# Cospectral mates for generalized
Johnson and Grassmann graphs

Aida Abiad Jozefien D’haeseleer Willem H. Haemers Robin Simoens
<EMAIL_ADDRESS>Department of Mathematics and Computer Science,
Eindhoven University of Technology, The Netherlands
Department of Mathematics: Analysis, Logic and Discrete Mathematics, Ghent
University, Belgium
Department of Mathematics and Data Science, Vrije Universiteit Brussel,
<EMAIL_ADDRESS>Department of Mathematics: Analysis,
Logic and Discrete Mathematics, Ghent University<EMAIL_ADDRESS>Department of Econometrics and Operations Research, Tilburg University, The
<EMAIL_ADDRESS>Department of Mathematics: Analysis, Logic
and Discrete Mathematics, Ghent University, Belgium

###### Abstract

We provide three infinite families of graphs in the Johnson and Grassmann
schemes that are not uniquely determined by their spectrum. We do so by
constructing graphs that are cospectral but non-isomorphic to these graphs.

MSC Subject: 05C50, 15A18
Keywords: Graph, Eigenvalues, Determined by spectrum, Switching

##  1 Introduction

A major problem in the field of spectral graph theory is to decide whether a
given graph is determined by its spectrum (the eigenvalues of its adjacency
matrix). This problem has been solved for many families of graphs; sometimes
by proving that the spectrum determines the graph, and sometimes by
constructing cospectral mates (non-isomorphic graphs with the same spectrum).
Although a conjecture by the third author states that almost all graphs are
determined by their spectrum, many well-known graphs seem to have cospectral
mates. This is, in particular, the case for several graphs in the Johnson and
Grassmann association schemes, see for instance [2, 3, 4, 6, 9]. We contribute
to this line of research by constructing cospectral mates for a number of open
cases in the Johnson and Grassmann schemes. For this, we use switching
techniques due to Godsil and McKay [5] and Wang, Qiu and Hu [11].

This paper is structured as follows. In Section 2, we introduce the necessary
definitions. Section 3 contains an overview of previous results on the
cospectrality of generalized Johnson and Grassmann graphs. In Section 4, we
prove that the generalized Johnson graph $J_{\\{2\\}}(n,4)$, $n\geq 8$, is not
determined by its spectrum, thereby solving the second open problem of [2]. In
Section 5, we show that the generalized Johnson graph
$J_{\\{1,2,\dots\frac{k-1}{2}\\}}(2k,k)$ with $k\geq 5$, $k$ odd, is not
determined by its spectrum. In Section 6, we prove that the $q$-Kneser graph
$K_{q}(n,k)$ is not determined by its spectrum if $q=2$. In Section 7 we
provide three specific generalized Johnson graphs that are not determined by
their spectrum and pose some open questions. Finally, the Appendix contains an
overview of all cospectrality results for small graphs in the aforementioned
schemes.

##  2 Preliminaries

Let $\Gamma=(V,E)$ denote a simple loopless graph. The _spectrum_ of $\Gamma$
is the multiset of eigenvalues of the adjacency matrix of $\Gamma$. Graphs are
_cospectral_ if they have the same spectrum. Two graphs that are cospectral
but not isomorphic, are called _cospectral mates_. A graph is _determined by
its spectrum_ (DS) if it has no cospectral mate. Otherwise, it is not
determined by its spectrum (NDS).

One of the most well-known tools for constructing cospectral graphs was
introduced by Godsil and McKay [5].

######  Theorem 1 (GM-switching [5]).

Let $\Gamma$ be a graph and $\\{C_{1},C_{2},\dots C_{k},D\\}$ a partition of
$V(\Gamma)$ such that the following hold for all $i,j\in\\{1,2,\dots,k\\}$:

  1. (i)

Any two vertices of $C_{i}$ have the same number of neighbours in $C_{j}$.

  2. (ii)

Every vertex in $D$ has exactly $0$, $\frac{1}{2}|C_{i}|$ or $|C_{i}|$
neighbours in $C_{i}$.

For all $i\in\\{1,2,\dots,k\\}$, every $u\in C_{i}$ and every $v\in D$ that
has exactly $\frac{1}{2}|C_{i}|$ neighbours in $C_{i}$, reverse the adjacency
between $u$ and $v$. The resulting graph is cospectral with $\Gamma$. We say
that it is obtained from $\Gamma$ by _GM-switching_ with respect to
$C_{1},C_{2},\dots C_{k}$.

Another switching technique to construct cospectral graphs was recently
introduced by Wang, Qui and Hu [11].

######  Theorem 2 (WQH-switching [11]).

Let $\Gamma$ be a graph and $C_{1},C_{2}$ disjoint subsets of $V(\Gamma)$ such
that the following hold:

  1. (i)

$|C_{1}|=|C_{2}|.$

  2. (ii)

There exists a constant $c$ such that for all $i,j\in\\{1,2\\}$, $i\neq j$ and
every vertex of $C_{i}$, the number of neighbours in $C_{i}$ minus the number
of neighbours in $C_{j}$ is equal to $c$.

  3. (iii)

Every vertex outside $C_{1}\cup C_{2}$ has either:

    1. (a)

$|C_{1}|$ neighbours in $C_{1}$ and $0$ neighbours in $C_{2}$,

    2. (b)

$0$ neighbours in $C_{1}$ and $|C_{2}|$ neighbours in $C_{2}$,

    3. (c)

the same number of neighbours in $C_{1}$ and $C_{2}$.

For every $u\in C_{1}\cup C_{2}$ and every $v\notin C_{1}\cup C_{2}$ for which
(a) or (b) holds, reverse the adjacency between $u$ and $v$. The resulting
graph is cospectral with $\Gamma$. We say that it is obtained from $\Gamma$ by
_WQH-switching_ with respect to $C_{1}$ and $C_{2}$.

We should mention that a more general version of WQH-switching was introduced
in [10].

Let $n$, $k$ be positive integers with $k\leq n$. We now define the graphs of
our interest, namely those in the Johnson and Grassmann schemes.

######  Definition 3.

Let $S\subseteq\\{0,1,\dots,k-1\\}$. The _generalized Johnson graph
$J_{S}(n,k)$_ has as vertices the $k$-subsets of $\\{1,\dots,n\\}$, where two
vertices are adjacent if their intersection size is in $S$.

In particular, $J_{\\{0\\}}(n,k)$ is the _Kneser graph_ $K(n,k)$ and
$J_{\\{k-1\\}}(n,k)$ is the _Johnson graph_ $J(n,k)$.

If we replace sets by subspaces and sizes by dimensions in Definition 3, we
obtain the definition of a generalized Grassmann graph. A vector space of
dimension $k$ is called a _$k$ -space_ for short. Furthermore, let
$\mathbb{F}_{q}$ denote the finite field of order $q$ ($q$ is a prime power).

######  Definition 4.

Let $S\subseteq\\{0,1,\dots,k-1\\}$. The _generalized Grassmann graph
$J_{q,S}(n,k)$_ has as vertices the $k$-subspaces of $\mathbb{F}_{q}^{n}$,
where two vertices are adjacent if their intersection dimension is in $S$.

In particular, $J_{q,\\{0\\}}(n,k)$ is the _$q$ -Kneser graph_ $K_{q}(n,k)$
and $J_{q,\\{k-1\\}}(n,k)$ is the _Grassmann graph_ $J_{q}(n,k)$.

From now on, for all the aforementioned graphs, we assume that $k\leq n/2$,
since $J_{S}(n,k)$ is isomorphic to $J_{\\{s+n-2k\mid s\in S\\}}(n,n-k)$. We
also assume that $|S|\leq k/2$ because the complement of $J_{S}(n,k)$ is
$J_{\\{1,\dots,n\\}\setminus S}(n,k)$, and a regular graph is DS if and only
if its complement is DS. Note that these observations also hold for Grassmann
graphs. Also note that if $S=\emptyset$, the graph is complete or edgeless and
therefore trivially DS. Therefore, we also assume that $|S|\geq 1$, and in
particular, $k\geq 2$.

##  3 Previous results

The following results about the cospectrality of generalized Johnson and
Grassmann graphs have been obtained by various authors. We present them in a
chronological order.

Let $q$ be a prime power and $1\leq k\leq n/2$.

######  Theorem 5 ([1, 7]).

$K(n,2)$ is DS if and only if $n\neq 8$.

######  Theorem 6 ([8]).

$K(2k+1,k)$ is DS.

######  Theorem 7 ([4]).

$J_{q}(2k+1,k)$ is NDS.

######  Theorem 8 ([3]).

$J(n,k)$ and $J_{q}(n,k)$ are NDS if $k\geq 3$.

######  Theorem 9 ([6]).

  1. (i)

$K(n,k)$ is NDS if there exists an $m<k-1$ for which
$\binom{n-k+m}{m}=2\binom{n-2k+m}{m}$.

  2. (ii)

$K_{\\{0,2,4,\dots\\}}(n,k)$ is NDS if $k\geq 3$.

######  Theorem 10 ([2]).

  1. (i)

$J_{\\{0,1,\dots,m\\}}(3k-2m-1,k)$ is NDS if $k\geq\max(m+2,3)$.

  2. (ii)

$J_{\\{0,1,\dots,m\\}}(n,2m+1)$ is NDS if $m\geq 2$ and $n\geq 4m+2$.

######  Theorem 11 ([9]).

$J_{q}(n,2)$ is NDS.

##  4 $\boldsymbol{J_{\\{2\\}}(n,4)}$ is NDS if $\boldsymbol{n\geq 8}$

The authors of [2] found, computationally, that the graph $J_{\\{2\\}}(8,4)$
has a cospectral mate by GM-switching with respect to switching sets of size
$8$, which implies that it is NDS. In this section, we place this graph in an
infinite family of graphs that are NDS by using the recent WQH-switching. This
solves the second open problem of [2].

We assume that $n\geq 8$, in accordance with the usual assumption that $k\leq
n/2$. Note that if $n=7$, $J_{\\{2\\}}(n,4)\cong J_{\\{1\\}}(7,3)$ is NDS by
Theorem 7. If $n=6$, then $J_{\\{2\\}}(n,4)\cong K(6,2)$ is DS by Theorem 5
and if $n\leq 5$, then $J_{\\{2\\}}(n,4)$ is trivially DS.

In the main result of this section, we use that $J_{\\{2\\}}(n,4)$ is edge-
regular (adjacent vertices have a constant number of common neighbours), while
the cospectral mate is not. Actually, every “elementary” generalized Johnson
graph $J_{\\{i\\}}(n,k)$ is edge-regular, because it is the distance-$(k-i)$
graph of the Johnson graph $J(n,k)$, which is distance-regular.

######  Lemma 12.

Adjacent vertices of $J_{\\{2\\}}(n,4)$ have $\frac{1}{2}n(n+3)-26$ common
neighbours.

###### Proof.

Choose any two adjacent vertices $v$ and $w$. By relabelling the elements of
$\\{1,2,\dots,n\\}$, we may assume that $v=\\{1,2,3,4\\}$ and
$w=\\{1,2,5,6\\}$. There are three types of common neighbours.

  1. (i)

Vertices that include $\\{1,2\\}$. The other two elements can be chosen from
$\\{7,8,\dots,n\\}$. So there are $\binom{n-6}{2}$ such vertices.

  2. (ii)

Vertices that intersect $\\{1,2\\}$ in one element. They must contain one
element of $\\{3,4\\}$ and one element of $\\{5,6\\}$, while the remaining
element should be in $\\{7,8,\dots,n\\}$. There are $2^{3}(n-6)$ such
vertices.

  3. (iii)

If a vertex is adjacent to $v$ and $w$ but does not contain $1$ or $2$, it
must be equal to $\\{3,4,5,6\\}$.

We can now conclude that $v$ and $w$ have
$\binom{n-6}{2}+2^{3}(n-6)+1=\frac{1}{2}n(n+3)-26$ common neighbours. ∎

######  Theorem 13.

$J_{\\{2\\}}(n,4)$ is not determined by its spectrum if $n\geq 8$.

###### Proof.

Define the sets $C_{1}:=\\{v_{1},v_{2},v_{3}\\}$ and
$C_{2}:=\\{v_{4},v_{5},v_{6}\\}$, where:
• $v_{1}:=\\{1,2,3,4\\}$ • $v_{2}:=\\{1,2,3,5\\}$ • $v_{3}:=\\{1,2,3,6\\}$ •
$v_{4}:=\\{1,4,5,6\\}$ • $v_{5}:=\\{2,4,5,6\\}$ • $v_{6}:=\\{3,4,5,6\\}$
$v_{1}$$v_{2}$$v_{3}$$v_{4}$$v_{5}$$v_{6}$$C_{1}$$C_{2}$
Notice the symmetry with respect to the sets $\\{1,2,3\\}$ and $\\{4,5,6\\}$.
With this in mind, we can prove that $C_{1}$ and $C_{2}$ form a WQH-switching
set of $J_{\\{2\\}}(n,4)$, by checking all conditions of Theorem 2. The first
two conditions are fulfilled by definition. In order to check the third
condition, select an arbitrary $4$-set $v\notin C_{1}\cup C_{2}$. We
distinguish seven cases, according to the intersection size of $v$ with
$\\{1,2,3\\}$ and $\\{4,5,6\\}$, see Table 1. Because of the symmetry of $C$
with respect to these sets, we assume that
$|v\cap\\{1,2,3\\}|\leq|v\cap\\{4,5,6\\}|$. Also note that
$|v\cap\\{1,2,3\\}|+|v\cap\\{4,5,6\\}|\leq 4$, and that these cardinalities
cannot be $1$ and $3$ simultaneously, since otherwise the vertex is one of the
vertices in $C_{1}\cup C_{2}$. In each case, the requirements of Theorem
2(iii) are met.

$|v\cap\\{1,2,3\\}|$ | 0 | 0 | 0 | 0 | 1 | 1 | 2
---|---|---|---|---|---|---|---
$|v\cap\\{4,5,6\\}|$ | 0 | 1 | 2 | 3 | 1 | 2 | 2
5 # neighbours in $C_{1}$ | 0 | 0 | 0 | 0 | 1 | 2 | 1
# neighbours in $C_{2}$ | 0 | 0 | 3 | 0 | 1 | 2 | 1
Table 1: Given $v\notin C_{1}\cup C_{2}$, the number of neighbours of $v$ in
$C_{1}$ and $C_{2}$ only depend on its intersection size with $\\{1,2,3\\}$
and $\\{4,5,6\\}$.

We are left to prove that the graph obtained by WQH-switching with respect to
$C_{1}$ and $C_{2}$, is not isomorphic to the original one.

Consider the vertices $v=\\{1,2,3,4\\}\in C_{1}$ and $w=\\{1,4,5,7\\}\notin
C_{1}\cup C_{2}$. The neighbours of $w$ are preserved by switching, since
$|w\cap\\{1,2,3\\}|=1$ and $|w\cap\\{4,5,6\\}|=2$ (see Table 1). In
particular, $v$ and $w$ are adjacent in both $\Gamma$ and $\Gamma^{\prime}$.
We show that they have more common neighbours in $\Gamma^{\prime}$ than in
$\Gamma$. For this, we only need to consider neighbours of $w$ for which the
adjacency with $v$ is changed during the switching process. We observe from
Table 1 that the switching only affects those vertices which have $2$ elements
in one of the sets $\\{1,2,3\\},\\{4,5,6\\}$ and none in the other. Neighbours
of $w$ that contain $2$ elements of $\\{1,2,3\\}$ and none of $\\{4,5,6\\}$,
are of the form $\\{1,2,7,a\\}$ or $\\{1,3,7,a\\}$ with $a\geq 8$. So there
are a total of $2(n-7)$ vertices that are adjacent to $v$ and $w$ in $\Gamma$
but not adjacent to $v$ in $\Gamma^{\prime}$. On the other hand, neighbours of
$w$ that contain no element of $\\{1,2,3\\}$ and $2$ elements of
$\\{4,5,6\\}$, are of the form $\\{4,5,a,b\\}$, $\\{4,6,7,a\\}$ or
$\\{5,6,7,a\\}$ with $a,b\geq 8$. They are not adjacent to $v$ in $\Gamma$,
but become adjacent to $v$ after switching. So $\binom{n-7}{2}+2(n-7)$ new
common neighbours are created.

We conclude that $v$ and $w$ have $\binom{n-7}{2}$ more common neighbours in
$\Gamma^{\prime}$ than any two adjacent vertices in $\Gamma$, which is a
strictly positive difference, except when $n=8$. In order to solve this one
specific case, we take a look at an other pair of vertices.

Assume $n=8$ and consider the vertices $v=\\{1,2,3,4\\}\in C_{1}$ and
$u=\\{5,6,7,8\\}\notin C_{1}\cup C_{2}$. They are not adjacent in $\Gamma$,
but become adjacent after switching, since $|u\cap\\{1,2,3\\}|=0$ and
$|u\cap\\{4,5,6\\}|=2$ (see Table 1). We give a lower bound on the number of
common neighbours of $v$ and $u$ in $\Gamma^{\prime}$. In the original graph,
$v$ and $u$ have $\binom{4}{2}^{2}=36$ common neighbours. Three of these are
the vertices of $C_{2}$, and are therefore lost after switching. All other
common neighbours of $v$ and $u$ in $\Gamma$ that are no longer common
neighbours in $\Gamma^{\prime}$, are those that contain $2$ elements of
$\\{1,2,3\\}$ and none of $\\{4,5,6\\}$, i.e. the three vertices
$\\{1,2,7,8\\}$, $\\{1,3,7,8\\}$ and $\\{2,3,7,8\\}$. We are left with at
least $36-3-3=30$ common neighbours, which is strictly more than
$\frac{1}{2}8(8+3)-26=18$, the number of common neighbours of any two vertices
in $\Gamma$ (Lemma 12).

We proved that there are always two neighbours in $\Gamma^{\prime}$ which have
strictly more common neighbours than any two neighbours in $\Gamma$. So
$\Gamma$ and $\Gamma^{\prime}$ cannot be isomorphic. ∎

##  5 $\boldsymbol{J_{\\{1,2,\dots\frac{k-1}{2}\\}}(2k,k)}$ is NDS if
$\boldsymbol{k\geq 5}$, $\boldsymbol{k}$ odd

The next result uses GM-switching with respect to two switching sets. Our
argument is inspired by the proof of Theorem 10(i), as
$J_{\\{1,2,\dots\frac{k-1}{2}\\}}(2k,k)$ can be seen as two copies of
$J_{\\{0,1,\dots\frac{k-3}{2}\\}}(2k-1,k-1)$, connected in a certain way.
While our first switching set corresponds to the one used to show Theorem
10(i), the second one is given by its complement.

######  Theorem 14.

$J_{\\{1,2,\dots\frac{k-1}{2}\\}}(2k,k)$ is NDS if $k\geq 5$, $k$ odd.

###### Proof.

Denote $\Gamma:=J_{\\{1,2,\dots\frac{k-1}{2}\\}}(2k,k)$. Let $C_{1}$ be the
set of $k$-subsets of $[2k]$ that contain $[k-1]$ and let $C_{2}$ be the set
of $k$-subsets of $[2k]$ that are disjoint with $[k-1]$. There is a bijection
from $C_{1}$ to $C_{2}$, given by the complement.

$[k-1]$ Figure 1: Elements of $C_{2}$ include $[k-1]$ and one other element.

We first check the conditions of Theorem 1 in order to show that $C_{1}$ and
$C_{2}$ form a GM-switching set of $\Gamma$. It is immediate that
$|C_{1}|=|C_{2}|=k+1$. Moreover, every two $k$-sets of $C_{1}$ (resp. $C_{2}$)
intersect in $k-1$ elements, which means that $C_{1}$ (resp. $C_{2}$) is a
coclique. Any vertex in $C_{1}$ has $k$ neighbours in $C_{2}$ and vice versa.
Choose an arbitrary vertex $v\notin C_{1}\cup C_{2}$ and let $l$ be the size
of its intersection with $[k-1]$. Then $v$ intersects $l+1$ vertices of
$C_{1}$ in $l$ elements and the other $k-l$ vertices in $l+1$ elements.
Similarly, $v$ intersects $l+1$ vertices of $C_{2}$ in $k-l$ elements and
$k-l$ vertices in $k-l-1$ elements. So if $l<\frac{k-1}{2}$, then $v$ is
adjacent to all vertices of $C_{1}$ and none of $C_{2}$. If $l=\frac{k-1}{2}$,
then it is adjacent to $\frac{k+1}{2}=\frac{1}{2}|C_{1}|$ elements of each
switching set, and if $l>\frac{k-1}{2}$, then it is adjacent with no vertices
of $C_{1}$ and all vertices of $C_{2}$. We conclude that $C_{1}$ and $C_{2}$
fulfil the GM-switching conditions.

Next, we show that the graph $\Gamma^{\prime}$ obtained from $\Gamma$ by GM-
switching with respect to $C_{1}$ and $C_{2})$, is not isomorphic to $\Gamma$.
Define $v:=[k]\in C_{1}$ and $w:=\\{2,3,\dots,k+1\\}\notin C_{1}\cup C_{2}$.

We claim that $v$ and $w$ have strictly less common neighbours after
switching. For this, it suffices to count the number of common neighbours that
are lost or added during the GM-switching process. Therefore, we only consider
those neighbours of $w$ that have $\frac{1}{2}|C_{1}|=\frac{k+1}{2}$
neighbours in $C_{1}$, i.e. those that contain exactly $\frac{k-1}{2}$
elements of $[k-1]$. First consider the common neighbours that are added. They
should not be adjacent to $v$ in the original graph, which means that they
must contain the element $k$. Adjacency with $w$ implies that they must also
contain the element $1$ and not the element $k+1$. Thus, there are

$\underbrace{\binom{k-2}{\frac{k-3}{2}}}_{\begin{subarray}{c}\text{$\frac{k-1}{2}$
elements of $[k-1]$,}\\\ \text{one of which is
$1$}\end{subarray}}\cdot\underbrace{\binom{k-1}{\frac{k-1}{2}}}_{\begin{subarray}{c}\text{$\frac{k+1}{2}$
elements not in $[k-1]$}\\\ \text{one of which is $k$}\end{subarray}}$

added common neighbours of $v$ and $w$. Now consider the common neighbours
that are lost. Since these vertices must originally be adjacent to $v$, they
may not contain $k$. If they contain the element $1$, then they will
automatically be adjacent to $w$. So there are at least

$\underbrace{\binom{k-2}{\frac{k-3}{2}}}_{\begin{subarray}{c}\text{$\frac{k-1}{2}$
elements of $[k-1]$,}\\\ \text{one of which is
$1$}\end{subarray}}\cdot\underbrace{\binom{k}{\frac{k+1}{2}}}_{\begin{subarray}{c}\text{$\frac{k+1}{2}$
elements}\\\ \text{not in $[k]$}\end{subarray}}$

common neighbours of $v$ and $w$ that are lost during switching, which is
already strictly more than the number of added common neighbours.

Now, assume by contradiction that $\Gamma\cong\Gamma^{\prime}$. Let
$\lambda(u,w)$ denote the number of common neighbours of $u$ and $w$, for any
vertex $u$ and $w$ defined as before. Since $\Gamma$ is vertex-transitive, the
multiset

$\Lambda(w)=\\{\lambda(u,w)\mid u\not\sim w\\}$

remains the same after switching. The value of $\lambda(u,w)$ stays the same
if $u\notin C_{1}\cup C_{2}$, so we can restrict ourselves to vertices in
$C_{1}\cup C_{2}$, i.e. the multiset

$\\{\lambda(u,w)\mid u\in C_{1}\cup C_{2},u\not\sim w\\}$

remains the same. Since $w$ is adjacent to all vertices of $C_{2}$ and none of
$C_{1}$, this is equivalent to saying that

$\\{\lambda(u,w)\mid u\in C_{1}\\}$

is invariant. We now apply the same reasoning as in [2]. Define
$v^{\prime}:=[k-1]\cup\\{k+1\\}$. The automorphism induced by the permutation
$(k+2\ k+3\ \cdots 2k)$ fixes $v$, $v^{\prime}$ and $w$, but shifts the other
vertices of $C_{1}$ cyclically. So $\lambda(u,w)$ is constant among all $u\in
C_{1}\setminus\\{v,v^{\prime}\\}$. Together with the observation that
$\lambda(v,w)=\lambda(v^{\prime},w)$, we see that $\lambda(v,w)$ is the same
for $\Gamma$ and $\Gamma^{\prime}$, a contradiction with the above.

We conclude that $\Gamma$ and $\Gamma^{\prime}$ are cospectral mates. Thus,
$\Gamma$ is NDS. ∎

##  6 $\boldsymbol{q}$-Kneser graphs are NDS if $\boldsymbol{q=2}$

In this section, we prove that the $q$-Kneser graph
$K_{2}(n,k)=J_{2,\\{0\\}}(n,k)$ is NDS (under the assumption that $k\geq 2$,
otherwise the graph is trivially DS) using GM-switching with respect to a
switching set of size $4$. This result can be seen as a $q$-analog of Theorem
9(i) in the case that $q=2$ (and without the extra condition).

######  Theorem 15.

$K_{2}(n,k)$ is not determined by its spectrum.

###### Proof.

Let $\Gamma:=K_{2}(n,k)$. Define the following $1$-spaces in
$\mathbb{F}_{2}^{n}$ (drawn projectively, as points in $PG(n-1,2)$).
• $p_{1}:=\langle(1,0,0,0,0,\dots,0)\rangle$ •
$p_{2}:=\langle(0,1,0,0,0,\dots,0)\rangle$ •
$p_{3}:=\langle(0,0,1,0,0,\dots,0)\rangle$ •
$p_{4}:=\langle(1,1,0,0,0,\dots,0)\rangle$ •
$p_{5}:=\langle(1,0,1,0,0,\dots,0)\rangle$ •
$p_{6}:=\langle(0,1,1,0,0,\dots,0)\rangle$
$p_{5}$$p_{4}$$p_{6}$$p_{3}$$p_{1}$$p_{2}$$\pi$$[k-2]$
Let $\pi$ be a $(k-2)$-space that has a trivial intersection with the
$3$-space (projective plane) spanned by these points. Define
$C:=\\{p_{1}p_{2}\pi,\;p_{1}p_{3}\pi,\;p_{2}p_{3}\pi,\;p_{4}p_{5}\pi\\}$.

We first prove that $C$ is a GM-switching set of $\Gamma$ by checking the
conditions of Theorem 1 (where $k=1$). Every element of $C$ meets every other
element of $C$ in a $k-1$-space, so $C$ is a coclique and therefore regular.
Choose an arbitrary $k$-space not in $C$ and let
$R:=\\{p_{1}\pi,p_{2}\pi,\dots,p_{6}\pi\\}$. If the chosen $k$-space
intersects all elements of $R$ trivially, then it also intersects all spaces
in $C$ trivially, which means that it is adjacent to all the corresponding
vertices of the $q$-Kneser graph. If it intersects just one of the elements of
$R$ non-trivially, then it is adjacent to exactly $2$ vertices of $C$. If the
$k$-space intersects at least two elements of $R$ non-trivially, then it
either intersects $\pi$ nontrivially, or it contains a $2$-space in the
$k+1$-space $p_{1}p_{2}p_{3}\pi$. Either case, it intersects every element of
$C$ non-trivially, therefore being adjacent to no element of $C$.

Let $\Gamma^{\prime}$ be the graph obtained from $\Gamma$ by GM-switching with
respect to $C$. We prove that $\Gamma$ and $\Gamma^{\prime}$ are non-
isomorphic. Let $\tau$ be a $(k-1)$-space that intersects $p_{1}p_{2}p_{3}\pi$
trivially and consider the pairwise non-adjacent $k$-spaces $p_{1}\tau$,
$p_{2}\tau$ and $p_{4}\tau$ in $\Gamma^{\prime}$. Then there is exactly one
$k$-space that is adjacent to $p_{1}\tau$ but not adjacent to both $p_{2}\tau$
and $p_{4}\tau$ in $\Gamma^{\prime}$. This is the $k$-space $p_{1}p_{3}\pi$.
Indeed, a $k$-space outside $C$ that intersects both $p_{2}\tau$ and
$p_{4}\tau$, must intersect $p_{1}\tau$ as well. So the only spaces that can
meet this property are in $C$. Since the switching reverses adjacency for
$p_{1}\tau$, $p_{2}\tau$ and $p_{4}\tau$, we get that $p_{1}p_{3}\pi$ is
adjacent to $p_{1}\tau$ and not adjacent to $p_{2}\tau$ and $p_{4}\tau$, while
the other elements of $C$ are not.

$p_{5}$$p_{4}$$p_{6}$$p_{3}$$p_{1}$$p_{2}$$\pi$$\tau$$[k-1]$$[k-2]$

Now consider three arbitrarily chosen non-adjacent $k$-spaces in $\Gamma$.
Call them $\alpha$, $\beta$ and $\gamma$. Non-adjacency here means that they
intersect one another. We prove that the number of $k$-spaces that intersects
$\alpha$ trivially but intersects both $\beta$ and $\gamma$ non-trivially, is
never equal to one. Let $\delta$ be such a space (if it does not exist, we are
done). Choose two $1$-spaces (projective points) $p\in\beta\cap\delta$ and
$q\in\gamma\cap\delta$. The $2$-space (projective line) $pq$ does not
intersect $\alpha$, as it lies in $\delta$. There are
$2^{k(k-2)}\genfrac{[}{]}{0.0pt}{}{n-k-1}{k-2}_{2}>1$ different $k$-spaces
through $pq$ that intersect $\alpha$ trivially, and they all meet the above
property.

$p$$q$$\alpha$$\beta$$\gamma$$\delta$

Since the number of $k$-spaces with this property is different for $\Gamma$
and $\Gamma^{\prime}$, while it should be invariant under isomorphism, we
conclude that $\Gamma$ and $\Gamma^{\prime}$ are not isomorphic. ∎

##  7 Open problems

Besides the three infinite families of graphs that are NDS, we also found
three “sporadic” generalized Johnson graphs with a cospectral mate. For each
of these three graphs, we provide a pair of switching sets that produces a
cospectral mate, see the table below. The fact that they are non-isomorphic
was checked by computer.

graph | method | switching sets
---|---|---
$J_{\\{1\\}}(11,4)$ | WQH |  $C_{1}=\left\\{\\{1,2,3,10\\},\\{4,5,6,10\\},\\{7,8,9,10\\}\right\\}$ $C_{2}=\left\\{\\{1,2,3,11\\},\\{4,5,6,11\\},\\{7,8,9,11\\}\right\\}$
$J_{\\{2,4\\}}(10,5)$ | GM |  $C_{1}=\left\\{\\{1,2,3,4,5\\},\\{1,2,3,6,7\\},\\{1,2,4,6,8\\},\\{1,2,5,7,8\\}\right\\}$ $C_{2}=\left\\{\\{6,7,8,9,10\\},\\{4,5,8,9,10\\},\\{3,5,7,9,10\\},\\{3,4,6,9,10\\}\right\\}$
$J_{\\{2,4\\}}(12,6)$ | GM |  $C_{1}=\left\\{\\{1,2,3,4,5,6\\},\\{1,2,3,4,7,8\\},\\{1,2,3,5,7,9\\},\\{1,2,3,6,8,9\\}\right\\}$ $C_{2}=\left\\{\\{7,8,9,10,11,12\\},\\{5,6,9,10,11,12\\},\\{4,6,8,10,11,12\\},\\{4,5,7,10,11,12\\}\right\\}$

It would be nice to see if any of these switching sets can be extended to an
infinite family of generalized Johnson graphs that are NDS. We believe that
$J_{\\{2,4,\dots\\}}(2k,k)$ is NDS if $k\geq 4$, following a similar argument
as in [6], but with two switching sets instead of one.

More generally, it remains an open problem to determine the cospectrality of
other graphs in the Johnson and Grassmann schemes (entries “?” in the tables
in the Appendix). The smallest open case is $K(9,3)$, see also [2, Section 5].

### Acknowledgements

Aida Abiad is partially supported by the Dutch Research Council through the
grant VI.Vidi.213.085 and by the Research Foundation Flanders through the
grant 1285921N. Jozefien D’haeseleer is supported by the Research Foundation
Flanders through the grant 1218522N.

## References

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  * [2] S.M. Cioabă, W.H. Haemers, T. Johnston and M. McGinnis, Cospectral mates for the union of some classes in the Johnson association scheme, Linear Algebra and its Applications 539 (2018), 219–228.
  * [3] E.R. van Dam, W.H. Haemers, J.H. Koolen and E. Spence, Characterizing distance-regularity of graphs by the spectrum, Journal of Combinatorial Theory, Series A 113 (2006), 1805–1820.
  * [4] E.R. van Dam and J.H. Koolen, A new family of distance-regular graphs with unbounded diameter, Inventiones Mathematicae 162 (2005), 189–193.
  * [5] C.D. Godsil and B.D. McKay, Constructing cospectral graphs, Aequationes Mathematicae 25 (1982), 257–268.
  * [6] W.H. Haemers and F. Ramezani, Graphs cospectral with Kneser graphs, Graphs and Combinatorics 531 (2010), 159–164.
  * [7] A.J. Hoffman, On the uniqueness of the triangular association scheme, The Annals of Mathematical Statistics 31 (1960), 492–497.
  * [8] T. Huang and C. Liu, Spectral characterization of some generalized Odd graphs, Graphs and Combinatorics 15 (1999), 195–209.
  * [9] F. Ihringer and A. Munemasa, New strongly regular graphs from finite geometries via switching, Linear Algebra and its Applications 580 (2019), 464–474.
  * [10] L. Qiu, Y. Ji, W. Wang, On a theorem of Godsil and McKay concerning the construction of cospectral graphs, _Linear Algebra and its Applications_ 603 (2020), 265–274.
  * [11] W. Wang, L. Qiu and Y. Hu, Cospectral graphs, GM-switching and regular rational orthogonal matrices of level $p$, Linear Algebra and its Applications 563 (2019), 154–177.

## Appendix: overview of results

This appendix provides a structured overview of the currently known
cospectrality results for some small generalized Johnson and Grassmann graphs.
New results are indicated with an asterisk (* or **).

Legend: | 0Trivial | 1Hoffman/Chang (1959)
---|---|---
2Huang, Liu (1999) | 3Van Dam et al. (2006) | 4Haemers, Ramezani (2010)
5Cioabă et al. (2018) | *Theorem 13 | **Sporadic (Section 7)

${J_{S}(n,2)}$ | $S$ | $|V|$
---|---|---
$\\{0\\}$
5 $n$ | 4 | DS0 | 6
5 | DS1 | 10
6 | DS1 | 15
7 | DS1 | 21
8 | NDS1 | 28
9 | DS1 | 36
Table 2: Cospectrality of small generalized Johnson graphs with $k=2$. ${J_{S}(n,3)}$ | $S$ | $|V|$
---|---|---
$\\{0\\}$ | $\\{1\\}$ | $\\{2\\}$
5 $n$ | 6 | DS0 | NDS4 | NDS3 | 20
7 | DS2 | NDS4 | NDS3 | 35
8 | NDS4 | NDS4 | NDS3 | 56
9 | ? | NDS4 | NDS3 | 84
10 | ? | NDS4 | NDS3 | 120
11 | ? | NDS4 | NDS3 | 165
Table 3: Cospectrality of small generalized Johnson graphs with $k=3$. ${J_{S}(n,4)}$ | $S$ | $|V|$
---|---|---
$\\{0\\}$ | $\\{1\\}$ | $\\{2\\}$ | $\\{3\\}$ | $\\{0,1\\}$ | $\\{0,2\\}$ | $\\{0,3\\}$
5 $n$ | 8 | DS0 | ? | NDS* | NDS3 | ? | NDS4 | ? | 70
9 | DS2 | ? | NDS* | NDS3 | NDS5 | NDS4 | ? | 126
10 | ? | ? | NDS* | NDS3 | ? | NDS4 | ? | 210
11 | NDS4 | NDS** | NDS* | NDS3 | ? | NDS4 | ? | 330
12 | ? | ? | NDS* | NDS3 | ? | NDS4 | ? | 495
13 | ? | ? | NDS* | NDS3 | ? | NDS4 | ? | 715
Table 4: Cospectrality of small generalized Johnson graphs with $k=4$.

Legend: | 0Trivial | 2Huang, Liu (1999) | 2Van Dam, Koolen (2005) | 3Van Dam et al. (2006)
---|---|---|---|---
4Haemers, Ramezani (2010) | 5Cioabă et al. (2018) | 6Ihringer, Munemasa (2019) | *Theorem 14 | **Sporadic (Section 7)

${J_{S}(n,5)}$ | $S$ | $|V|$
---|---|---
$\\{0\\}$ | $\\{1\\}$ | $\\{2\\}$ | $\\{3\\}$ | $\\{4\\}$ | $\\{0,1\\}$ | $\\{0,2\\}$ | $\\{0,3\\}$ | $\\{0,4\\}$ | $\\{1,2\\}$ | $\\{1,3\\}$ | $\\{1,4\\}$ | $\\{2,3\\}$ | $\\{2,4\\}$ | $\\{3,4\\}$
5 $n$ | 10 | DS0 | ? | ? | ? | NDS3 | ? | ? | ? | ? | NDS* | NDS4 | ? | ? | NDS** | NDS5 | 252
11 | DS1 | ? | ? | ? | NDS3 | ? | ? | ? | ? | ? | NDS4 | ? | ? | ? | NDS5 | 462
12 | ? | ? | ? | ? | NDS3 | NDS5 | ? | ? | ? | ? | NDS4 | ? | ? | ? | NDS5 | 792
13 | ? | ? | ? | ? | NDS3 | ? | ? | ? | ? | ? | NDS4 | ? | ? | ? | NDS5 | 1287
14 | NDS4 | ? | ? | ? | NDS3 | ? | ? | ? | ? | ? | NDS4 | ? | ? | ? | NDS5 | 2002
15 | ? | ? | ? | ? | NDS3 | ? | ? | ? | ? | ? | NDS4 | ? | ? | ? | NDS5 | 3003
Table 5: Cospectrality of small generalized Johnson graphs with $k=5$. ${J_{q,S}(n,2)}$ | $q=2$ | $q=3$ | $q=4$
---|---|---|---
$S$ | $|V|$ | $S$ | $|V|$ | $S$ | $|V|$
$\\{0\\}$ | $\\{0\\}$ | $\\{0\\}$
5 $n$ | 4 | NDS6 | 35 | NDS6 | 130 | NDS6 | 357
5 | NDS6 | 155 | NDS2 | 1210 | NDS2 | 5797
6 | NDS6 | 651 | NDS6 | 11 011 | NDS6 | 93 093
7 | NDS6 | 2667 | NDS6 | 99 463 | NDS6 | $\approx 10^{6}$
8 | NDS6 | 10 795 | NDS6 | $\approx 10^{6}$ | NDS6 | $\approx 10^{7}$
9 | NDS6 | 43 435 | NDS6 | $\approx 10^{7}$ | NDS6 | $\approx 10^{8}$
Table 6: Cospectrality of small generalized Grassmann graphs with $k=2$ and
$q=2,3,4$.

Legend: | 1Van Dam et al. (2006) | *Theorem 15
---|---|---

${J_{q,S}(n,3)}$ | $q=2$ | $q=3$ | $q=4$
---|---|---|---
$S$ | $|V|$ | $S$ | $|V|$ | $S$ | $|V|$
$\\{0\\}$ | $\\{1\\}$ | $\\{2\\}$ | $\\{0\\}$ | $\\{1\\}$ | $\\{2\\}$ | $\\{0\\}$ | $\\{1\\}$ | $\\{2\\}$
5 $n$ | 6 | NDS* | ? | NDS1 | 1395 | ? | ? | NDS1 | 33 880 | ? | ? | NDS1 | 376 805
7 | NDS* | ? | NDS1 | 11 811 | ? | ? | NDS1 | $\approx 10^{6}$ | ? | ? | NDS1 | $\approx 10^{7}$
8 | NDS* | ? | NDS1 | 97 155 | ? | ? | NDS1 | $\approx 10^{7}$ | ? | ? | NDS1 | $\approx 10^{9}$
9 | NDS* | ? | NDS1 | $\approx 10^{6}$ | ? | ? | NDS1 | $\approx 10^{9}$ | ? | ? | NDS1 | $\approx 10^{11}$
10 | NDS* | ? | NDS1 | $\approx 10^{7}$ | ? | ? | NDS1 | $\approx 10^{10}$ | ? | ? | NDS1 | $\approx 10^{13}$
11 | NDS* | ? | NDS1 | $\approx 10^{8}$ | ? | ? | NDS1 | $\approx 10^{11}$ | ? | ? | NDS1 | $\approx 10^{14}$
Table 7: Cospectrality of small generalized Grassmann graphs with $k=3$ and
$q=2,3,4$.

${J_{q,S}(n,4)}$ | $q=2$ | $q=3$ | $q=4$
---|---|---|---
$S$ | $|V|$ | $S$ | $|V|$ | $S$ | $|V|$
$\\{0\\}$ | $\\{1\\}$ | $\\{2\\}$ | $\\{3\\}$ | $\\{0,1\\}$ | $\\{0,2\\}$ | $\\{0,3\\}$ | $\\{0\\}$ | $\\{1\\}$ | $\\{2\\}$ | $\\{3\\}$ | $\\{0,1\\}$ | $\\{0,2\\}$ | $\\{0,3\\}$ | $\\{0\\}$ | $\\{1\\}$ | $\\{2\\}$ | $\\{3\\}$ | $\\{0,1\\}$ | $\\{0,2\\}$ | $\\{0,3\\}$
5 $n$ | 8 | NDS* | ? | ? | NDS1 | ? | ? | ? | 200 787 | ? | ? | ? | NDS1 | ? | ? | ? | $\approx 10^{8}$ | ? | ? | ? | NDS1 | ? | ? | ? | $\approx 10^{10}$
9 | NDS* | ? | ? | NDS1 | ? | ? | ? | $\approx 10^{6}$ | ? | ? | ? | NDS1 | ? | ? | ? | $\approx 10^{10}$ | ? | ? | ? | NDS1 | ? | ? | ? | $\approx 10^{12}$
10 | NDS* | ? | ? | NDS1 | ? | ? | ? | $\approx 10^{8}$ | ? | ? | ? | NDS1 | ? | ? | ? | $\approx 10^{12}$ | ? | ? | ? | NDS1 | ? | ? | ? | $\approx 10^{14}$
11 | NDS* | ? | ? | NDS1 | ? | ? | ? | $\approx 10^{9}$ | ? | ? | ? | NDS1 | ? | ? | ? | $\approx 10^{13}$ | ? | ? | ? | NDS1 | ? | ? | ? | $\approx 10^{17}$
12 | NDS* | ? | ? | NDS1 | ? | ? | ? | $\approx 10^{10}$ | ? | ? | ? | NDS1 | ? | ? | ? | $\approx 10^{15}$ | ? | ? | ? | NDS1 | ? | ? | ? | $\approx 10^{19}$
13 | NDS* | ? | ? | NDS1 | ? | ? | ? | $\approx 10^{11}$ | ? | ? | ? | NDS1 | ? | ? | ? | $\approx 10^{17}$ | ? | ? | ? | NDS1 | ? | ? | ? | $\approx 10^{22}$

Table 8: Cospectrality of small generalized Grassmann graphs with $k=4$ and
$q=2,3,4$.


††thanks<EMAIL_ADDRESS>
# Granular Segregation in Tapered Rotating Drums

S. González  L. Orefice  Research Center Pharmaceutical Engineering GmbH,
Inffeldg. 13, Graz, Austria

###### Abstract

We study the granular segregation in a tapered rotating drum by means of
simulations. In this geometry, both radial and axial segregation appear, the
strength of which depends on the filling fraction. We study the effect of the
drum’s tapering angle on the segregation speed and show that, coherently with
several recent studies, the axial segregation is due to the combined effect of
the radial segregation and the shape of the drum. We show that the axial
segregation behaves in a way analogous to the one found in chute flows, and
corresponds to previous theories for shallow gravity driven surface flows with
diffusion. By means of this analogy, we show that tapered drums could be the
simplest experimental set-up to obtain the free parameters of the theory.

###### pacs:

45.70.Mg

It has been hinted that axial segregation in rotating drums is mainly the
result of the shape of the free-surface, the thin layer of flowing particles
in the drum Arntz et al. (2012); Pohlman and Paprocki Jr. (2012); Zaman et al.
(2013); Chen et al. (2011). The free-surface is not flat, but slightly curved
along the axial direction, with peaks near the parallel walls of the drum. Due
to the friction with them, the particles on the sides are lifted more than the
particles in the centre, thus creating a convective flow in the drum that
induces the axial segregation. How to harvest this relation between flow and
the segregation pattern remains an open question: a question whose answer
could have far-reaching practical implications.

Motivated by the results of Kawaguchi et al. (2006); Yada et al. (2010) –
where axial segregation of binary mixtures in a tapered rotating drum is
reported albeit not explained – and how they relate to our own results
Gonzalez et al. (2014) – where the shape of the rotating drum is used to speed
up and control the axial segregation – we study the segregation pattern in
tapered rotating drums by means of simulations, and see to what extent it is
possible to control it by modifying the drum’s shape.

It has recently been shown that, contrary to the common belief that 3D
rotating drums are the sum of independent quasi-2D slices interacting only via
diffusion, the shape of the drum does influence the surface flow creating a
slow axial drift Zaman et al. (2013). This drift is caused by the axial slope
of the drum’s walls, which determines the 3D structure of the flow. If a bi-
disperse mixture is used, the particles will follow the flow structure and
will segregate accordingly.

In the particular case of a tapered drum, when mixed particles enter the
flowing layer in the upper part of the avalanche, gravity drives them towards
the small side of the drum before returning them to the large one. Small
particles can percolate trough the voids created by the big ones leaving the
flowing layer before it changes its direction. This leads to a net transport
of small particles to the small side of the drum and of large particles to the
big side: axial segregation occurs. Since the shape of the flowing layer
depends on the shape of the drum, one would also expect it to affect the
segregation. In this paper we show that this is indeed the case and we study
how it happens.

The coupling of axial segregation with the velocity profile has been already
reported for the classical rotating drum Chen et al. (2011). In the tapered
rotating drum the same coupling is in play but with a fundamental difference:
the shape of the drum increases the curvature of the flow lines on the flowing
layer thus transporting more small particles than in a parallel drum, and in
contrast to the latter, in a definite direction.

The structure of the paper is as follow. First, the drum geometry and and
simulation details are explained. Second, we compare our results for half-
filled drums with experiments to validate our model. Third, to minimize the
effect of radial segregation, a lower filling fraction is used and total
segregation is observed. The effect of the drum’s tapering angle on the speed
of the segregation is also studied. Finally, comparisons will be drawn with
simulations of mixtures in 2 and 3D chutes and with a theory for shallow
gravity driven surface flows with diffusion.

Figure 1: (Color online) Snapshots for half-filled system at different
positions along the rotation axis (from left to right) and different times
(from top to bottom, the number of revolutions indicated in the left). Small
particles dark, large particles light. One can see that first the radial
segregation appears, and this slowly and continuously transforms into axial
segregation, that here is visible in the dominance of small particle in the
small side of the drum.

##  I Simulations

Our simulations are in a slightly different geometry than that of Kawaguchi et
al. (2006); Yada et al. (2010) (shorter drum) to speed up the simulation time.
We use an equal volume mixture of 5 and 10mm radii particles in a tapered drum
150mm length, with caps of radii $r=50$ mm (at $z=0$mm) and variable R (at
$z=150$mm), in the left and right side, respectively. Their relation define
the angle of the drum, such that $R=r(1+\tan(\alpha))$. In the simulations,
particles’ contact forces are represented using a standard spring dash-pot
model Cundall and Strack (1979) as implemented in Thornton et al. (2012a). The
angular velocity is set to $\pi\ s^{-1}$, i.e. 30rpm, which implies an average
Froude number of $Fr=\omega^{2}(r+R)/2g\sim 0.1$. The system is either at 50
or 25$\%$ filling fraction. As initial condition the particles are arranged in
a square lattice, alternating one big particle and eight small to obtain an
initial mixed state. Snapshots of the system for different times and axial
positions are shown in Fig. 1 (cf. Fig. 5 in Ref. Yada et al. (2010) for an
experimental comparison).

The simulations were performed using Mercury-DPM Thornton et al. (2012a) with
the same material properties used in our previous research Gonzalez et al.
(2014), since they proved to reproduce the experimental results.

###  I.1 Shape of the Free Surface

In this section we show the three-dimensional structure of the flow. We focus
on three different characteristics: the level of the surface along the axis of
rotation (Fig. 2), the velocity field on the free surface (Fig. 3) and the
density profile, as seen by a vertical cut in the centre of the drum (Fig. 4).
Together, they provide a rather complete way to visualise the structure. Data
are averaged for 30 consecutive snapshots (one every 0.8s) and coarse-grained
to obtain a continuous function following the procedure of Ref. Weinhart et
al. (2012). The weight function is a Lucy polynomial Lucy (1977) with a width
of $2d$. Unless otherwise noted, every coarse-graining presented here is done
with the same parameters. We measured at different times during the evolution
of the system, and the results do not significantly change, so we conclude
that what we observe depends only on the geometry and not in the degree of
segregation.

The free surface along the axial direction shows level differences. This
phenomenon, to the best of our knowledge, was first described by Zik et al.
for a drum modulated along the axial direction with a sinusoidal function Zik
et al. (1994). This can be seen in our simulations from Fig. 1, or looking at
the iso-lines of the density at different axial positions, in Fig. 2. This
level difference along the axis makes particles to move along the gradient of
the surface, that is as particles go down the avalanche, they move forwards
and backwards in the axis perpendicular to the page.

Particles enter the flowing layer with a small velocity and they are mainly
directed towards the small side of the drum. This is due to the tapered
geometry: when entering the flowing layer, the particles in the large side of
the drum are higher than those in the small side of the drum, thus creating a
flow towards the small side. This is inverted when particles reach the lower
part of the avalanche: particles in the large side of the drum are lower than
those in the small side. Increasing the angle of the drum, increases the
curvature and the magnitude of this flow. This can be seen in Fig. 3 for three
different angles. The surface is defined as the plane whose projection can be
seen in Fig. 2 (solid line).

When a mixture of particles enters the avalanche in the upper part, they will
segregate as they travel downslope. If the drum is large enough, some small
particles will reach the core before the flow changes its direction, creating
a net transport of small particles. (Large enough here means in relation to
the segregation speed of the mixture. For size-driven segregation, the smaller
the size ratio of the particles, the slower segregation is, and thus the drums
needs to be larger.) The large particles, since they tend to stay on the
surface, can flow back to the large side of the drum, following the flow
lines. In this way, axial segregation occurs.

Figure 2: Contour lines for a density $\rho^{*}=\rho_{particles}/10$ at three
different axial positions for the same systems as in Fig. 1. The level
difference can be seen mostly in the upper part of the avalanche. One should
think of the curves as being displaced in the direction normal to the page.
Compare the qualitative agreement with the model of Ref. Zik et al. (1994).
Note that since the flow velocity increases along $z$, the blue curve shows a
slightly more S-shaped profile. Solid line shows the plane where the velocity
is measured for Fig. 3 and dashed vertical line the plane where the density is
measure for Fig. 4. Figure 3: Average velocity field of the particles’ free
surface for the tapered drum (from left to right, tapering angle
$\alpha=10^{\circ},15^{\circ},20^{\circ}$), the colour representing the
particles’ mean velocity in $m/s$. The length of the component of the vector
field parallel to the axis of rotation is increased to stress the effect of
the tapered geometry on the flow. As expected, the convective flows increases
its amplitude and reach with the angle of tapering.

Finally, the packing fraction of the system gives us an insight on the inner
structure of the flow, and shows the dependence of the free surface on the
drum’s angle, Fig. 4. Despite the presence of segregation, the density is
uniform during the evolution of the system, with the densest region at the
bottom of the drum. The density is mostly homogeneous along the axial
direction with increased spatial fluctuations as $\alpha$ increases.

The density remains constants during the segregation process, and is maximum
in the lower region of the drum for all the angles studied. We confirmed that
the density does not vary by measuring it at different times during the
segregation process.

Figure 4: (Color online) Packing fraction on the middle plane of the drum for
increasing drum angles $\alpha=10^{\circ},15^{\circ},20^{\circ}$ , from left
to right. The increasing slope of the free surface can be seen from left to
right, as $\alpha$ increases.

###  I.2 Radial and Axial Segregation

To corroborate that the proposed mechanism is responsible of the axial
segregation, we measure the radial and axial segregation in the drum as a
function of time. In order to have order parameters for both, the radial
segregation is defined as follows:

$q_{\rm radial}=N_{s<z^{*}}/N_{Ts}+N_{L>z^{*}}/N_{TL}-1~{},$ (1)

where $N_{s<(L>)r^{*}}$ is the number of small (large) particles whose radial
coordinate is smaller (larger) than the half volume radius $r^{*}$ (note that
this radius depend on the axial coordinate), $N_{Ts(L)}$ being the total
number of small (large) particles. That is, we count the fraction of particles
that are in the expected side of the drum. When $q_{\rm radial}=1$ all the
large particles are in the outside and the small in the core of the drum. It
must be noted that, due to the flowing layer, the radial segregation can never
reach the maximum value. To measure the axial segregation we use a quantity
that also works with banding (which appeared in one of the systems studied)
Taberlet et al. (2004):

$q_{axial}(t)=\frac{1}{l}\int_{0}^{l}\frac{|c_{s}(z,t)-\overline{c}_{s}(z)|}{\overline{c}_{s}(z)}dz~{},$
(2)

where $c_{s}(z,t)$ is the number of small particles along the axis, and
$\overline{c}_{s}(z)$ the number of small beads that correspond to a slice of
the drum at position $z$ for the totally mixed state. It must be noted that
since we are simulating a tapered rotating drum, the mean number of small
particles changes with $z$ since the cross section of the drum increases
accordingly.

Figure 5: Axial (solid-lines) and radial (dashed-lines) segregation as a
function of time for the tapered rotating drum. From light gray to black in
increasing angle of tapering. As it can be seen, there is no correlation
between the angle and the radial segregation. However, axial segregation shows
a clear correlation with $\alpha$.

Figure 5 shows the temporal evolution for each kind of segregation for three
different angles. As expected, the radial segregation sets in quite fast,
essentially after four turns, but it saturates soon to a roughly constant
value. On the other hand, the axial segregation is slower but steadily
increases beyond the point of saturation for the radial segregation.
Eventually (around the 10th revolution) the axial segregation causes the value
of $q_{radial}$ to decrease slightly, since in the small side of the drum the
number of large particles decreases greatly.

If $\alpha$ is increased the segregation speed should increase since the level
difference on the free surface is also increasing Zik et al. (1994). This is
indeed the case as Kawaguchi et al. (2006) shows experimentally (see in
particular Fig. 8). The same principle can be seen in other configurations
such as the spherical drum Naji and Stannarius (2009), or the double cone
configuration Alexander et al. (2001).

To better understand the role of geometry on the segregation process, several
simulations were run for different $\alpha$. It is expected that the radial
segregation will not depend on the geometry of system. However, for the axial
segregation, since it depends on the velocity profile and on the geometry of
the flowing layer, one expects to find a strong dependence on the angle of the
drum since this determines the level difference in the flow. Indeed this is
the case as Fig. 5 shows.

###  I.3 Less than Half Filled Drum

Our previous result shed light on the relation between $\alpha$ and the axial
segregation, but this is strong influenced by the radial segregation in half
filled drum. So large in fact, that at this filling fraction some systems
segregate the other way around, that is, large particles in the small side of
the drum, as reported in Kawaguchi et al. (2006) and confirmed by some of our
simulations (data not shown). We ran simulations at $25\%$ filling fraction,
for different $\alpha$, and we observe indeed that the axial segregation is
bolder. What is more, the radial segregation disappears almost completely
since the mixing at this filling fraction is maximum Metcalfe et al. (1995).
In this configuration, a pure phase of small particles appears in the small
side of the drum and a steady state for the segregation is reached, while
increasing the angle of the drum increases the segregation speed (see Fig. 6).

This behaviour is strikingly similar to that of bi-disperse chute flows,
either in 2D Staron and Phillips (2014) or 3D Thornton et al. (2012b).
Numerically, those are simulated with a box with periodic boundary conditions
in the flowing plane, and a rough inclined bottom. Depending on the
inclination of the bottom with respect to gravity, a steady flow can be
obtained. This is a straightforward way to study segregation for a fixed
velocity profile, however their experimental realisation is rather complex.

Figure 6 shows the displacement of the centre of mass of the large particles
normalized by its asymptotic value, for three different drum angles

$\Delta z=\frac{z_{cm}(\infty)-z_{cm}(t)}{z_{cm}(\infty)-z_{cm}(0)}~{},$ (3)

where $z_{cm}(t)$ is the centre of mass position on the axial direction for
the large particles. As $\alpha$ increases, the system segregates faster since
the shear rate increases with it. One can see that the drum’s angle influences
the segregation in the same way as the size ratio between the particles
Thornton et al. (2012b); Marks et al. (2012) or the volume fraction between
the two species Staron and Phillips (2014).

Figure 6: Displacement of the centre of mass of large particles for systems
with different $\alpha$, normalized by the asymptotic value $\Delta z$ in a
log-linear scale. Straight lines correspond to the exponential fit. Compare
with Ref. Staron and Phillips (2014). If plotted in a linear scale, one
obtains results as in Ref. Thornton et al. (2012b).

This is valid for a wide range of drum angles. We define the segregation speed
as the inverse of the exponent of the exponential fit $\Delta
z=\exp(-t/\tau)$. In general, the segregation speed is a monotonic function of
the angle, as one can see in Fig. 7, the line being a linear eye-guide. A
simple estimation tells us that the distance a particle travels on the free
surface is $L_{s}\propto\tan(\alpha)$. If the particles travel twice as much
during one avalanche they will segregate twice as fast, and this would explain
the linear behaviour of the segregation speed with the angle.

Since we have this qualitative agreement between two completely different
systems, the obvious next step is to see how well the theoretical predictions
developed for the chute flow work on the tapered drum.

Figure 7: Normalised segregation speed measured as the inverse of the typical
time, $\tau_{0}/\tau$ obtained from the centre of mass displacement on Fig. 6
for different tapering angles ($\tau_{0}$ is the typical time for
$\alpha=2^{\circ}$). The solid line is a linear eye guide.

###  I.4 Comparison with theory

Let us look at the phenomenon from another point of view. Since the free
surface is at an angle with respect to the horizontal (going up towards the
large side of the drum, see Fig. 2) the projection of gravity on the free
surface is non-zero. This translates into an effective gravity towards the
small side of the drum. From the theory for shallow granular avalanches Gray
and Thornton (2005); Gray and Ancey (2011) we know the time evolution for the
concentration of each species, and we should be able to apply this result to
the tapered drum.

Whereas the chute flow develops vertical segregation as it moves downslope,
the rotating drum develops axial segregation as time passes. This is due to
the structure of the flow as seen in Fig. 3. The tapered drum turns the
kinetic sieving along the flow lines into axial segregation at the system’s
level. According to this simple analogy, small particles should go to the
small side of the drum, while large particle will go towards the large side.
Furthermore, since the free-surface is a geometric characteristic of the flow
and does not depend on the degree of segregation of the mixture, the tapered
drum will segregate the particles into stable configurations in the same
manner that a chute flow does.

The temporal evolution of the concentration is something particularly
difficult to measure experimentally and, to the best of our knowledge, there
is only one realisation Wiederseiner et al. (2011). Our results show a
strikingly good fit with the theory, Fig. 8. For the sake of space, we refer
the reader to the original paper for the derivation of the solution Gray and
Chugunov (2006) and to the appendix for the explicit formula we use. Note that
for a tapered drum the half volume height is at $z_{50}=L/\sqrt{(}2)$, the
spatial coordinate of the theoretical solution has been scaled to account for
this.

The results are remarkably similar concerning both the evolution of the
concentration and the spatial distribution. However, we cannot fit both the
temporal evolution and the steady state solution. We chose to fit the Péclet
number $Pe$ (a dimensionless number that quantifies how large the segregation
is compared to diffusive remixing) to reproduce the evolution of the system
rather than the steady state. Does this mean that the $Pe$ in the tapered drum
changes with time? Answering this question goes beyond the scope of this
paper.

It must be noted that we are comparing the results of only one realisation,
and our data points are not averaged on time, which would decrease the noise
and give a better fit, but we prefer to show the raw results to highlight the
quality of the data that is possible to obtain with this set-up. We confirmed
that the same qualitative behaviour holds for different angles and drum
lengths (data not shown).

Figure 8: Left panel, concentration of small particles along the normalized
rotation axis as a function of time for our numerical experiments with
$\alpha=10^{\circ}$ and $L=200$mm. Right panel, theoretical solution using the
results of Ref. Gray and Chugunov (2006) with $Pe=14$.

Finally, it must be noted that Gray & Chugonov’s theory can be used to explain
the centre of mass movement reported in Staron and Phillips (2014); Thornton
et al. (2012b), together with our results. Consider the solution for the
concentration of small particles in the chute, $\phi(x,t)$. To obtain the
centre of mass of the small particles one integrates the concentration times
the position, $z_{cm}(t)=\int_{0}^{1}z~{}\phi(z,t)~{}dz~{},$ which can be
solved numerically in a very straightforward way. It is easy to verify that
the predicted solution shows an exponential decay for small values of $Pe\sim
4$, but seems to show two different slopes for larger values of it . In this
way, $Pe$ can be measured not only for the steady state solution, as done in
Thornton et al. (2012b) but from the evolution of the system. The results
presented here indicate that both may be different.

A detailed study of $Pe$ for different system parameters is beyond the scope
of this paper. Our objective being to show how well the theory describe the
evolution of the system and how the tapered drum could become a trivial
experimental set-up where to measure the Péclet number. A detailed study of
this should be hopefully a subject to be addressed both experimentally and
theoretically in the future.

##  II Conclusions

We have studied the size-driven segregation of bi-disperse mixtures in tapered
rotating drums. The relation between velocity field, free surface, and the
segregation pattern was studied. The results confirm that level differences on
the free surface create axial flows, which in turn cause axial segregation.
For the first time we report total axial segregation in this particular
geometry and we studied its dependence on different parameters; in particular,
we showed that the axial segregation speeds up with the angle of the drum.

We have shown that tapered drums can be used to determine the Péclet number
used in previous theories from a straightforward experimental set-up. This has
been done before but the experiments are very difficult to set up “with both
well controlled initial conditions and a steady uniform bulk flow field”
Wiederseiner et al. (2011). And indeed, to the best of our knowledge, the
results from Wiederseiner et al. are the only ones that give the Péclet number
experimentally. On the contrary, to obtain well controlled results on a
tapered drum is extremely easy, for example, using the same set-up that
motivated this study Kawaguchi et al. (2006).

Besides these theoretical implications, the results presented here have direct
applications in the separation and sorting of granulates in areas ranging from
powder handling in the pharmaceutical industry to ore milling in the mining
industry and shed light on the important role of geometry in segregation
processes, not only for rotating drums but everywhere tridimensional flows
appear.

## ACKNOWLEDGMENTS

This work was supported by the IPROCOM Marie Curie initial training network,
funded through the People Programme (Marie Curie Actions) of the European
Union’s Seventh Framework Programme FP7/2007-2013/ under REA grant agreement
No. 316555. We would like to thank J. Kihnast, N. Rivas, W. den Otter, S.
Luding and A.R. Thornton for fruitful discussions, and the kind hospitality of
A. Michrafy at the Ecole de Mines, Albi, France, where part of work was
realised. The simulations performed for this paper were undertaken in Mercury-
DPM. It is primarily developed by T. Weinhart, A.R. Thornton and D. Krijgsman
at the University of Twente.

## Appendix I

The general solution for the concentration of small particles can be written
as

$\phi=\frac{1}{2}\left(1-\frac{2}{\omega}\frac{\partial\omega}{\partial\xi}\right),$

where $\omega$ is decomposed onto a steady and a time-varying part
$\omega=\omega_{s}+\omega_{\tau}$. Their analytical forms is given by

$\omega_{s}=\chi_{0}(\xi_{0})\frac{\sinh(\xi/2)}{\sinh(\xi_{0}/2)}-\frac{\sinh((\xi-\xi_{0})/2)}{\sinh(\xi_{0}/2)}~{},$

for the steady solution, while the temporal part is given by

$\omega_{\tau}=\sum_{n=1}^{\infty}A_{n}\exp\left(-(\frac{n^{2}\pi^{2}}{\xi_{0}^{2}}+\frac{1}{4})\tau\right)\sin(\frac{n\pi\xi}{\xi_{0}})~{}.$

where the transformed time and space variables are given by
$\tau=(S_{r}^{2}/D_{r})t$ and $\xi=(S_{r}/D_{r})z$ and $\xi_{0}=Pe$. For an
equal volume mixture, as in our case, $\chi_{0}(\xi_{0})=1$. Using Eq. (5.46)
from Gray and Chugunov (2006) for our mean concentration $\phi_{m}=0.5$, we
find that the coefficients in the Fourier sine series are simplified into

$A_{n}=\frac{2Pe^{2}}{n\pi(Pe^{2}+4n^{2}\pi^{2})}\left(1-(-1)^{n}\right).$

In our solution, we take only the first hundred terms of the series.

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