# Extreme and statistical properties of eigenvalue indices of simple connected
graphs

Soňa Pavlíková Alexander Dubček University of Trenčín, Slovakia Daniel
Ševčovič Comenius University in Bratislava, Slovakia Jozef Širáň Slovak
Technical University in Bratislava, Slovakia

###### Abstract

We analyze graphs attaining the extreme values of various spectral indices in
the class of all simple connected graphs, as well as in the class of graphs
which are not complete multipartite graphs. We also present results on density
of spectral gap indices and its nonpersistency with respect to small
perturbations of the underlying graph. We show that a small change in the set
set of edges may result in a significant change of the spectral index like,
e.g., the spectral gap or spectral index. We also present a statistical and
numerical analysis of spectral indices of graphs of the order $m\leq 10$. We
analyze the extreme values for spectral indices for graphs and their small
perturbations. Finally, we present the statistical and extreme properties of
graphs on $m\leq 10$ vertices.

Keywords: Graph spectrum; spectral index; extreme properties of eigenvalues;
distribution of eigenvalues; complete multipartite graphs;

2000 MSC: 05C50 05B20 05C22 15A09 15A18 15B36

##  1 Introduction

In theoretical chemistry, biology, or statistics, spectral indices and
properties of graphs representing the structure of chemical molecules or
transition diagrams for finite Markov chains play an important role (cf.
Cvetković [11, 12], Brouwer and Haemers [8] and references therein). In the
past decades, various graph energies and indices have been proposed and
analyzed. For example, the sum of absolute values of eigenvalues is called the
matching energy index (cf. Chen and Liu [29]), the maximum of the absolute
values of the least positive and largest negative eigenvalue is related to the
HOMO-LUMO index (see Mohar [34, 35], Li et al. [30], Jaklić et al. [26],
Fowler et al. [20]), their difference is related to the HOMO-LUMO separation
gap (cf. Gutman and Rouvray [22], Li et al. [30], Zhang and An [45], Fowler et
al. [19]).

The spectrum $\sigma(G_{A})\equiv\sigma(A)$ of a simple nonoriented connected
graph $G_{A}$ on $m$ vertices is given by the eigenvalues of its adjacency
matrix $A$:

$\lambda_{max}\equiv\lambda_{1}\geq\lambda_{2}\geq\dots\geq\lambda_{m}\equiv\lambda_{min}.$

For a simple graph (without loops and multiple edges) we have $A_{ii}=0$, and
so $\sum_{i=1}^{m}\lambda_{i}=trace(A)=0$. Hence
$\lambda_{1}>0,\lambda_{m}<0$.

In what follows, we shall denote $\lambda_{+}(A)$, and $\lambda_{-}(A)$ the
least positive and largest negative eigenvalues of a symmetric matrix $A$
having positive and negative real eigenvalues. Let us denote by
$\Lambda^{gap}(A)=\lambda_{+}(A)-\lambda_{-}(A)$ and
$\Lambda^{ind}(A)=\max(|\lambda_{+}(A)|,|\lambda_{-}(A)|)$ the spectral gap
and the spectral index of a symmetric matrix $A$. Furthermore, we define the
spectral power $\Lambda^{pow}(A)=\sum_{k=1}^{m}|\lambda_{k}|$. Clearly, all
three indices $\Lambda^{gap},\Lambda^{ind}$, and $\Lambda^{pow}$ depend on
positive $\sigma_{+}(A)=\\{\lambda\in\sigma(A),\lambda>0\\}$, and negative
$\sigma_{-}(A)=\\{\lambda\in\sigma(A),\lambda<0\\}$ parts of the spectrum of
the matrix $A$. In fact, $\lambda_{+}(A)=\min\sigma_{+}(A),\
\lambda_{-}(A)=\max\sigma_{-}(A)$, and
$\Lambda^{pow}(A)=\sum_{\lambda\in\sigma_{+}(A)}\lambda-\sum_{\lambda\in\sigma_{-}(A)}\lambda$.

In the past decades, various concepts of introducing inverses of graphs based
on inversion of the adjacency matrix have been proposed. In general, the
inverse of the adjacency matrix does not need to define a graph again because
it may contain negative elements (cf. [23]). Godsil [21] proposed a successful
approach to overcome this difficulty, which defined a graph to be (positively)
invertible if the inverse of its nonsingular adjacency matrix is diagonally
similar (cf. [44]) to a nonnegative integral matrix representing the adjacency
matrix of the inverse graph in which positive labels determine edge
multiplicities. In the papers [36, 37], Pavlíková and Ševčovič extended this
notion to a wider class of graphs by introducing the concept of negative
invertibility of a graph.

In chemical applications, the spectral gap $\Lambda^{gap}$ of a structural
graph of a molecule is related to the so-called HOMO-LUMO energy separation
gap of the energy of the highest occupied molecular orbital (HOMO) and the
lowest unoccupied molecular orbital (LUMO). Following Hückel’s molecular
orbital method [25], eigenvalues of a graph that describes an organic molecule
are related to the energies of molecular orbitals (see also Streitwieser [42,
Chapter 5.1]). Finally, according Aihara [1, 2], it is energetically
unfavorable to add electrons to a high-lying LUMO orbital. Hence, a larger
HOMO-LUMO gap implies a higher kinetic stability and low chemical reactivity
of a molecule. Furthermore, the HOMO-LUMO energy gap generally decreases with
the number of vertices in the structural graph (cf. [3]).

In this paper, we analyze the extreme and statistical properties of the
spectrum of all simple connected graphs. It includes the analysis of maximal
and minimal eigenvalues, as well as indices such as, e.g., spectral gap,
spectral index, and the power of spectrum. We analyze graphs that attain
extreme values of various indices in the class of all simple connected graphs,
as well as in the class of graphs that are not complete multipartite graphs.
We also present results on the density of spectral gap indices and its
nonpersistency with respect to small perturbations of the underlying graph. We
show that a small change in the set set of edges may result in a significant
change of the spectral gap or spectral index. We also present a statistical
and numerical analysis of indices of graphs of order $m\leq 10$.

The paper is organized as follows. In Section 2 we first recall the known
results on extreme values of maximal and minimal eigenvalues of adjacency
matrices. We also report the number of all simple connected graphs due to
McKay [33]. Next, we analyze the extreme values for indices for completed
multipartite graphs and their small perturbations. In Section 3 we focus our
attention on the statistical and extreme properties of graphs on $m\leq 10$
vertices.

##  2 Extreme properties of indices

Figure 1: The numbers $c_{m}$ of all simple connected as a function of number
of vertices (blue solid line), and its approximation by means of the
approximation formula (1) (red dashed line).

Denote by $c_{m}$ the number of simple non-isomorphic connected graphs on $m$
vertices. According to the McKay’s list of all simple connected graphs [33]
the numbers $c_{m},m\leq 10$, are summarized in Table 1.

Table 1: Numbers of all simple connected graphs on $m\leq 10$ vertices. $m$ | $2$ | $3$ | $4$ | $5$ | $6$ | $7$ | $8$ | $9$ | $10$
---|---|---|---|---|---|---|---|---|---
total # | $1$ | $2$ | $6$ | $21$ | $112$ | $853$ | $11117$ | $261080$ | $11716571$

Although there exists an approximation formula for the number of labelled
simple connected graphs of the given order $m$ and number of edges (cf.
Bender, Canfield, and McKay [6]) for small values of $m$ the number $c_{m}$
can be approximated by the following compact the quadratic exponential
function:

$c_{m}\approx\omega_{0}10^{\omega_{1}(m-9)+\omega_{2}(m-9)^{2}},\quad\text{where}\
\omega_{0}=261080,\ \ \omega_{1}=1.4,\ \ \omega_{2}=0.09.$ (1)

This formula is exact for $m=9$ and gives good approximation results for other
orders $m\leq 10$ (see Fig. 1).

Recall the following well-known facts: the maximal value of
$\lambda_{max}=\lambda_{1}$ over all simple connected graphs on the $m$
vertices is equal to $m-1$, and it is attained by the complete graph $K_{m}$.
The minimal value of $\lambda_{max}$ is equal to $2\cos(\pi/(m+1))$, and it is
attained for the path graph $P_{m}$. Furthermore, the lower bound for the
minimal eigenvalue $\lambda_{min}=\lambda_{m}\geq-\sqrt{\lfloor
m/2\rfloor\lceil m/2\rceil}$ was independently proved in [10, 24, 38]. The
lower bound is attained for the complete bipartite graph $K_{m_{1},m_{2}}$
where $m_{1}=\lceil m/2\rceil,m_{2}=\lfloor m/2\rfloor$. The maximum value of
$\lambda_{min}$ on all simple connected graphs on the $m$ vertices is equal to
$-1$, and it is attained for the complete graph $K_{m}$.

###  2.1  Indices for complete multipartite graphs and their perturbations

The aim of this section is to analyze indices and their extreme values for
simple connected graphs on the $m$ vertices.

######  Proposition 1.

Let us denote $K_{m_{1},\dots,m_{k}}$ the complete multipartite graph where
$1\leq m_{1}\leq\dots\leq m_{k}$ denote the sizes of parts,
$m_{1}+\dots+m_{k}=m$, and $k\geq 2$ is the number of parts. Then the spectrum
of the adjacency matrix $A$ of $K_{m_{1},\dots,m_{k}}$ satisfies
$\sigma(A)\subseteq[-m_{k},m-m/k]$. If $m_{i}<m_{i+1}$ then there exists a
single eigenvalue $\lambda\in(-m_{i+1},-m_{i})$. If
$m_{i}=m_{i+1}=\dots=m_{i+j}$ then $\lambda=-m_{i}$ is an eigenvalue of $A$
with multiplicity $j$.

Finally, $0<\lambda_{+}(A)\leq m-m/k$ and $-m/k\leq\lambda_{-}(A)<0$. As a
consequence, $\Lambda^{gap}(A)\leq m,\Lambda^{ind}(A)\leq m-1$,
$\Lambda^{pow}(A)\leq 2(m-m/k)$. The equalities for the indices
$\Lambda^{gap}(A),\Lambda^{ind}(A)$ are reached by the complete graph
$G_{A}=K_{m}$.

###### Proof.

The adjacency matrix $A$ of $K_{m_{1},\dots,m_{k}}$ has the block form:

$A={\bf 1}{\bf 1}^{T}-diag(D_{1},\dots,D_{k}),$

where ${\bf 1}=(1,\dots,1)^{T}\in\mathbb{R}^{m}$, and $D_{i}$ is the
$m_{i}\times m_{i}$ matrix consisting of ones. Now, if $\lambda$ is a nonzero
eigenvalue of $A$ with an eigenvector $x=(x_{1},\dots,x_{m})^{T}$ then

$\alpha-\alpha_{p}=\lambda x_{l},\quad\text{for each}\ \
l=\mu_{p-1}+1,\dots,\mu_{p},\
\quad\alpha_{p}=\sum_{l=1+\mu_{p-1}}^{\mu_{p}}x_{l},\quad\mu_{p}=\sum_{r=1}^{p}m_{r},$
(2)

for $p=1,\dots,k$. Here $\alpha=\sum_{p=1}^{k}\alpha_{p}=\sum_{j=1}^{m}x_{j}$.
For example, if $p=1$ then $\sum_{j=1}^{m_{1}}x_{j}=\alpha
m_{1}/(\lambda+m_{1})$ provided that $\lambda\not=-m_{1}$. Similarly, we can
proceed with the remaining parts $m_{2},\dots,m_{k}$. In the case $\alpha=0$
we have $\lambda\in\\{-m_{1},\dots,-m_{k}\\}$. In the case $\alpha\not=0$ we
conclude $\lambda\not\in\\{-m_{1},\dots,-m_{k}\\}$, and the eigenvalue
$\lambda$ satisfies the rational equation:

$\psi(\lambda)=1,\quad\text{where}\
\psi(\lambda)=\sum_{i=1}^{k}\frac{m_{i}}{\lambda+m_{i}}.$ (3)

Conversely, if $\lambda\not\in\\{-m_{1},\dots,-m_{k}\\}$ satisfies
$\psi(\lambda)=1$ then it is easy to verify that the nontrivial vector
$x\in\mathbb{R}^{m}$,

$x=(\underbrace{y_{1},\dots,y_{1}}_{m_{1}\
\text{times}},\underbrace{y_{2},\dots,y_{2}}_{m_{2}\
\text{times}},\dots,\underbrace{y_{k},\dots,y_{k}}_{m_{k}\
\text{times}})^{T},\quad\text{where}\ y_{i}=\frac{m_{i}}{\lambda+m_{i}},$

is an eigenvector of $A$, i.e. $Ax=\lambda x$.

In what follows, we shall derive necessary bounds on eigenvalues of $A$.
Suppose to the contrary that $\lambda<-m_{k}$ is an eigenvalue of $A$. Then
$\lambda+m_{i}\leq\lambda+m_{k}<0$ for any $i=1,\dots,k$, and so
$\psi(\lambda)<0<1$. Therefore, $\lambda\geq-m_{k}$ for any eigenvalue
$\lambda\in\sigma(A)$. To derive an upper bound for the positive eigenvalue of
$A$ we introduce an auxiliary function
$\phi(\xi_{1},\dots,\xi_{k})=\sum_{i=1}^{k}\frac{\xi_{i}}{\lambda+\xi_{i}}$
where $\lambda>0$ is fixed. The function $\phi:\mathbb{R}^{k}\to\mathbb{R}$ is
concave. Using the Lagrange function
$\mathscr{L}(\xi,\mu)=\phi(\xi_{1},\dots,\xi_{k})-\mu\sum_{i=1}^{k}\xi_{i}$ it
is easy to verify that $\phi$ achieves the unique constrained maximum in the
set $\\{\xi\in\mathbb{R}^{k},\sum_{i=1}^{k}\xi_{i}=m\\}$ at the point
$\hat{\xi}=(m/k,\dots,m/k)^{T}$. Therefore, for any $\lambda>0$ we have

$\psi(\lambda)=\sum_{i=1}^{k}\frac{m_{i}}{\lambda+m_{i}}=\phi(m_{1},\dots,m_{k})\leq\phi(m/k,\dots,m/k)=\frac{m}{\lambda+m/k}.$

If $\lambda>0$ is a positive eigenvalue of $A$ then $\psi(\lambda)=1$ and so
$\lambda+m/k\leq m$, that is, $0<\lambda\leq m-m/k$. Therefore,
$\sigma(A)\subset[-m_{k},m-m/k]$.

In the trivial case of an equipartite graph $K_{m_{1},\dots,m_{k}}$ with
$m_{1}=\dots=m_{k}=m/k$ we obtain $\lambda_{-}(A)\geq-m_{k}=-m/k$ and
$\lambda_{+}(A)\leq m-m/k$. Thus, $\Lambda^{gap}\leq m$, and
$\Lambda^{ind}\leq m-m/k\leq m-1$. This estimate also follows from the results
of [17] and [15]. Therefore, for any $1\leq l<k$ we conclude that
$\Lambda^{gap}(A)=\lambda_{+}(A)-\lambda_{-}(A)\leq m-m/k-(-m/k)=m$.
Similarly, $\Lambda^{ind}(A)\leq m-1$.

Now, consider a non-equipartite graph $K_{m_{1},\dots,m_{k}}$ with
$m_{1}=\dots=m_{l}<m_{l+1}\leq\dots\leq m_{k}$ where $1\leq l<k$. Suppose that
$l=1$, that is, $1\leq m_{1}<m_{2}\leq\dots\leq m_{k}$. The function $\psi$ is
strictly decreasing in the interval $(-m_{2},-m_{1})$ with infinite limits
$\pm\infty$ when $\lambda\to-m_{2}$ and $\lambda\to-m_{1}$, respectively.
Therefore, there exists a unique eigenvalue $\lambda\in(-m_{2},-m_{1})$ of the
matrix $A$. We have $m_{1}+(k-1)m_{2}\leq\sum_{i=1}^{k}m_{i}=m$. Define
$\tilde{\lambda}=-m_{1}/k-m_{2}(k-1)/k$. Then $\tilde{\lambda}\geq-m/k$. In
what follows we shall prove that $\psi(\tilde{\lambda})\geq 1$. The function
$\xi\mapsto\xi/(\tilde{\lambda}+\xi)$ decreases for $\xi>-\tilde{\lambda}$.
Therefore

$\displaystyle\psi(\tilde{\lambda})$ $\displaystyle\geq$
$\displaystyle\frac{m_{1}}{\tilde{\lambda}+m_{1}}+(k-1)\frac{m_{2}}{\tilde{\lambda}+m_{2}}=-\frac{k}{k-1}\frac{m_{1}}{m_{2}-m_{1}}+k(k-1)\frac{m_{1}}{m_{2}-m_{1}}$
$\displaystyle=$
$\displaystyle\frac{k}{k-1}\frac{(k-1)^{2}m_{2}-m_{1}}{m_{2}-m_{1}}\geq\frac{k}{k-1}>1,$

because $k\geq 2$. Since $\psi$ is strictly decreasing in the interval
$(-m_{2},-m_{1})$ we have $-m/k\leq\tilde{\lambda}<\lambda$ because
$\psi(\lambda)=1$.

In the case $l\geq 2$ we can apply a simple perturbation argument. Indeed, let
us perturb the adjacency matrix $A$ by a small parameter $0<\varepsilon\ll 1$
as follows:

$A^{\varepsilon}={\bf 1}{\bf
1}^{T}-diag((1-\varepsilon)D_{1},D_{2},\dots,D_{l-1},(1+\varepsilon)D_{l},D_{l+1},\dots,D_{k}).$

It corresponds to the perturbation
$m_{1}^{\varepsilon}=(1-\varepsilon)m_{1},m_{l}^{\varepsilon}=(1+\varepsilon)m_{l}$.
All remaining $m_{i}$ remain unchanged for $i\not=1$ and $i\not=l$. Then for
the corresponding perturbed function $\psi^{\varepsilon}$ there exists a
solution $\lambda^{\varepsilon}\in(m_{1}-\varepsilon,m_{1})$ of the equation
$\psi^{\varepsilon}(\lambda^{\varepsilon})=1$. Since the spectrum of
$A^{\varepsilon}$ depends continuously on the parameter $\varepsilon\to 0$, we
see that $\lambda^{\varepsilon}\to\lambda=-m_{1}=\dots=-m_{l}$ is an
eigenvalue of the graph $G_{A}$ provided that $l\geq 2$. In this case
$\lambda=-m_{1}\geq-m/k$.

A complete multipartite graph $G_{A}=K_{m_{1},m_{2},\dots,m_{k}}$ has exactly
one positive eigenvalue $\lambda_{1}>0$ (cf. Smith [14]). Since
$\sum_{i=1}^{m}\lambda_{i}=0$ we have
$\Lambda^{pow}(A)=\sum_{i=1}^{m}|\lambda_{i}|=2\lambda_{1}\leq 2(m-m/k)$. The
spectrum of the complete graph $K_{m}$ consists of eigenvalues $m-1$, and $-1$
with multiplicity $m-1$. Therefore, $\Lambda^{gap}=m,\Lambda^{ind}=m-1$, as
claimed. ∎

######  Remark 1.

The main idea of the proof of Proposition 1 is a non-trivial generalization of
the interlacing theorem [17, Theorem 1] due to Esser and Harary. It is based
on a solution $\lambda$ to the dispersion equation (3), that is
$\psi(\lambda)=1$ (see [17, Eq. (9)]). In [17, Corollary 1] they showed that
$\sigma(A)\subseteq[-m_{k},m-m_{1}]$. Because
$km_{1}\leq\sum_{i=1}^{k}m_{i}=m$, we obtain $m-m/k\leq m-m_{1}$. Using the
concavity of the function $\phi:\mathbb{R}^{k}\to\mathbb{R}$ and the
constrained optimization argument, we were able to improve this estimate. We
derived the estimate $\sigma(A)\subseteq[-m_{k},m-m/k]$ which yields optimal
bounds $\Lambda^{gap}\leq m,\Lambda^{ind}\leq m-1$ derived in Proposition 1.
Furthermore, we introduced a novel analytic perturbation technique to handle
the case when the sizes $m_{1}=\dots=m_{l}$ of parts coincide.

######  Remark 2.

It follows from the proof of Proposition 1 that $\lambda$ is an eigenvalue of
$A$ if and only if the vector
$z=(\alpha_{1},\dots,\alpha_{k})^{T}\in\mathbb{R}^{k}$ (see (2)) is an
eigenvector of the $k\times k$ matrix $\mathscr{A}$, i.e.
$\mathscr{A}z=\lambda z$, where $\mathscr{A}_{ij}=m_{i}$ for $i\not=j$,
$\mathscr{A}_{ii}=0$.

As a consequence, the spectrum of the complete bipartite graph
$K_{m_{1},m_{2}}$ consists of $m_{1}+m_{2}-2$ zeros and
$\pm\sqrt{m_{1}m_{2}}$. Therefore,
$\Lambda^{gap}(K_{m_{1},m_{2}})=\Lambda^{pow}(K_{m_{1},m_{2}})=2\sqrt{m_{1}m_{2}}$,
and $\Lambda^{ind}(K_{m_{1},m_{2}})=\sqrt{m_{1}m_{2}}$. Furthermore, if $m$ is
even, then $\Lambda^{gap}(K_{m/2,m/2})=m=\Lambda^{gap}(K_{m})$, i.e., the
complete bipartite graph $K_{m/2,m/2}$ as well as the complete graph $K_{m}$
maximize the spectral gap $\Lambda^{gap}$. The smallest example is the
complete graph $K_{4}$ with eigenvalues $\\{3,-1,-1,-1\\}$ and the circle
$C_{4}\equiv K_{2,2}$ with eigenvalues $\\{2,0,0,-2\\}$ that yields the same
maximum value of $\Lambda^{gap}=4$.

Similarly, one can derive the equation for spectrum of the complete tripartite
graph $K_{m_{1},m_{2},m_{3}}$. It leads to the following depressed cubic
equation $\lambda^{3}+r\lambda+s=0$ with
$r=-(m_{1}m_{2}+m_{2}m_{3}+m_{1}m_{3}),s=-2m_{1}m_{2}m_{3}$. However, the
discriminant $\Delta=-(4r^{3}+27s^{2})$ is positive for a non-equipartite
graph, and there are three real roots of the depressed cubic. With regard to
Galois theory, roots cannot be expressed by an algebraic expression, and
Cardano’s formula leads to ”casus irreducibilis”.

######  Proposition 2.

Let us consider the class of all simple connected graphs on $m$ vertices. The
following statements regarding the indices $\Lambda^{gap},\Lambda^{ind}$ and
$\Lambda^{pow}$ hold.

  * a)

If $G_{A}$ is not a complete multipartite graph of order $m$, then
$\Lambda^{gap}(A)\leq m-1,\Lambda^{ind}(A)\leq m/2$ for $m$ even, and
$\Lambda^{gap}(A)\leq m-3/2,\Lambda^{ind}(A)\leq\sqrt{m^{2}-1}/2$ for $m$ odd.

  * b)

The maximum value of $\Lambda^{pow}$ on the $m\leq 7$ vertices is equal to
$2m-2$, and it is attained for the complete graph $K_{m}$. For $m=7$ there are
two maximizing graphs with $\Lambda^{pow}=12$ \- the complete graph $K_{7}$
and the noncomplete graph shown in Fig. 4. Starting $m\geq 8$ the maximal
$\Lambda^{pow}$ is attained by noncomplete graphs depicted in Fig. 5 for
$8\leq m\leq 10$.

###### Proof.

According to Smith [14], a simple connected graph has exactly one positive
eigenvalue (i.e. $\lambda_{2}(A)\leq 0$) if and only if it is a complete
multipartite graph $K_{m_{1},\dots,m_{k}}$ where $1\leq m_{1}\leq\dots\leq
m_{k}$ denotes the sizes of parts, $m_{1}+\dots+m_{k}=m$, and $k\geq 2$ is the
number of parts (see [14, Theorem 6.7]).

To prove a), let us consider a graph $G_{A}$ different from any complete
multipartite graph $K_{m_{1},\dots,m_{k}}$. Therefore, $\lambda_{2}(A)>0$. We
combine this information with the result due to D. Powers regarding the second
largest eigenvalue $\lambda_{2}(A)$. According to [38] (see also [39], [40]),
for a simple connected graph $G_{A}$ on $m$ vertices we have the following
estimate for the second largest eigenvalue $\lambda_{2}(A)$:

$-1\leq\lambda_{2}(A)\leq\lfloor m/2\rfloor-1$

(see also Cvetković and Simić [13]). Since $\lambda_{2}(A)>0$ we have
$0<\lambda_{+}(A)\leq\lambda_{2}(A)\leq\lfloor m/2\rfloor-1$, and
$-\sqrt{\lfloor m/2\rfloor\lceil
m/2\rceil}\leq\lambda_{min}(A)\leq\lambda_{-}(A)<0$. Hence the spectral gap
$\Lambda^{gap}=\lambda_{+}(A)-\lambda_{-}(A)\leq\sqrt{\lfloor m/2\rfloor\lceil
m/2\rceil}+\lfloor m/2\rfloor-1$. If $m$ is even, it leads to the estimate
$\Lambda^{gap}\leq m-1$. If $m$ is odd, then it is easy to verify
$\Lambda^{gap}\leq m-3/2$. Analogously, $\Lambda^{ind}\leq m/2$ if $m$ is
even, and $\Lambda^{ind}\leq\sqrt{m^{2}-1}/2$ if $m$ is odd.

The part b) is contained in Section 3 dealing with statistical properties of
eigenvalue indices. ∎

Recall that for the complete bipartite graph $K_{m,m}$ the spectrum consists
of zeros and $\pm m$. As a consequence
$\lim_{m\to\infty}\Lambda^{gap}(K_{m,m})=\infty$. The next result shows that a
small change in a large graph $K_{m,m}$ caused by the removal of a single edge
may result in a huge change in the spectral gap.

######  Proposition 3.

Let us denote by $K_{m,m}^{-e}$ the bipartite noncomplete graph constructed
from the complete bipartite graph $K_{m,m}$ by deleting exactly one edge. Then
its spectrum consists of $2m-4$ zeros and four real eigenvalues

$\lambda^{\pm,\pm}=\pm\left(1-m\pm\sqrt{m^{2}+2m-3}\right)/2.$ (4)

For the spectral gap we have
$\Lambda^{gap}(K_{m,m}^{-e})=1-m+\sqrt{m^{2}+2m-3}$, and

$2\sqrt{1-2/(m+1)}<\Lambda^{gap}(K_{m,m}^{-e})<2\sqrt{1-1/m}.$

As a consequence, $\lim_{m\to\infty}\Lambda^{gap}(K_{m,m}^{-e})=2$.

###### Proof.

Without loss of generality, we may assume that the adjacency matrix $A$ of the
graph $K_{m,m}^{-e}$ has the form

$A=\left(\begin{array}[]{cc}0&\mathbf{1}\mathbf{1}^{T}\\\
\mathbf{1}\mathbf{1}^{t}&0\end{array}\right)-\left(\begin{array}[]{c}0\\\
e_{1}\end{array}\right)(e_{1},0)-\left(\begin{array}[]{c}e_{1}\\\
0\end{array}\right)(0,e_{1}),$

where $\mathbf{1}=(1,\dots,1)^{T},e_{1}=(1,0,\dots,0)^{T}\in\mathbb{R}^{m}$.
Assume that $\lambda$ is an eigenvalue of $A$, and
$(0,0)\not=(x,y)\in\mathbb{R}^{m}\times\mathbb{R}^{m}$ is an eigenvector.
Denote $\alpha=\sum_{i=1}^{m}x_{i},\ \beta=\sum_{i=1}^{m}y_{i}$. Then

$\beta-y_{1}=\lambda x_{1},\quad\alpha-x_{1}=\lambda y_{1},\quad\beta=\lambda
x_{i},\quad\alpha=\lambda y_{i},\quad i=2,\dots,m.$

Assuming $\lambda=\pm 1$ leads to an obvious contradiction, as it implies
$\alpha=\beta=0$, and $x=0,y=0$. The matrix $A$ has zero eigenvalue
$\lambda=0$, with $2(m-1)$ dimensional eigenspace
$\\{(x,y)\in\mathbb{R}^{m}\times\mathbb{R}^{m},x_{1}=y_{1}=0\\}$. Therefore,
for $\lambda\not=\pm 1,0$ we have
$x_{1}=(\alpha-\beta\lambda)/(1-\lambda^{2}),\
y_{1}=(\beta-\alpha\lambda)/(1-\lambda^{2})$, and
$x_{2}=\beta/\lambda,y_{i}=\alpha/\lambda,,\quad i=2,\dots,m$. It results in a
system of two linear equations for $\alpha,\beta$:

$\alpha=\frac{m-1}{\lambda}\beta+\frac{\alpha-\beta\lambda}{1-\lambda^{2}},\quad\beta=\frac{m-1}{\lambda}\alpha+\frac{\beta-\alpha\lambda}{1-\lambda^{2}},$

which has a non-trivial solution $(\alpha,\beta)\not=(0,0)$ provided that
$\lambda\not=\pm 1,0,$ is a solution of the following dispersion equation:

$\left(\frac{1}{1-\lambda^{2}}-1\right)^{2}-\left(\frac{m-1}{\lambda}-\frac{\lambda}{1-\lambda^{2}}\right)^{2}=0.$

After rearranging terms, $\lambda$ is a solution of the cubic equation

$\pm\lambda^{3}+m\lambda^{2}-m+1=0,$

having roots $\mp 1$ (which are not eigenvalues of $A$), and four other roots
$\lambda^{\pm,\pm}$ given as in (4), as claimed. The rest of the proof easily
follows. ∎

A similar property to the result of Proposition 3 regarding indices can be
observed when adding one edge to a complete bipartite graph, that is,
destroying the bipartiteness of the original complete bipartite graph by small
perturbation.

######  Proposition 4.

Let us denote by $G_{A}=K_{m,m}^{+e}$ a graph of the order $2m$ constructed
from the complete bipartite graph $K_{m,m}$ by adding exactly one edge to the
first part. Then its spectrum consists of $2m-4$ zeros and four real
eigenvalues $\lambda^{(1),(2),(3),(4)}$ where
$\lambda^{(4)}=\lambda_{-}(A)=-1$, and three other roots
$\lambda^{(3)}<-1<0<\lambda^{(2)}<\lambda^{(1)}$ solve the cubic equation
$\lambda^{2}(1-\lambda)-m(m-2-m\lambda)=0$. The smallest positive eigenvalue
has the form $\lambda_{+}(A)\equiv\lambda^{(2)}=1-2/m-2/m^{3}+O(m^{-4})$ as
$m\to\infty$. As a consequence,
$\lim_{m\to\infty}\Lambda^{gap}(K_{m,m}^{+e})=2$, and
$\lim_{m\to\infty}\Lambda^{ind}(K_{m,m}^{+e})=1$.

###### Proof.

It is similar to the proof of the previous Proposition 3. Arguing similarly as
before, one can show that $\lambda^{(4)}=-1$ is an eigenvalue with
multiplicity one. The other nonzero eigenvalues are roots of the cubic
equation $\lambda^{2}(1-\lambda)-m(m-2-m\lambda)=0$ which can be transformed
into a depressed cubic equation with a positive discriminant $\Delta$. Thus,
it has three distinct real eigenvalues $\lambda^{(1),(2),(3)}$. Performing the
standard asymptotic analysis, we conclude
$\lambda_{+}(A)=\lambda^{(2)}=1-2/m-2/m^{3}+O(m^{-4})$ as $m\to\infty$, as
claimed. ∎

######  Remark 3.

In [18] it is shown that for a bipartite graph $K_{m_{1},m_{2}}$ of the order
$m=m_{1}+m_{2}$ and the average valency $d$ of vertices, one has
$\lambda_{m/2}-\lambda_{1+m/2}\leq\sqrt{d}$.

We end this section with the following statement regarding the density of
values of the spectral index $\Lambda^{gap}$ in the class of complete
bipartite graphs.

######  Proposition 5.

For every pair of real numbers $0\leq\delta<\gamma<1$, there exist an order
$m$ and a complete bipartite graph $K_{m_{1},m_{2}}$ of the order
$m=m_{1}+m_{2}$ such that $m-\gamma\leq\Lambda^{gap}(K_{m_{1},m_{2}})\leq
m-\delta$.

###### Proof.

Recall the known fact (see, e.g. [16]) that the set of fractional parts
$\sqrt{m}-[\sqrt{m}]$ of roots of all positive integers $m$ is dense in the
interval $[0,1)$. Hence, there exists an integer $m_{2}$, such that
$\sqrt{\delta}\leq\sqrt{m_{2}}-[\sqrt{m_{2}}]\leq\sqrt{\gamma}$. Take
$m_{1}:=[\sqrt{m_{2}}]^{2}\leq m_{2}$. Then
$\sqrt{\delta}\leq\sqrt{m_{2}}-\sqrt{m_{1}}\leq\sqrt{\gamma}$. By squaring and
rearranging terms, we obtain $(m_{1}+m_{2})-c\leq
2\sqrt{m_{1}m_{2}}\leq(m_{1}+m_{2})-d$. Now we take the bipartite graph
$K_{m_{1},m_{2}}$, of order $m=m_{1}+m_{2}$. Since
$\Lambda^{gap}(K_{m_{1},m_{2}})=2\sqrt{m_{1}m_{2}}$ the claim follows. ∎

###  2.2  Indices for noncomplete graphs

The purpose of this section is to analyze indices for noncomplete multipartite
graphs.

######  Proposition 6.

If $G_{A}$ is a bipartite but not complete bipartite graph, with the average
vertex degree $d$, and the multiplicity of the zero eigenvalue of the order
$k$, then

$\Lambda^{gap}(G_{A})\leq 2\sqrt{\frac{d(m-2d)}{m-k-2}}\ .$ (5)

###### Proof.

Let $G_{A}$ be a bipartite but not complete bipartite graph with adjacency
matrix $A$ having null space of dimension $k$. Since $G_{A}$ is not complete
bipartite, we have $k\leq m-4$. It follows that $m$ and $k$ have the same
parity, so that $m-k=2r$ for some positive integer $r\geq 2$. By bipartiteness
of $G_{A}$ we may assume that its eigenvalues have the form
$\lambda_{1}\geq\lambda_{2}\geq\ldots\geq\lambda_{r}>0=\lambda_{r+1}=\ldots=\lambda_{r+k}>-\lambda_{r}\geq\ldots\geq-\lambda_{2}\geq-\lambda_{1}$,
so that $\lambda_{+}=\lambda_{r}$ and $\lambda_{-}=-\lambda_{r}$. The earlier
used fact that $\lambda_{1}\geq d$ trivially implies that

$\sum_{i=1}^{r}\lambda_{i}^{2}\geq d^{2}+(r-1)\lambda_{+}^{2}\ \ .$ (6)

It is well known that the sum of squares
$\sum_{i=1}^{m}\lambda_{i}^{2}=trace(A^{2})=md$, where $d$ is the average
valency of vertices of $G_{A}$, that is, $md/2$ is the number of edges in the
graph $G_{A}$ (cf. Bapat [5]). Combined with the inequality $\lambda_{1}\geq
d$ used earlier, we obtain

$md=2\sum_{i=1}^{r}\lambda_{i}^{2}\geq
2d^{2}+2(r-1)\lambda_{+}^{2}=2d^{2}+(m-k-2)\lambda_{+}^{2}$ (7)

and evaluation of $\lambda_{+}(A)$ from (7) gives
$\lambda_{+}(A)=-\lambda_{-}(A)\leq\sqrt{\frac{d(m-2d)}{m-k-2}}$ which implies
the inequality (5) in our statement. ∎

######  Remark 4.

The estimate (5) is nearly optimal. For example, for the graph
$K_{m_{1},m_{1}}^{-e}$ we have $m=2m_{1}$, $d=m_{1}-\frac{1}{m_{1}}$ and
$k=m-4$, and (5) for these values gives
$\Lambda^{gap}(K_{m_{1},m_{1}}^{-e})\leq 2\sqrt{1-4/m^{2}}$, which is a
slightly worse estimate than the one derived in the analysis of the spectrum
of $K_{m_{1},m_{1}}^{-e}$.

Finally, we show that the maximal (minimal) eigenvalue can increase (decrease)
by adding one vertex to the original graph.

######  Proposition 7.

Assume $G_{A}$ is a simple connected graph on the vertices $m$ with the
maximal and minimal eigenvalues $\lambda_{max}(A)$, and $\lambda_{min}(A)$.
Then there exists a graph $G_{\mathscr{A}}$ on the $m+1$ vertices constructed
from $G_{A}$ by adding one vertex connected to each of the vertices $G_{A}$
that has the maximal eigenvalue such that

$\lambda_{max}(\mathscr{A})\geq\frac{\lambda_{max}(A)+\sqrt{(\lambda_{max}(A))^{2}+4}}{2}.$

Similarly, there exists a vertex $i_{0}$ of $G_{A}$ such that the graph
$G_{\mathscr{A}}$ on $m+1$ vertices constructed from $G_{A}$ by adding a
pendant vertex to the vertex $i_{0}$ has the minimal eigenvalues satisfying
the estimate

$\lambda_{min}(\mathscr{A})\leq\frac{\lambda_{min}(A)-\sqrt{(\lambda_{min}(A))^{2}+4/m}}{2}.$

###### Proof.

The sum of all eigenvalues of the symmetric matrix $A$ is zero because the
trace of $A$ is zero. Hence $\lambda_{min}(A)<0<\lambda_{max}(A)$. Let
$\mathscr{A}$ be the $(m+1)\times(m+1)$ adjacency matrix of the graph
$G_{\mathscr{A}}$ obtained from $G_{A}$ by adding a vertex connected to a
subset of vertices of $G_{A}$. Its adjacency matrix $\mathscr{A}$ has the
block form

$\mathscr{A}=\left(\begin{array}[]{cc}A&e\\\ e^{T}&0\end{array}\right),$ (8)

where $e=(e_{1},\dots,e_{m})^{T}$, $e_{i}\in\\{0,1\\}$. The maximal eigenvalue
$\lambda_{max}(\mathscr{A})$ can be computed by means of the Rayleigh ratio,
i.e.

$\lambda_{max}(\mathscr{A})=\max_{x\in\mathbb{R}^{m},\xi\in\mathbb{R}}\frac{(x^{T},\xi)\left(\begin{array}[]{cc}A&e\\\
e^{T}&0\end{array}\right)\left(\begin{array}[]{c}x\\\
\xi\end{array}\right)}{|x|^{2}+\xi^{2}}=\max_{x\in\mathbb{R}^{m},\xi\in\mathbb{R}}\frac{x^{T}Ax+2(e^{T}x)\xi}{|x|^{2}+\xi^{2}},$

where $|x|$ is the Euclidean norm of the vector $x$. Let $\hat{x}$ be an
eigenvector for corresponding to the maximal eigenvalue $\lambda_{max}(A)$,
that is, $A\hat{x}=\lambda_{max}(A)\hat{x}$. Then

$\lambda_{max}(\mathscr{A})\geq\max_{\xi\in\mathbb{R}}\frac{\lambda_{max}(A)+2(e^{T}\hat{x})\xi}{1+\xi^{2}}=\lambda_{max}(A)\max_{\xi\in\mathbb{R}}\frac{1+\alpha\xi}{1+\xi^{2}},$

where $\alpha=2(e^{T}\hat{x})/\lambda_{max}(A)$. Let us introduce the
auxiliary function $\psi:\mathbb{R}\to\mathbb{R}$,
$\psi(\xi)=(1+\alpha\xi)/(1+\xi^{2})$, where $\alpha\in\mathbb{R}$ is a
parameter. Using the first-order necessary condition it is easy to verify that
the maximum of the function $\psi$ is attained at
$\xi=(-1+\sqrt{1+\alpha^{2}})/\alpha$. As a consequence, we have

$\max_{\xi}\frac{1+\alpha\xi}{1+\xi^{2}}=\frac{1+\sqrt{1+\alpha^{2}}}{2}>0.$

Notice that the adjacency matrix contains only nonnegative elements. With
regard to the Perron-Frobenius theorem, an eigenvector corresponding to the
maximal eigenvalue $\lambda_{max}(A)$ is nonnegative, i.e. $\hat{x}\geq 0$.
Consider the vector $e=(1,\dots,1)^{T}$ consisting of ones. It corresponds to
the new vertex connected to all the vertices of $G_{A}$. Then
$(e^{T}\hat{x})^{2}=(\hat{x}_{1}+\dots+\hat{x}_{m})^{2}\geq|\hat{x}|^{2}=1$
because all $\hat{x}_{i}\geq 0$ are nonnegative. Inserting the parameter
$\alpha^{2}=4(e^{T}\hat{x})^{2}/(\lambda_{max}(A))^{2}\geq
4/(\lambda_{max}(A))^{2}$ we obtain
$\lambda_{max}(\mathscr{A})\geq\frac{1}{2}(\lambda_{max}(A)+\sqrt{(\lambda_{max}(A))^{2}+4})$,
as claimed.

Similarly, let $\bar{x}$ be the unit eigenvector corresponding to the minimal
eigenvalue $\lambda_{min}(A)$, that is,
$A\bar{x}=\lambda_{min}(A)\bar{x},|\bar{x}|=1$. Let $i_{0}$ be the index such
that $|\hat{x}_{i_{0}}|=\max_{i}|\hat{x}_{i}|$. Since $|\hat{x}|=1$ we have
$|\hat{x}_{i_{0}}|\geq 1/\sqrt{m}$. Assume that the graph $G_{\mathscr{A}}$ is
constructed from $G_{A}$ by adding one vertex connected to the vertex $i_{0}$.
That is $e=(e_{1},\dots,e_{m})^{T}$, $e_{i_{0}}=1$, and $e_{i}=0$ for
$i\not=i_{0}$. Then $(e^{T}\hat{x})^{2}=(\hat{x}_{i_{0}})^{2}\geq 1/m$. Hence

$\lambda_{min}(\mathscr{A})=\min_{x\in\mathbb{R}^{m},\xi\in\mathbb{R}}\frac{x^{T}Ax+2(e^{T}x)\xi}{|x|^{2}+\xi^{2}}\leq\min_{\xi\in\mathbb{R}}\frac{\lambda_{min}(A)+2(e^{T}\bar{x})\xi}{1+\xi^{2}}=\lambda_{min}(A)\max_{\xi\in\mathbb{R}}\frac{1+\alpha\xi}{1+\xi^{2}}$

because $\lambda_{min}(A)<0$. Here $\alpha=2(e^{T}\bar{x})/\lambda_{min}(A)$.
Consider the index $i_{0}$ for which $|x_{i_{0}}|$ is maximal. Then
$(\bar{x}_{0})^{2}\geq 1/m$, and

$\lambda_{min}(\mathscr{A})\leq\lambda_{min}(A)\frac{1+\sqrt{1+\alpha^{2}}}{2}\leq\frac{\lambda_{min}(A)-\sqrt{(\lambda_{min}(A))^{2}+4/m}}{2},$

and the proof of the proposition follows. ∎

##  3 Statistical properties of indices

The purpose of this section is to report statistical results on maximal
(minimal) eigenvalues, and indices for the class of all simple connected
graphs on $m\leq 10$ vertices. In Table 2 the operators
$E,\sigma,{\mathcal{S}}$ and ${\mathcal{K}}$ represent the mean value,
standard deviation, skewness and kurtosis of the corresponding sets of
eigenvalues $\lambda_{max}$, and $\lambda_{min}$, respectively. For larger $m$
the skewness ${\mathcal{S}}(\lambda_{max})$ approaches zero and the kurtosis
${\mathcal{K}}(\lambda_{max})$ tends to $3$ meaning that the distribution of
maximal eigenvalues of all simple connected graphs on the $m$ vertices becomes
normally distributed as $m$ increases. The skewness
${\mathcal{S}}(\lambda_{min})<0$ is negative and the kurtosis
${\mathcal{K}}(\lambda_{min})>3$ meaning that the distribution of minimal
eigenvalues of connected graphs on the $m$ vertices is skewed to the left. It
has fat tails (leptokurtic distribution) because it has positive excess
kurtosis ${\mathcal{K}}(\lambda_{min})-3>0$ as $m$ increases. We employed the
list of all simple connected graphs due to B. McKay which is available at the
repository [33]. We calculated the spectra for all graphs and the
corresponding indices. Calculating indices for $m=10$ is a computationally
complex task, since the number $11716571$ of all simple connected graphs is
very large. To our knowledge, a consolidated list of connected nonisomorphic
graphs is not available for orders $m\geq 11$.

Table 2: Descriptive statistics of the maximal(minimal) eigenvalues $\lambda_{max}$ ($\lambda_{min}$), spectral gap $\Lambda^{gap}$, spectral index $\Lambda^{ind}$, and spectral power $\Lambda^{pow}$ for all simple connected graphs on $m\leq 10$ vertices. $m$ | $2$ | $3$ | $4$ | $5$ | $6$ | $7$ | $8$ | $9$ | $10$
---|---|---|---|---|---|---|---|---|---
total # | $1$ | $2$ | $6$ | $21$ | $112$ | $853$ | $11117$ | $261080$ | $11716571$
$E(\lambda_{max})$ | 1 | 1.7071 | 2.1802 | 2.6417 | 3.0582 | 3.4856 | 3.9288 | 4.4001 | 4.8895
$\sigma(\lambda_{max})$ | 0 | 0.4142 | 0.5228 | 0.5968 | 0.6368 | 0.6562 | 0.6595 | 0.6529 | 0.6471
${\mathcal{S}}(\lambda_{max})$ | - | 0 | 0.5096 | 0.5171 | 0.4142 | 0.2855 | 0.1536 | 0.0608 | 0.0132
${\mathcal{K}}(\lambda_{max})$ | - | 1 | 1.9715 | 2.6351 | 2.9901 | 3.0804 | 3.0578 | 3.0313 | 3.0096
$\max(\lambda_{max})$ | 1 | 2 | 3 | 4 | 5 | 6 | 7 | 8 | 9
$\min(\lambda_{max})$ | 1 | 1.4142 | 1.6180 | 1.7321 | 1.8019 | 1.8478 | 1.8794 | 1.9021 | 1.9190
$E(\lambda_{min})$ | -1 | -1.2071 | -1.5655 | -1.7911 | -2.0302 | -2.2264 | -2.4191 | -2.6018 | -2.7756
$\sigma(\lambda_{min})$ | 0 | 0.2929 | 0.3305 | 0.2981 | 0.3012 | 0.2995 | 0.2994 | 0.2915 | 0.2832
${\mathcal{S}}(\lambda_{min})$ | - | 0 | 0.5740 | 0.2506 | -0.4079 | -0.5438 | -0.4937 | -0.4121 | -0.3927
${\mathcal{K}}(\lambda_{min})$ | - | 1 | 2.7899 | 4.2278 | 4.1917 | 3.5318 | 3.3933 | 3.3626 | 3.3289
$\max(\lambda_{min})$ | -1 | -1 | -1 | -1 | -1 | -1 | -1 | -1 | -1
$\min(\lambda_{min})$ | -1 | -1.4142 | -2 | -2.4495 | -3 | -3.4641 | -4 | -4.4721 | -5
$\max(\Lambda^{gap})$ | 2 | 3 | 4 | 5 | 6 | 7 | 8 | 9 | 10
$\min(\Lambda^{gap})$ | 2 | 2.8284 | 1.2360 | 1.0806 | 0.7423 | 0.6390 | 0.3468 | 0.2834 | 0.1565
$\max(\Lambda^{ind})$ | 1 | 2 | 3 | 4 | 5 | 6 | 7 | 8 | 9
$\min(\Lambda^{ind})$ | 1 | 1.4142 | 0.6180 | 0.6180 | 0.4142 | 0.3573 | 0.1826 | 0.1502 | 0.0841
$\max(\Lambda^{pow})$ | 2 | 4 | 6 | 8 | 10 | 12 | 14.3253 | 17.0600 | 20
$\min(\Lambda^{pow})$ | 2 | 2.8284 | 3.4642 | 4.0000 | 4.4722 | 4.8990 | 5.2916 | 5.6568 | 6.0000

$m=7$ $m=8$ $m=9$

$m=7$ $m=8$ $m=9$

Figure 2: Histograms of distribution of maximal (top row) and minimal (bottom
row) eigenvalues for all simple connected graphs on $7\leq m\leq 9$ vertices.
For their statistical properties, see Table 2.

$m=7$ $m=8$ $m=9$

Figure 3: Histograms of distribution of $\Lambda^{gap}$ (top row),
$\Lambda^{ind}$ (middle row), and $\Lambda^{pow}$ (bottom row) for all simple
connected graphs on $7\leq m\leq 9$ vertices. For their statistical
properties, see Table 2. Figure 4: The noncomplete graph on $m=7$ vertices
with eigenvalues $\\{5,1,-1,-1,-1,-1,-2\\}$ maximizing the value
$\Lambda^{pow}=12$ in the class of all simple connected graphs of the degree
$m=7$.

$m=8$ $m=9$ $m=10$

Figure 5: noncomplete graphs on $8\leq m\leq 10$ vertices maximizing
$\Lambda^{pow}$ which is greater than the value $=2m-2$ attained by the
complete graph $K_{m}$. For values of $\Lambda^{pow}$ see Table 2.

Interestingly enough, for the values of $m\leq 7$ the maximum value of
$\Lambda^{pow}$ is achieved for the complete graph $K_{m}$ with the
eigenvalues $\\{m-1,-1,\dots,-1\\}$ and the maximal value
$\Lambda^{pow}=2m-2$. For $m=7$ there are exactly two graphs with the same
maximal value $\Lambda^{pow}=12$. The noncomplete maximizing graph with
eigenvalues $\\{5,1,-1,-1,-1,-1,-2\\}$ is shown in Fig. 4. Starting from the
degree $m=8$ the maximal value of $\Lambda^{pow}$ is attained for noncomplete
graphs shown in Fig. 5. In Fig. 6 we show graphs on $5\leq m\leq 10$
minimizing $\Lambda^{gap}$. Path graphs $P_{m}$ minimize $\Lambda^{gap}$ and
$\Lambda^{ind}$ for $m=2,3,4$ (see Table 2). In Fig. 7 we show graphs on
$m=6,7,9,10$ minimizing $\Lambda^{ind}$. For $m=5,8$ the minimizing graphs are
the same as those for $\Lambda^{gap}$ shown in Fig. 7 (see Table 2).

$m=5$ $m=6$ $m=7$

$m=8$ $m=9$ $m=10$

Figure 6: Graphs on $5\leq m\leq 10$ minimizing $\Lambda^{gap}$. For values of
$\Lambda^{pow}$ see Table 2.

######  Remark 5.

According to Caporossi et al. [9, Theorem 2], for a general simple connected
graph $G_{A}$ we have $\Lambda^{pow}(G_{A})\geq 2\sqrt{m-1}$. The unique
minimal value of $\Lambda^{pow}=2\sqrt{m-1}$ is attained by the star graph
$S_{m}\equiv K_{m-1,1}$. For related results, we refer to Stanic [41, (2.11),
p. 33] and McClelland [32].

$m=6$ $m=7$ $m=9$ $m=10$

Figure 7: Graphs on $5\leq m\leq 10$ minimizing $\Lambda^{ind}$. For values of
$\Lambda^{pow}$ see Table 2.

##  4 Conclusions

In this paper we analyzed the spectral properties of all simple connected
graphs. We focus our attention to the class of graphs which are complete
multipartite graphs. We also present results on density of spectral gap
indices and its nonpersistency with respect to small perturbations of the
underlying graph. We also analyzed the spectral properties of graphs different
from those of complete multipartite graphs. We presented statistical and
numerical analysis of the indices $\Lambda^{gap},\Lambda^{ind}$, and
$\Lambda^{pow}$ of graphs of order $m\leq 10$.

## Acknowledgments

Support of the Slovak Research and Development Agency under the projects
APVV-19-0308 (SP, JS), and APVV-20-0311 (DS) is kindly acknowledged.

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longtable

# Quality Control of Image Sensors using Gaseous Tritium Light Sources

David McFadden1,2,3 Brad Amos4 Rainer Heintzmann1,2,3

###### Abstract

We propose a practical method for radiometrically calibrating cameras using
widely available gaseous tritium light sources (_betalights_). Along with the
gain (conversion factor) and read noise level, the predictable photon flux of
the source allows us to gauge the quantum efficiency. The design is easily
reproducible with a 3D printer (three-dimensional printer) and three
inexpensive parts. Suitable for common image sensors, we believe that the
method has the potential to be a useful tool in microscopy facilities and
optical labs alike.

1 Institute of Physical Chemistry and Abbe Center of Photonics, Friedrich-
Schiller-University, Jena, Germany
2 Leibniz Institute of Photonic Technology, Albert-Einstein-Straße 9, 07745
Jena, Germany
3 Jena Center for Soft Matter (JCSM), Friedrich Schiller University Jena,
Jena, Germany
4 Medical Research Council, MRC, Laboratory of Molecular Biology, Cambridge,
United Kingdom

## Introduction

Photodetectors and cameras are common instruments used for acquiring
scientific data. Scientists are often particularly concerned about the read
noise performance and the photon conversion factor, which may vary from device
to device. Manufacturers therefore often include such information in quality
control reports provided to the customers upon delivery. The detectors may,
however, exhibit damage or ageing effects which can affect the quality and
reproducibility of data. Various robust methods for end-user calibration exist
[1]. But despite this, many labs do not regularly perform quantitative quality
control checks on their instruments [2]. It is even less common to measure the
radiometric quantum efficiency in a laboratory setting. One explanation could
be a relative lack of convenient and low-cost calibration sources.

## Existing methods

Measuring the gain and noise characteristics of a detector requires a stable
light source. In order to measure the quantum efficiency, the light source
must furthermore be calibrated to a radiometric standard. This is the main
application for calibrated radiometric transfer standards [3]. In the visible
spectrum, these are realised using quartz-tungsten-halogen lamps, which have
an irradiance uncertainty in the range of 1% [4]. They offer a limited
lifetime, require a long start-up period and a regular recalibration is also
recommended [5]. These lamps, in turn, require a calibrated power supply,
resulting in a combined setup that is both bulky and expensive, with prices on
the order of 1000 EUR to 6000 EUR. For most labs, a calibration tool that
costs as much as a camera itself is unlikely to be considered a good return on
investment.

Alternatively, one could perform a control experiment using a calibrated
detector and correct for the different detector geometries, but this adds
complexity to the procedure and introduces a potential for human error. This
is the method outlined in the industry-standard EMVA1288[6].

X-rays from radioactive sources, such as Fe55, are sometimes used to measure
the gain of visible range image sensors[7 ch. 2.3,8]. The method can be
challenging to use in situ, though, as the glass windows of cooled cameras
absorb X-rays. It is also not suited to measure the visible range quantum
efficiency as this requires visible light.

## Gaseous Tritium Light Sources

Gaseous Tritium Light Sources (GTLSs, or betalights), used in niche products
such as high-end watches, gun sights and fishing tackle, are relatively
inexpensive (approximate cost: 5-20 EUR) and emit a stable and predictable
light over long periods of time, ideally suited as a standard light source.
They rely on free electrons from the beta-decay of tritium (hydrogen-3) gas to
excite light-emitting phosphors that coat the inside layer of a glass tube
(figure 1a).

(a)

(b)

Figure 1: Left: View of a 2 mm $\times$ 12 mm tube. Right: Photograph of the
assembled source.

Compared to thermal lamps, they offer some attractive properties: They can be
made very compact, being smaller than the hands on a watch. And they are self-
powered, independent of an electrical power supply.

The idea of using radioluminescent light sources as a low-light radiometric
standard is not new [9,10]. But while radioluminescence-based sensor quality-
control has been performed by astronomical observatories [11],
radioluminescence based calibration methods have not to our knowledge been
widely adopted in microscopy facilities.

While the use of radioactive materials initially raises safety concerns, the
quantities used in small GTLSs are considered safe and unlikely to result in a
significant radiation exposure, even in the worst-case scenario [12,13].

The regulatory situation is an important consideration for any application.
But while radionuclides, including tritium, are a controlled substance across
the world, many jurisdictions have exemptions for devices up to a certain
level of activity that would usually include small GTLSs. For instance, in
Germany, this limit is set at one gigabecquerel [14].

## Stability

To assess the stability, we measured the signal that a green GTLS produced on
a calibrated photodiode (Thorlabs S130C sensor head with PM100D power meter)
at a fixed position over time. While work by Mikhalchenko et al. [15] has
demonstrated in principle the stability of tritium radioluminescent sources,
and others have praised their stable properties [11], [16] , we found little
quantitative data, and it is unclear how these results would apply to
generically sourced GTLSs, given their proprietary formulations and
manufacturing methods.

We purchased a green-emitting GTLS from a commercial vendor 111Mad Nuclear
Scientist Ltd., London, United Kingdom and monitored the optical power over
178 days (see section “Optical Power measurement” in the methods supplement).
The device was found to have excellent predictability. The decay is in good
agreement with the expected decay of tritium activity (half-life 12.3 years
[17])(figure 2). The output power decreased by 3.9% over the duration of the
measurement, whereas the expected decay in tritium activity over this time is
2.7%. This corresponds to a half-life of 8.4 years. We cannot rule out that
this discrepancy is caused by a long-term drift in the photodiode or the
current meter. (cf. “Power meter reliability” in the methods supplement).

Barring any nonlinear ageing process in the phosphor coating, these results
imply that the source could be used as a standard for low-light optical power
over many years.

It has been noted that the temperature dependence of GTLSs is a drawback [18],
but we consider the mentioned $0.3\%/^{\circ}C$ to be comparatively small for
our intended application.

Figure 2: A plot showing the continuous decrease in optical power over time.
Each dot represents one day’s measurement. The sudden signal drop by 0.6% at
129 days and 144 days can likely be attributed to a temperature decrease in
the lab over the holiday period which can, in turn, affect the photodiode. The
root-mean-square deviation of a linear fit is $1.3\times 10^{-3}$. If we
exclude the aforementioned temperature drop as outlier measurements, this
decreases to $4.8\times 10^{-4}$. The orange line represents the decay that we
would expect from the literature value for the radioactive decay of tritium
(half-life: 12.3 years).

## Design

With this knowledge, we conceptualised a radiometric calibration tool for
cameras. The goal is that it should be reliable, reproducible and affordable.
Using it to calibrate a camera should be a simple plug and play procedure.

We designed a 3D (three-dimensional) printable mount (figure 1b) for the
aforementioned GTLS. Fused filament fabrication 3D printers were used for
development and manufacture (Ultimaker 2/3, Prusa i3).

The design mounts the GTLS in a fixed position, and two circular apertures
define a divergent beam which exits from the detector-pointing side (figure
3a). The apertures are dimensioned specifically to shape a beam that under-
fills the detector’s active area at a chosen distance (figure 3b). The
extended size of the first aperture results in a feathered edge for the beam
spot on the image sensor, yielding a continuous range of pixel intensities.
This is an ideal image pattern for the camera calibration method discussed in
the methods supplement (section “Image analysis”).

Under-filling the detector also greatly simplifies the comparison of quantum
efficiencies between different detectors, as it is not necessary to correct
for different geometries if all of them are under-filled.

There are no optical elements between the source and the camera, eliminating
potential sources of misalignment and contamination. To ensure a high-quality
light seal and repeatable attachment, we fastened it in a conventional c-mount
adapter.

A correctly chosen geometry will also restrict the light’s angle of incidence
on the detector, which is useful to ensure that the measurements are
representative of typical imaging applications.

For printing, we used both PLA (polylactic acid) and ABS (acrylonitrile
butadiene styrene), two common 3D printing materials (brand: BASF Ultrafuse,
black colour option). Both materials have a higher transmission in the near-
infrared than they do in the visible range (cf. methods supplement, section
“Light absorption of printed materials”). This could bias measurements, as
silicon-based detectors are sensitive up to a wavelength of about 1200 nm.
While we observed that the ABS material had an excellent overall optical
extinction, the PLA material leaked a small amount of infrared light even
through very thick layers. Based on this, we would recommend that ABS be
preferred for printed parts. These absorption characteristics may vary from
vendor to vendor, however. Whilst PLA can still be used, measurements should
be performed away from NIR sources such as daylight and incandescent lamps,
and with the room lighting turned off as a precaution. This should be enough
to limit light leakage to very low levels. All printed parts should be
visually inspected for holes and defects.

The design is freely available at https://github.com/mcfaddendavid/betalight-
calibration[19]. Parameters can be tuned so that the spot best matches various
detector geometries. The bill of materials for non-printed parts consists of:

()  Number |  Description |  Approx. Price
---|---|---
() |  |
1 |  Thorlabs C-mount retaining ring, Part number: CMRR |  11.52 EUR
1 |  Thorlabs CS- to C-Mount Extension Adapter, Part number: CML05 |  16.78 EUR
1 |  Cylindrical GTLS light source 2 mm x 12 mm (green colour suggested) |  8.80 EUR
() |  |

(a)

(b)

Figure 3: Left: Cross-section showing the mounted light source attached to a
camera. The divergent light beam (NA approximately 0.1) impinges upon the
image sensor. Apertures between the GTLS tube and the detector plane restrict
the beam size to under-fill the detector surface. Right: The beam profile on
the image sensor (physical size 16.4 mm $\times$ 16.4 mm). Essentially all of
the light is confined within the sensor area whilst covering a good portion of
the available space.

## Camera calibration and charge carriers

First, the mounted source is screwed onto the camera lens mount, and a stack
of images at constant exposure is acquired (cf. figure 3b). Afterwards, the
source is replaced with a camera body cap and we acquire a stack of dark
exposures with identical camera settings.

The typical approach for calibrating gain is the photon transfer method
[1,20,21]. Our implementation is based on a mean-variance curve (figure 4) and
is described in the electronic supplementary material. We use it to obtain
estimates for the read noise, the offset, and the camera gain, which relates
the arbitrary analog-digital-units (ADU) in the output image to a specific
number of charge carriers (photoelectrons) and is expressed in electrons per
ADU. Expressing the noise in terms of effective charge carriers is the only
meaningful way to compare cameras with different designs.

Figure 4: A photon transfer curve for an sCMOS camera generated by the
described method using the GTLS calibration source (exposure time 20 ms). The
blue dots represent the pixel bins, and the fitted line is in orange.
Surrounding the fit are lines representing the standard deviation of the
variance estimates. The gain estimate is 0.45 electrons per digital unit. The
read noise is 1.69 electrons root-mean-square.

Subtracting the mean projection along the time axis of the dark exposure stack
from the mean projection of the bright exposure gives us an unbiased average
frame. We sum the signal $S(i)$ of all pixels $i$ and multiply the result with
the camera gain. This yields a value for the average number of detected
photoelectrons per exposure $N_{e^{-}}$

$N_{e^{-}}=\sum_{i}S(i)\times\textrm{gain}$

Dividing this by the exposure time gives us a rate of effective photoelectrons
$\Phi(e^{-})$:

$\hypertarget{eq:electron_flux}{}{\Phi(e^{-})=\frac{N_{e^{-}}}{\textrm{exposure
time}}}$ (1)

## Source characterisation

A radiometrically calibrated spectrometer yields the spectral power
distribution of the source. The diffuse emission and low light level are an
additional challenge for some low-sensitivity uncooled spectrometers. Fibre-
based spectrometers may benefit from using a fibre-based mount for the GTLS
(figure 5b). Using careful background subtraction, we were nevertheless able
to obtain a satisfactory spectrum with a compact uncooled spectrometer (figure
5a) using only free-space propagation (figure 6).

(a)

(b)

Figure 5: Left: The source is attached as close as possible to the input of a
spectrometer (Qmini, RGB Photonics GmbH). Right: A mount designed to couple
light from the GTLS to an optical fibre. This usually improves the signal
level for fibre-based spectrometers. Figure 6: Normalised power density
spectra obtained from three different GTLSs, two green (green and orange
curves) and one blue (blue curve), with power weighted mean wavelengths at 537
nm, 541 nm and 461 nm respectively. The green spectra closely resemble each
other with the results differing by only 0.6%. Repeated measurements on the
same source achieved a standard deviation of 0.7%. Additionally, we measured
the green source after four months and could not detect a significant change
in the result, the measurements differing by 0.8%. (Integration time 12
seconds, averaging over ca. 514 acquired spectra. Analysed range: 370nm to
750nm)

A calibrated spectrometer will output a spectral power distribution
$\textrm{SPD}(\lambda)$. In SI units, this will be in W/nm. Using this as a
weighting function allows us to characterise the source by a power-weighted
mean wavelength $\bar{\lambda}$:

$\bar{\lambda}=\frac{\int_{\lambda}\lambda\>\textrm{SPD}(\lambda)d\lambda}{\int_{\lambda}\textrm{SPD}(\lambda)d\lambda}$

The assembled calibration source is then mounted pointing towards a power-
calibrated photodiode. The distance to the photodiode surface is exactly the
same that the source would otherwise be from the surface of the image sensor
of a c-mount camera body. (see methods supplement).

We measure the photocurrent that the beam induces on the under-filled
detector, and by correcting for the spectral responsivity of the photodiode as
well as the spectral power distribution of the source $\textrm{SPD}(\lambda)$
(see methods supplement, section “Optical Power measurement”), we obtain the
expected optical power $P$ of the beam.

The power-weighted mean wavelength can be associated with a photon energy:

$E_{ph}=\frac{hc}{\bar{\lambda}}$

Dividing the power $P$ by this photon energy yields the expected rate of
impinging photons $\Phi(\textrm{ph})$:

$\hypertarget{eq:photon_flux}{}{\Phi(\textrm{ph})=\frac{P}{E_{\textrm{ph}}}}$
(2)

Note that using the photon energy of the power-weighted mean wavelength to
calculate the photon rate in this way is valid for any temporally stable
spectral shape. If, however, the spectral shape experiences a shift over time,
then it will result in a different photon rate. Such a change might occur in
GTLSs with multi-coloured or blended phosphors.

We found that our photodiode power meter delivered satisfactory and repeatable
results, with a total optical power $E_{ph}$ of 118 pW. With a $\bar{\lambda}$
of 537 nm, this corresponds to a photon flux $\Phi(\textrm{ph})$ of
$3.20\times 10^{8}s^{-1}$.

The quantum efficiency is simply the ratio between the rate of detected
photoelectrons $\Phi(e^{-})$ (eq. 1) and the photon flux
$\Phi(\textrm{ph})$(eq. 2):

$\textrm{Quantum efficiency}=\frac{\Phi(e^{-})}{\Phi(ph)}$

## Results

Testing the procedure on three scientific cameras, we obtain quantum
efficiency values that agree very well with the quantum efficiencies quoted by
the manufacturers222The data was read manually from graphs provided by the
manufacturers. The manufacturers do not provide error estimates. We assume a
3% relative uncertainty in the calibrations as well as a 2% absolute
uncertainty in the rendering of the quantum efficiency graphs. There are more
potential unknown errors which we cannot account for (see methods supplement,
section “Manufacturer stated quantum efficiency”). For its working range, the
manufacturer quotes the uncertainty of the power meter as ±3% above 450nm, and
±5% below 450nm [22]. However, it should be noted that we are operating the
photodiode below its specified power range (Our source has an optical power of
118 pW, whereas the specified minimum power of the photodiode is 500pW, cf.
methods supplement, section “Power meter reliability”). The spectral
distribution of the source is another crucial input, and measurements on the
green source suggest an uncertainty on the order of 1%. We add this to a 3%
photodiode uncertainty to obtain a relative error of 4% for our measurements.,
once the spectral power distribution of the sources is taken into account. Two
of these are based on scientific CMOS (sCMOS) sensors. CMOS sensors are active
pixel sensors named after the complementary metaloxide-semiconductor (CMOS)
fabrication method, and the “scientific” is commonly used to distinguish high-
performance scientific cameras from low-performance sensors common in consumer
devices. The third sensor is an electronmultiplying charge coupled device
(EMCCD) sensor operated in its non-amplified mode. Table LABEL:tbl:qe_tab
shows values obtained using both blue and green coloured GTLSs. We find that
all but one of the measurements (denoted with “*“) fall within the estimated
uncertainty intervals of the manufacturer-specified values.

Table 2: Quantum efficiency values obtained from the calibration routine for three cameras along with values quoted by their manufacturers. All but one of the measurements (denoted with “*“) fall within the estimated uncertainty intervals of the manufacturer-specified values. The central wavelengths of the blue and green sources were at 461 nm and 537 nm respectively. () | Colour | Measured | Quoted | Difference
---|---|---|---|---
() |  |  |  |
EMCCD camera | Green | 88.6%±3.5% | 89.3%±4.7% | -0.7%
| Blue | 73.8%±3.0% | 74.1%±4.3% | +1.2%
sCMOS Camera no. 1 | Green | 69.0%±2.8% | 70.8%±4.2% | -1.8%
| Blue | 53.7%±2.1% | 55.5%±3.7% | -1.8%
sCMOS Camera no. 2 | Green | 70.5%±2.8% | 78.5%±4.4% | -8.0% (*)
| Blue | 66.8%±2.7% | 73.0%±4.2% | -6.2%
() |  |  |  |

Ultimately the results will also hinge upon the validity of the photon
transfer method. It is therefore paramount to scrutinise the results of a
photon transfer curve before using them in further processing. Some of our
scientific cameras produced repeatable curves across a wide dynamic range, and
we would have more confidence in such results. Figure 4 is such a curve. Other
cameras produced nonlinear curves in a way that would clearly invalidate gain
calibration results. (see methods supplement).

## Conclusion

We have demonstrated an affordable and suitably accurate tool for calibrating
cameras in a plug-and-play fashion in situ. The design requires only minimal
parts and a 3D printer, which allows it to be quickly disseminated to labs
where it might be useful.

Absolute quantum efficiency measurement was also demonstrated, but this
requires additional spectral information obtained from a radiometrically
calibrated spectrometer. Note that a radiometrically accurate spectrum is
generally required for conventional quantum-efficiency measurement methods,
and the EMVA1288 standard specifically recommends verifying the source
spectrum using a (radiometrically calibrated) spectrometer [6]. The standard
does, however, allow users to forgo this verification if an accurate spectrum
can be provided by the light source manufacturer.

We found no significant difference in the spectrum between two different green
GTLSs and were able to reproduce the same spectrum four months later. This
suggests that sufficiently accurate prior knowledge of the source spectrum
could be provided. But as our GTLSs are unbranded, we cannot know what quality
control mechanisms they underwent and the vendor also supplied very limited
documentation. As such, we can’t claim to have confidence that our spectral
data will apply to GTLSs purchased by others. We would therefore recommend
that anybody wishing to reproduce the method measure the spectrum of the GTLS
upon delivery and assembly, and periodically thereafter in order to monitor
for changes.

Compared to thermal calibration lamps, the optical power of GTLSs is very
weak, which prevents us from measuring across the full dynamic range of most
non-amplified detectors. Nevertheless, it is likely to be the appropriate
power range for many life-science applications, especially single-molecule-
localisation microscopy.

The relatively broad spectrum (FWHM $\approx$ 100 nm) of the source means that
we are averaging over a wider spectral range than the 50 nm permitted by
EMVA1288 [6]. Conversely, for some applications, the spectral width might be
considered too narrow or not at the correct location. The availability of
GTLSs with phosphors in many different colours [23] suggests, however, that
the method could be applied to other spectral ranges. These sources would need
to be evaluated individually for their stability, and such an approach might
not have any inherent advantage over existing tools for relative polychromatic
calibration, such as described by JM Beach [24].

As we do not expose the entire image sensor, local defects at the edges go
undetected, though these can also be easily found with conventional light
sources or by using a mount without baffles for the GTLS.

It should be noted that our method is especially useful for assessing the
long-term stability of detectors. The calibration source should be stored in a
dust-free container to ensure that this is not thwarted by dirt or dust
blocking the light path.

The current design also requires the camera to be detached from any lens
mount, which is a drawback as many microscope users do not wish to remove the
cameras from their setups once they have been assembled.

We are currently working on a design that could alleviate this by mounting the
source in an alternative location in the optical path of an instrument. For
microscopes, this could be at the sample plane or the microscope objective
mount. This would allow us to assess the long term stability of the entire
optical path of the instrument.

### Supplementary information

Electronic supplementary material is available online at
https://doi.org/10.6084/m9.figshare.c.5768142

### Data accessibility

The data, as well as the analysis script that was used to produce the results
of this article, are available on the Dryad data repository with the following
DOI: https://doi.org/10.5061/dryad.cvdncjt5f The code for the image analysis,
as described in the methods supplement, is maintained in the freely available
“NanoImagingPack” library: https://gitlab.com/bionanoimaging/nanoimagingpack/
3D files for the printed parts of the calibration source, as well as assembly
instructions, are available at: https://github.com/mcfaddendavid/betalight-
calibration

### Contributions

David McFadden designed, coordinated and performed the experiments, developed
the designs, analysed the results and drafted the manuscript. Brad Amos came
up with the idea of using betalights for camera calibration and revised the
manuscript. Rainer Heintzmann participated in the analysis and revised the
manuscript. All authors gave final approval for publication and agree to be
held accountable for the work performed therein.

### Competing interests

We declare we have no competing interests.

### Funding

This work was supported by the Deutsche Forschungsgemeinschaft (1278
Polytarget, Project C04).

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