions of $r$ and denoting by
$\overline{K}_{p,r}(A)$ and $\overline{K}_{p,r}(B)$ the equilateral metric
graph associated with the discrete graphs $\widehat{K}_{r,p}(A)$ and
$\widehat{K}_{p,r}(B)$. Then the graphs $\overline{K}_{p,r}(A)$ and
$\overline{K}_{p,r}(B)$ are non-isomorphic and isospectral for the Kirchhoff
Laplacian.

###  4.3. Edge subdivision

For presenting the following class of examples we first need to recall an
important operation on a graph $G=(V,E)$. The edge subdivision of
$e=\\{u,v\\}\in E$ consists in the deletion of $e=\\{u,v\\}$ from $G$, the
addition of a new vertex $w$ and two new edges $e_{1}=\\{u,w\\}$ and
$e_{2}=\\{w,v\\}$. We denote by $S(G)$ the graph where all edges of $G$ are
subdivided. Let $\widehat{K}_{5}$ be the complete graph with a pendent edge
decoration at each vertex of ${K}_{5}$ as in Figure 1 and consider the
corresponding edge subdivision graph shown in Figure 3. Since all graphs
involved are bipartite and connected we will always have $0$ and $2$ as simple
eigenvalues in the corresponding spectra.

$S(\widehat{K}_{5})$

$S(\widehat{K}_{5})(A)$

$S(\widehat{K}_{5})(B)$

$S({K}_{6})$

Figure 3. The figures above correspond to the auxiliary graph
$S(\widehat{K}_{6})$, the isospectral graphs $S(\widehat{K}_{5})(A)$ and
$S(\widehat{K}_{6})(B)$ for $A=(2,3)$ and $B=(1,4)$, and finally the auxiliary
graph $S(K_{6})$. Note that $S(\widehat{K}_{6})(A)$ and
$S(\widehat{K}_{6})(B)$ are obtained from $S(\widehat{K}_{5})$ by contracting
the pendant vertices according to $A$ and $B$, and that $S(K_{6})$ is obtained
from either $S(\widehat{K}_{6})(A)$ or $S(\widehat{K}_{6})(B)$ by contracting
all formerly pendant vertices once again into one vertex.

We will show using spectral diagrams that $S(\widehat{K}_{5})(A)$ and
$S(\widehat{K}_{5})(B)$ are isospectral. As before we consider the first graph
and to determine the nontrivial $20$ eigenvalues of $S(\widehat{K}_{5})(A)$ we
begin with the auxiliary graph $S(\widehat{K}_{5})$. By construction we have
the spectral relations

$S(\widehat{K}_{5})\preccurlyeq S(\widehat{K}_{5})(A)\stackrel{{\scriptstyle
3}}{{\preccurlyeq}}S(\widehat{K}_{5}),$

with shrinking number $t_{1}:=r-s=3$ using Proposition 3.5. Hence, for the
corresponding eigenvalues we have the following spectral diagram

$S(\widehat{K}_{5})$ |  |  |  |  | 0 | $w_{-}$ | $w_{-}$ | $w_{-}$ | $w_{-}$ | $z_{-}$ | $\widehat{w}_{-}$ | $\widehat{w}_{-}$ | $\widehat{w}_{-}$ | $\widehat{w}_{-}$ | 1 | 1 | 1 | 1 | 1 | $\widehat{w}_{+}$ | $\widehat{w}_{+}$ | $\widehat{w}_{+}$ | $\widehat{w}_{+}$ | $z_{+}$ | $w_{+}$ | $w_{+}$ | $w_{+}$ | $w_{+}$ | 2
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---
$S(\widehat{K}_{5})(A)$ |  |  |  |  | 0 | $w_{-}$ | ? | ? | ? | ? | $\widehat{w}_{-}$ | ? | ? | ? | 1 | 1 | ? | ? | ? | $\widehat{w}_{+}$ | ? | ? | ? | ? | $w_{+}$ | 2 |  |  |
$S(\widehat{K}_{5})$ |  | 0 | $w_{-}$ | $w_{-}$ | $w_{-}$ | $w_{-}$ | $z_{-}$ | $\widehat{w}_{-}$ | $\widehat{w}_{-}$ | $\widehat{w}_{-}$ | $\widehat{w}_{-}$ | 1 | 1 | 1 | 1 | 1 | $\widehat{w}_{+}$ | $\widehat{w}_{+}$ | $\widehat{w}_{+}$ | $\widehat{w}_{+}$ | $z_{+}$ | $w_{+}$ | $w_{+}$ | $w_{+}$ | $w_{+}$ | 2 |  |  |

where we denote for simplicity
$w_{\pm}=1\pm\left(\frac{1}{2}\sqrt{\frac{1}{5}\left(9+\sqrt{21}\right)}\right)$,
$\widehat{w}_{\pm}=1\pm\left(\frac{1}{2}\sqrt{\frac{1}{5}\left(9-\sqrt{21}\right)}\right)$
and $z_{\pm}=1\pm\sqrt{\frac{2}{5}}$. This bracketing determines six
eigenvalues (highlighted in light gray in the diagram) where we exploit the
fact that $w_{\pm}$ and $\widehat{w}_{\pm}$, both have multiplicity
$4-(6-3)=1$ and $1$ has multiplicity $5-(6-3)=2$.

To determine the remaining $14$ eigenvalues we take $G_{2}=S(K_{6})$ as an
auxiliary graph. In this case the shrinking number is $t_{2}=1$ and we obtain
the relation

$S(K_{6})\stackrel{{\scriptstyle
1}}{{\preccurlyeq}}S(\widehat{K}_{5})(A)\preccurlyeq S(K_{6})$

which can be represented by the following spectral diagram that fixes the
remaining eigenvalues which are highlighted in dark gray.

$S(K_{6})$ |  |  | 0 | $z_{-}$ | $z_{-}$ | $z_{-}$ | $z_{-}$ | $z_{-}$ | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | $z_{+}$ | $z_{+}$ | $z_{+}$ | $z_{+}$ | $z_{+}$ | 2
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---
$S(\widehat{K}_{5})(A)$ |  | 0 | $w_{-}$ | $z_{-}$ | $z_{-}$ | $z_{-}$ | $z_{-}$ | $\widehat{w}_{-}$ | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | $\widehat{w}_{+}$ | $z_{+}$ | $z_{+}$ | $z_{+}$ | $z_{+}$ | $w_{+}$ | 2
$S(K_{6})$ |  | 0 | $z_{-}$ | $z_{-}$ | $z_{-}$ | $z_{-}$ | $z_{-}$ | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | $z_{+}$ | $z_{+}$ | $z_{+}$ | $z_{+}$ | $z_{+}$ | 2 |

Again, only the number pendant paths and the length of the partitions are
relevant. Therefore, the partitions $A=\\{2,3\\}$ and $B=\\{1,4\\}$ give
isospectral graphs $S(\widehat{K}_{5})(A)$ and $S(\widehat{K}_{5})(B)$ which
are not isomorphic because the degree lists are different:

$\deg
S(\widehat{K}_{5})(A)=(\underbrace{2,\dots,2}_{16},3,5,5,5,5,5)\qquad\text{and}\qquad\deg
S(\widehat{K}_{5})(B)=(1,\underbrace{2,\dots,2}_{15},4,5,5,5,5,5)$

######  Example 4.7.

In a similar vein, the discrete fuzzy bipartite graphs below are also
isospectral and, clearly, non-isomorphic using the edge subdivision for all
the edges on the graph (see Figure 4).

$S(\widehat{K}_{2,5}(A))$

$S(\widehat{K}_{2,5}(B))$

Figure 4.  Examples of isospectral graphs $S(\widehat{K}_{2,5}(A))$ and
$S(\widehat{K}_{2,5}(B))$, obtained by edge subdivision of the fuzzy complete
bipartite construction $\widehat{K}_{2,5}(A)$ and $\widehat{K}_{2,5}(B)$ for
$A=(4,1)$ and $B=(2,3)$.

Finally, as in Corollary 4.4, we can extend isospectrality to the metric graph
scenario since the subdivision graphs are connected, and the number of edges
and vertices are preserved in the merging along the different partitions.
Hence all the graphs constructed in this section are also isospectral as
equilateral metric graphs for the Kirchhoff Laplacian.

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# Preventing sinking of a disk by leveraging the Parametric Hydraulic Jump
phenomenon

Jan Turczynowicz12  Radost Waszkiewicz12  Łukasz Gładczuk13<EMAIL_ADDRESS>1 Fenix Science Club, Aleja Stanów Zjednoczonych 24, 03-964 Warsaw 2 Institute
of Theoretical Physics, Faculty of Physics,
University of Warsaw, L. Pasteura 5, 02-093 Warsaw, Poland 3 Department of
Physics, Clarendon Laboratory,
University of Oxford, Oxford OX1 3PU, United Kingdom

###### Abstract

Although it is commonly expected that a metal disk placed on the surface of
water will sink, our investigation has revealed a surprising phenomenon: a
vertical jet directed onto the disk from above can allow it to remain afloat.
This result defies intuition, as one would assume that the force of the jet’s
impact would cause the disk to sink. We have discovered that this phenomenon
occurs as a result of water displacement from the top of the disk caused by
the impacting jet, operating through a mechanism similar to a classical
hydraulic jump. This displacement generates a difference in hydrostatic
pressures, resulting in an upward buoyancy force capable of balancing the
force of gravity. In contrast to the classical case, here the jump radius is
fixed by the geometric parameters of a disk, a phenomenon we refer to as the
_parametric hydraulic jump_. To further explore this effect, we have presented
a theoretical model based on scaling laws, which provides the conditions
required for the disk to float. The dimensionless scaling constant’s value was
determined through an independent experiment. Finally, we conducted
experiments on the disk’s floating and sinking, which showed a good match with
the proposed theory.

###### keywords:

Jets, Thin films, Wakes

##  1 Introduction

Placing a thin metal disk on the surface of water and directing a vertical
water jet onto the disk reveals an unusual phenomenon. Despite the disk being
denser than water, in some cases, the weight of water displaced from the
surface of the disk is sufficient for the buoyancy force to balance both the
impact force of the jet and the weight of the disk. This balance allows the
disk to remain afloat.

To understand the necessary conditions for this phenomenon to occur, it is
essential to study the water flow dynamics on the upper surface of the disk.
Due to the cylindrical symmetry of the flow and the very thin layer of fast-
moving liquid, this process is reminiscent of the classical hydraulic jump
problem.

The hydraulic jump has been a research topic at least since works of Rayleigh
(1914). Today, even with different variants of the geometry studied (Ivanova &
Gavrilyuk, 2019; Teymourtash & Mokhlesi, 2015; Wang & Khayat, 2019, 2018),
details of the phenomenon still remain a debated issue cf. (Bhagat et al.,
2018; Duchesne & Limat, 2022). Qualitatively, when a jet hits a plate, a
region of higher pressure at the centre of the disk necessarily appears. In
the region very close to the centre of the disk, where the boundary layer flow
is not yet developed (Baayoun et al., 2022), flow pattern is similar to the
one of an inviscid flow where high pressure at the centre accelerates the
liquid outwards. As the fluid moves further from the centre a combination of
outward acceleration and increased circumference leads to rapid decrease of
the film thickness $h$ (with $h\sim 1/r^{2}$, where $r$ denotes the distance
from the centre of the disk; Landau & Lifshitz 1987, 10 problem 5.1) until the
film is so thin that the viscous forces become significant. This thin layer
proves to be unstable and an abrupt transition to a subcritical, thicker
layer, happens at the distance $R_{\mathrm{J}}$ from the centre of the system.
Works of Watson (1964) and Bohr et al. (1993) based on boundary layer theories
show an experimentally validated scaling of $R_{\mathrm{J}}$ for a jet of
volumetric flux $Q$, kinematic viscosity $\nu$ and density $\rho$ to be

$R_{\mathrm{J}}\sim Q^{5/8}\nu^{-3/8}g^{-1/8},$ (1)

where $g$ is the gravitational acceleration. A recent work of Bhagat et al.
(2018) presented a different, also experimentally validated, scaling law

$R_{\mathrm{J}}\sim Q^{3/4}\rho^{1/4}\nu^{-1/4}\sigma^{-1/4},$ (2)

where $\sigma$ is the surface tension coefficient, claiming $g$ has no
influence on the location of the hydraulic jump. This was later contested in
(Duchesne et al., 2019) and (Bohr & Scheichl, 2021).

Both approaches indicate that the radius of the hydraulic jump increases with
an increase in the volumetric flux ($Q$). This relationship holds true for the
investigated finite disk only when $R_{\mathrm{J}}\lesssim R_{\mathrm{disk}}$.
However, when $Q$ reaches values that would cause the radius of the jump to
exceed the radius of the disk, the jump position becomes fixed to the radius
of the disk and the above expressions are no longer valid.

Experimental investigations revealed that, as the disk’s submersion depth
increases, a transition between two distinct regimes occurs (Figure 1b, c). We
refer to the two regimes as type I and type IIa jumps, following the
nomenclature of Ellegaard et al. (1996) and subsequent works of Yokoi & Xiao
(1999); Wang & Khayat (2019); Teymourtash & Mokhlesi (2015); Bush et al.
(2006). In the type I jump the water continues to flow away from the disk
after crossing the jump boundary. However, with greater submersion of the
disk, the water encounters a larger mass of non-moving water outside the disk
boundaries, significantly slowing down. This change in dynamics causes water
re-circulation, resulting in water flowing back onto the top of the disk (type
IIa jump).

We noticed that the type of jump coincides with the ability of the disk to
float. The disk would float only when a type I jump, similar to the one
depicted in Figure 1b, was very pronounced.

Figure 1:  Sketch (a) shows the flow pattern on the top of the disk. Jet,
characterised by flux $Q$ and cross-sectionally averaged velocity $U$, impacts
the disk creating a supercritical layer of thickness $h$ and velocity
$u_{\textrm{sup}}$. Near the disk’s edge, a parametric hydraulic jump occurs,
leading to a rapid increase in the liquid layer thickness to $H$ and a
reduction in liquid velocity to $u_{\textrm{low}}$. Depending on the value of
$H$ two types of jumps are observed. At $H=2.5\,\mathrm{mm}$ a type I jump (b)
occurs – where the entire disk is visible with a characteristic high wave
formed close to its edge. For a larger value of $H=5.5\,\mathrm{mm}$ a type
IIa jump (c) is observed – the edge of the disk is covered by the water, and
water close to the jump edge is moving _towards_ the centre of the disk. The
experiments shown in photos (b) and (c) used a disk with a radius of
$R=6\,\mathrm{cm}$, a jet flux of $Q=104\,\mathrm{ml/s}$, and a velocity of
$U=1.7\,\mathrm{m/s}$.

##  2 Parametric hydraulic jump

As the first step of our investigation, we focused on the flow dynamics,
independent of the floating dynamics. By fixing the disk to the bottom of the
liquid container with a variable-height pillar, we precisely controlled the
submersion depth of the disk. Changing this depth reproduced the two hydraulic
jump regime patterns, type I and type IIa. As shown in the inset of the Figure
1b, when the submersion depth was small, water on surface flowed outward. With
an increase in the submersion depth, past a critical value
$h_{\textrm{crit}}$, a region of recirculation appeared near the edge of the
disk (Figure 1c), causing the water to flow back onto the top surface of the
disk.

Figure 2:  Sketch (a) illustrates the experimental setup utilised to determine
the critical depth of the regime transition, $h_{\textrm{crit}}$. The disk’s
bottom was attached to a nut that could be moved along a vertical, threaded
rod, affixed to the bottom of a large vessel. This arrangement enabled
variation in the disk’s submersion depth ($H$). The radius of the jet ($r$)
was measured by taking a side photo of the jet and performing image analysis,
as shown in (b). The top image displays the original photograph before
analysis, while the bottom image, with enhanced contrast for better
readability, shows the detected jet boundaries. Figure 3:  The measurement
series (a) was obtained by varying the jet flux while maintaining a constant
nozzle height, resulting in a variable but measured jet radius. For each level
of jet flux, the disk ($R=6\,\textrm{cm}$) was slowly moved vertically to
determine the boundary of each flow regime. The orange and blue symbols in the
graph represent observations of type I and type IIa flow patterns,
respectively, observed around the disk. A master-curve (b) was constructed
from multiple measurement series by calculating $h_{\mathrm{crit}}$ and
$H^{*}$. In the graph, various symbols denote disks with different radii. The
horizontal dashed line marks the capillary length of $2.7\,\mathrm{mm}$.
Measuring the flow pattern below this capillary length presented notable
challenges. A linear fit, shown as a solid line, was applied to all data where
the transition depth exceeded this capillary length.

###  2.1 Scaling analysis

Hydraulic jumps are frequently modelled using scaling laws (e.g. Bhagat et al.
(2018)). In line with these methodologies, we develop a model for the critical
disk submersion depth $h_{\textrm{crit}}$ which marks the transition between
type I and type IIa hydraulic jump. In our specific scenario, illustrated in
Figure 1a, the jet impacts the centre of the disk with a constant flux $Q$.
This impact generates a supercritical flow layer atop the disk, characterised
by a height $h$ and a radial surface velocity $u_{\textrm{sup}}$. At the
disk’s periphery (radius $R$), this supercritical layer interacts with the
adjacent water, which, due to its significantly larger volume, possesses
greater inertia. Consequently, the fluid’s velocity near the edge reduces to a
subcritical level, $u_{\textrm{low}}$. This deceleration results in the
elevation of the water’s surface level to a height $H$, indicating a
conversion of the fluid’s kinetic energy into an increase in potential energy.

Assuming that the viscous drag balances the radial flow (Bhagat et al., 2018;
Landau & Lifshitz, 1987, ch. IV), leads to the following scaling:

$\frac{u_{\textrm{sup}}}{R}\sim\frac{\nu}{h^{2}}.$ (3)

Guided by our experimental observations, we assume that $u_{\textrm{sup}}\gg
u_{\textrm{low}}$ and that $h\ll H$. The flow height ($h$) is estimated by
applying the supercritical flow model described by Watson (1964), yielding
typical values around $h\sim 0.1\,\textrm{mm}$ and $H\sim 5\,\textrm{mm}$. The
ratio $u_{\textrm{low}}/u_{\textrm{sup}}$ can be estimated using the
conservation of mass, which suggests a typical value on the order of
$10^{-2}$.

Considering the transition between the supercritical and the subcritical
region, one can neglect the hydrostatic pressure influence before the jump and
the kinetic energy after it. Thus, the energy conservation for a thin slice of
the liquid yields

$\frac{1}{2}\rho u_{\textrm{sup}}^{2}h\sim\frac{1}{2}\rho gH^{2}.$ (4)

As the continuity implies $Q=2\pi u_{\textrm{sup}}Rh$, the maximal value of
$H$ which can be supported at the boundary scales according to

$H^{2}\sim\frac{Q^{3}}{8\pi^{3}g\nu R^{4}}.$ (5)

Defining the characteristic height

$H^{*}=\sqrt{Q^{3}/(8\pi^{3}g\nu R^{4})}\,,$ (6)

the expression for $h_{\textrm{crit}}$ reduces to

$h_{\textrm{crit}}=\gamma H^{*},$ (7)

with a dimensionless scaling constant $\gamma$. The value of the scaling
constant was determined in experiment described below.

###  2.2 Experiments

To accurately measure the transition depth, a reliable way of controlling the
position of the disk relative to the water surface is needed. This was
achieved by attaching the disk’s bottom to a nut, which could move on a
vertical threaded rod fixed to the vessel’s bottom, as shown in Figure 2a.
Turning the disk by $10^{\circ}$ raised or lowered the disk by $54\mathrm{\mu
m}$, providing a vertical travel of $8\,\mathrm{mm}$.

The jet originated from water directed through a straight vertical pipe, $1$
meter in length. This allowed for flow laminarisation and produced a smooth
jet impacting the disk. Water flow ($Q$) was regulated by a valve and
quantified using a flow meter. The width of the jet was varied by changing the
nozzle radius or the distance between the nozzle and the disk. The jet’s
radius was experimentally determined by photographing the jet from the side.
After enhancing the contrast of the photo, computer image analysis was
employed to identify the jet’s edges and subsequently measure its radius. This
process is illustrated in Figure 2b. For each measurement, 5 photos were
taken, and the results were averaged. The mean value had an estimated
uncertainty of $1\%$. Knowing the water flow and jet radius ($r$), the cross
sectionally averaged velocity of the jet was found using relation:

$Q=\pi r^{2}U.$ (8)

The experiment is sensitive to the downstream boundary condition (cf. for
example Bush et al. (2006)). We utilised a large cylindrical vessel, measuring
$31\,\textrm{cm}$ in height and $50\,\textrm{cm}$ in diameter, which was
carefully levelled. This ensured minimal water height deviations at the
container’s edge (less than $3\,\textrm{mm}$). It was difficult to achieve
better accuracy due to surface tension effects at the edge of the container.
As water was overflowing the edge of the vessel, the submersion depth of the
disk remained constant during the experiment.

The setup was used to determine the submersion depth of the disk at which the
type I and type IIa jumps occurred across various values of $Q$ and $U$. Each
series was executed while maintaining a constant nozzle height above the water
(note that $U$ was variable in these series). The transition between different
jump types did not occur simultaneously across the entire disk. Instead,
during these transitions, we observed a type I jump on some parts of the disk
and a type IIa jump on others. For each measurement, we recorded the greatest
depth for which type I jump was pronounced and the shallowest depth for which
type IIa jump was pronounced. The experimentally measured critical height,
denoted as $h_{\mathrm{crit}}$, was calculated as the average of these extreme
values. An illustrative measurement series for jump height is depicted in
Figure 3a. Utilising the presented scaling analysis (7), we were able to
combine all measurement series into a single master curve, as shown in Figure
3b.

Within the studied parameter range, the critical height $h_{\textrm{crit}}$
adheres to the predicted linear relationship (7). A deviation from linear
dependency can be noticed for $h_{\textrm{crit}}\lesssim 2.5\,\mathrm{mm}$
which aligns with the capillary length of water $l_{\textrm{cap}}\approx
2.7\,\mathrm{mm}$. This implies that surface tension phenomena might influence
the value of $h_{\textrm{crit}}$ at smaller $H^{*}$ values.

##  3 Unsinkable disk

The phenomenon, referred to as the _unsinkable disk_ , emerges when a
vertically directed jet impinges upon the centre of a freely floating metallic
disk. Under specific conditions, a disk that would ordinarily submerge can
instead achieve a floating equilibrium. Remarkably, this phenomenon exhibits
self-stabilising characteristics; when the disk is disturbed, either by a
change in inclination or position, it returns to a stable configuration.

Initial observations revealed that the phenomenon of the unsinkable disk
manifests exclusively under type I jump conditions. Conversely, the disk sinks
when a type IIa jump is observed. This insight suggests that the system
parameters needed for the disk to float are congruent with those that give
rise to a type I jump. This correlation can be intuitively understood. In the
case of a type IIa jump, a water vortex forms— as depicted in Figure 1c —
which redirects the water back onto the disk. This change diminishes the
effective buoyant force, ultimately causing the disk to sink.

###  3.1 Force balance

For the disk to achieve a state of equilibrium, in which it floats, the net
force exerted upon it must be zero,

$0=F_{\textrm{b}}-F_{\textrm{j}}-F_{\textrm{g}}.$ (9)

The forces contributing to this equilibrium include: the force generated by
the impinging jet, $F_{\textrm{j}}$; the gravitational force acting on the
disk, $F_{\textrm{g}}$; and the hydrostatic pressure exerted on the disk’s
underside, which manifests as an upward buoyancy force $F_{\textrm{b}}$. The
force exerted by the water jet with radius $r$ and flux $Q$ can be expressed
as (Stephens & Ward, 1972)

$F_{\textrm{j}}=\rho\frac{Q^{2}}{\pi r^{2}}.$ (10)

The buoyant force acting on the disk can be decomposed into two distinct
components: the first component arises from the buoyancy attributable to the
volume $V$ displaced by the disk itself; the second component depends on the
volume of air located above the disk, which, due to negligible thickness of
the water layer, is a function of the disk’s floating depth
$h_{\textrm{float}}$ and its surface area $S=\pi R^{2}$:

$F_{\textrm{b}}=\rho g(V+h_{\textrm{float}}S).$ (11)

Utilising equations (10) and (11), and by introducing the effective mass of
the disk as $m_{\textrm{eff}}=m_{\textrm{disk}}-\rho V$, the floating depth
can be formulated as a function of the relevant parameters:

$h_{\textrm{float}}=\frac{m_{\textrm{eff}}}{\rho S}+\frac{Q^{2}}{gS\pi
r^{2}}.$ (12)

The depth at which the disk floats shows a quadratic increase in response to
the rising water flux ($Q$). On the other hand, the disk can only float if the
jet has enough momentum to displace the necessary amount of water. Following
the observation that the disk floats only when type I jump occurs
$h_{\mathrm{float}}<h_{\mathrm{crit}}$, a condition for the disk to float is
found:

$\gamma\sqrt{\frac{\rho^{2}Q^{3}g}{8\pi\nu}}<gm_{\mathrm{eff}}+\rho QU$ (13)

which establishes a relationship between the water flux ($Q$) and the jet
velocity ($U$), required for achieving disk flotation. In our theoretical
calculations we used density of water $\rho=997\,\textrm{kg}/\textrm{m}^{3}$,
and its kinematic viscosity $\nu=10^{-6}\,\textrm{m}^{2}/\textrm{s}$ as
reported by Cooper & Dooley (2008).

###  3.2 Experiments

Figure 4:  The diagram (a) illustrates the experimental setup used to
investigate whether the disk would float or sink. The flow rate $Q$ and the
jet radius $r$ were independently manipulated by changing the diameter of
source pipe and adjusting the distance of the nozzle from the water surface.
$Q$ and $r$ were used to compute the jet velocity $U$. The results (b-d) from
the floating/sinking measurement experiments, alongside the theoretical
prediction of the float/sink boundary (indicated by the dashed black line)
show good quantitative agreement. The parameters of the disks used in these
experiments are summarised in the charts.

To test the necessary conditions for the disk to float, a similar experimental
setup to the one described in the Section 2.2 was used. This time, the disk
was unrestricted and free to move in all directions, as shown in Figure 4a.
The disks used in the experiment were carefully positioned beneath the jet and
subsequently released. We observed that the disks sank almost immediately when
the jet flux $Q$ and the jet velocity $U$ were insufficient. The disk was
considered to be floating if it remained on the surface for at least 15
seconds.

The disks used in the experiments were approximately $0.1\,\mathrm{mm}$ thick
and made of aluminium. To adjust the weight of the disks, and to overcome the
effects of the surface tension, copper washers were glued to their underside.
The additional buoyancy and weight resulting from these modifications have
been taken into account in the comparison with theoretical predictions.

We studied $U$ in range from $0.6\,\mathrm{m/s}$ to $2.5\,\mathrm{m/s}$ and
focused specifically on values of the water flux $Q$, that are close to the
threshold between the states of floating and sinking. Measured behaviour of
the disks, along with the theoretically predicted float/sink boundary, are
depicted in Figure 4b-d. We noticed that when values of flux $Q$ were
sufficiently high the disk would remain afloat. Conversely, when the values of
$Q$ were too low, the disk would gradually sink by increasing its floating
depth. This process continued until the hydraulic jump collapsed, allowing
water to flow onto the disk.

The theoretical model generally aligns well with most of the cases studied.
However, deviations from the theoretical predictions were observed for higher
values of $U$. In these instances, despite the theory suggesting that the disk
should remain afloat, it sank during our experiments. In these instances, the
disk exhibited oscillations of increasing amplitude, characterised by lateral
movement and tilting, where the jet notably forced the impacted segment of the
disk downward. At a certain point, the tilt became so pronounced, and the
submersion of the disk’s lower side so deep, that the water jump could no
longer prevent ambient water from overflowing onto the disk, leading to its
eventual sinking.

While the analysis of disk rotational stability falls outside the scope of
this paper, it is worth noting that in the instances where the disk was
floating, any slight disturbance in its position relative to the jet resulted
in a consistent return to its stable position. Interestingly, our static
theoretical analysis predicted floating even in cases of oscillatory sinking.
This discrepancy highlights the necessity of incorporating disk dynamics into
considerations for a comprehensive understanding of this phenomenon.

##  4 Conclusions

Our study demonstrates that directing a vertical jet onto a disk from above
can prevent the disk from sinking. This effect is attributed to the phenomenon
of the parametric hydraulic jump, which displaces water from the top of the
disk, thereby generating an additional upward buoyancy force. We identified
two distinct types of hydraulic jumps, type I and IIa, and established that
maintaining the disk’s flotation is feasible under type I jump conditions.

The parametric hydraulic jump differs from classical consideration, as the
position of the water jump is fixed by the geometry of the setup. In this
specific case, we utilised a scaling law to determine the critical submersion
depth at which the transition from type I to type IIa jump occurs. The scaling
constant of the model, $\gamma=0.624\pm 0.010$, was determined in an
independent experiment aimed at measuring the submersion depth corresponding
to this critical transition between jump types.

The phenomenon was studied experimentally across a broad range of parameters,
including disk’s radius, jet’s flux, and jet’s radius. Results were compared
against the theoretical model, demonstrating substantial agreement in nearly
all the cases examined. Notably, at high values of the jet flux and velocity,
the disk displayed signs of instability, characterised by progressively
intensifying oscillations. This observation indicates that while the proposed
model establishes necessary conditions for disk flotation, it does not yet
encompass all the sufficient conditions required for a comprehensive
understanding of the phenomenon.

###### Acknowledgements.

We extend our sincere gratitude to Rajesh Bhagat for his invaluable guidance
on the theoretical aspects of this publication. The research presented herein
was conducted in part during the the preparations of the Fenix Science Club’s
(Klub Naukowy Fenix) Polish team for the International Young Physicist
Tournament (IYPT) 2022. The team comprised Jan Turczynowicz, Maciej Dąbkowski,
Mikołaj Czarnecki, Igor Kumela, Rafał Bryl, with Łukasz Gładczuk and Radost
Waszkiewicz serving as team leaders. This particular work addresses the
problem _4\. Unsinkable Disk_ posed at the IYPT 2022. Work of Jan Turczynowicz
was supported by Ochota na Naukę project ‘Ochota na Fizykę (Spring 2022)’.
Work of Radost Waszkiewicz was supported by the National Science Centre of
Poland grant Sonata no. 2018/31/D/ST3/02408.

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