On a version of the spectral excess theorem

TL;DR

This paper characterizes when a graph's distance matrix is a polynomial in its adjacency matrix using spectral properties and extends the spectral excess theorem to more general graphs.

Contribution

It provides a new spectral characterization for the polynomial relation between distance and adjacency matrices, generalizing the spectral excess theorem.

Findings

Characterization of when the distance matrix is a polynomial in the adjacency matrix.

Extension of the spectral excess theorem to general graphs using Laplacian spectrum.

Conditions involving the spectrum and mean distances for such polynomial relations.

Abstract

Given a regular (connected) graph $\Gamma=(X,E)$ with adjacency matrix $A$, $d+1$ distinct eigenvalues, and diameter $D$, we give a characterization of when its distance matrix $A_D$ is a polynomial in $A$, in terms of the adjacency spectrum of $\Gamma$ and the arithmetic (or harmonic) mean of the numbers of vertices at distance $\le D-1$ of every vertex. The same results is proved for any graph by using its Laplacian matrix $L$ and corresponding spectrum. When $D=d$ we reobtain the spectral excess theorem characterizing distance-regular graphs.

Equations

$$⟨f,g{⟩}_A=\frac{1}{n}\text{mbox}\mathrm{tr}(f(A)g(A))=\frac{1}{n}\text{mbox}\mathrm{sum}(f(A)\circ g(A))=\frac{1}{n}\sum _{i=0}^{d}m({\lambda }_i)f({\lambda }_i)g({\lambda }_i),$$

$$p_d({\lambda }_0)=n{\left(\sum _{i=0}^d\frac{{\varphi }_0^2}{m_i{\varphi }_i^2}\right)}^{-1},$$

$$⟨f,g{⟩}_L=\frac{1}{n}\text{mbox}\mathrm{tr}(f(L)g(L))=\frac{1}{n}\text{mbox}\mathrm{sum}(f(L)\circ g(L))=\frac{1}{n}\sum _{i=0}^{d}m({\theta }_i)f({\theta }_i)g({\theta }_i),$$

$$⟨x,r_0{⟩}_L=\frac{1}{n}\text{mbox}\mathrm{tr}(L)=\frac{1}{n}\sum _{x\in X}k(x)=\overline{k}\text{mbox}and\Vert r_0{\Vert }_L^2=\frac{1}{n}\text{mbox}\mathrm{tr}(I^2)=1.$$

$$\Vert t{\Vert }_L^2$$

$$r_d(0)=n{\left(\sum _{i=0}^d\frac{{\psi }_0^2}{m({\theta }_i){\psi }_i^2}\right)}^{-1},$$

$$\left(\sum _{y\notin {\Gamma }_D(x)}(C_{xy}{)}^2\right)\left(\sum _{y\notin {\Gamma }_D(x)}{1}^2\right)\ge {\left(\sum _{y\notin {\Gamma }_D(x)}{C}_{xy}\right)}^2.$$

$$\sum _{y\notin {\Gamma }_D(x)}(C_{xy}{)}^2\ge \frac{1}{|X|-|{\Gamma }_D(x)|}{\left(\sum _{y\notin {\Gamma }_D(x)}{C}_{xy}\right)}^2,$$

$$\frac{|X|}{{\sum }_{x\in X}\frac{1}{|X|-|{\Gamma }_D(x)|}}\ge q_{D-1}(k)=|X|-\sum _{i=D}^d{p}_i(k),$$

$$q_{D-1}(k)$$

$$\frac{|X|}{{\sum }_{x\in X}\frac{1}{|X|-|{\Gamma }_D(x)|}}\ge q_{D-1}(k).$$

$$\Vert q_{D-1}{\Vert }^2=\frac{1}{n}\text{mbox}\mathrm{tr}(q_{D-1}(A{)}^2)=\frac{1}{n}{\alpha }^2\text{mbox}\mathrm{sum}\left(\sum _{i=0}^{D-1}{A}_i\right)=n_{D-1}{\alpha }^2=n_{D-1}\alpha ,$$

$$\frac{1}{|X|}\sum _{x\in X}(|X|-|{\Gamma }_D(x)|)\ge |X|-\sum _{i=D}^d{p}_i(k)$$

$$|{\Gamma }_D(x)|\le \sum _{i=D}^d{p}_i(k).$$

$$\frac{|X|}{{\sum }_{x\in X}\frac{1}{|X|-|{\Gamma }_d(x)|}}=q_{d-1}(k)=|X|-p_d(k)=n\left[1-{\left(\sum _{i=0}^d\frac{{\varphi }_0^2}{m_i{\varphi }_i^2}\right)}^{-1}\right].$$

$$\frac{1}{|X|}\sum _{x\in X}(|X|-|{\Gamma }_D(x)|)=p_d(k)=n{\left(\sum _{i=0}^d\frac{{\varphi }_0^2}{m_i{\varphi }_i^2}\right)}^{-1}.$$

$$AM=\frac{a_1+a_2+\cdots +a_n}{n}\text{mbox}andHM=\frac{n}{\frac{1}{a_1}+\frac{1}{a_2}+\cdots +\frac{1}{a_n}}$$

$$\frac{|X|}{{\sum }_{x\in X}\frac{1}{|X|-|{\Gamma }_D(x)|}}\ge s_{D-1}(0)=|X|-\sum _{i=D}^d{r}_i(0),$$

$$s_{D-1}(0)$$

$$\frac{|X|}{{\sum }_{x\in X}\frac{1}{|X|-|{\Gamma }_D(x)|}}\ge s_{D-1}(0)=|X|-\sum _{i=D}^d{r}_i(0).$$

$$\Vert s_{D-1}{\Vert }_L^2=\frac{1}{n}\text{mbox}\mathrm{tr}(s_{D-1}(L{)}^2)=\frac{1}{n}{\alpha }^2\text{mbox}\mathrm{sum}\left(\sum _{i=0}^{D-1}{A}_i\right)=n_{D-1}{\alpha }^2=n_{D-1}\alpha ,$$

$$\frac{|X|}{{\sum }_{x\in X}\frac{1}{|X|-|{\Gamma }_d(x)|}}=s_{d-1}(0)=|X|-r_d(0)=n\left[1-{\left(\sum _{i=0}^d\frac{{\psi }_0^2}{m({\theta }_i){\psi }_i^2}\right)}^{-1}\right].$$

$$|{\Gamma }_D(x)|\le p_d(k)\forall x\in X.$$

Full text

MSC: 05C50, 05E30

Keywords: Graph, adjacency algebra, spectrum, harmonic mean, distance-regular graph, Laplacian.

1 Preliminaries

Let $\Gamma=(X,E)$ be a (simple and connected) graph on $n=|X|$ vertices, with adjacency matrix $A$, and spectrum ${\mbox{\rm sp\,}}\Gamma=\{\lambda_{0}^{m(\lambda_{0})},\lambda_{1}^{m(\lambda_{1})},\ldots,\lambda_{d}^{m(\lambda_{d})}\}$, where $\lambda_{0}>\lambda_{1}>\cdots>\lambda_{d}$ are the distinct eigenvalues, and the superscripts stand for their multiplicities $m_{i}=m(\lambda_{i})$. If $\Gamma$ has diameter $D$, we denote by $\Gamma_{i}(x)$ the set of vertices at distance $i=1,\ldots,D$ from $x\in X$, and $k_{i}(x)=|\Gamma_{i}(x)|$. We abbreviate $k_{1}(x)$ by $k(x)$, the degree of vertex $x$.

Given to square matrices $M,N\in{\mbox{\rm Mat}}_{n\times n}({\mathbb{R}})$, let ${\mbox{\rm sum}}(M)$ denote the sum of all entries of $M$, so that ${\mbox{\rm tr}}(MN)={\mbox{\rm sum}}(M\circ N)$, where ‘$\circ$’ stand for the Hadamard (or entrywise) product. The predistance polynomials $p_{0},p_{1},\ldots,p_{d}$ of $\Gamma$, introduced in [9], are a sequence of orthogonal polynomials with respect to the scalar product

[TABLE]

normalized in such a way that $\|p_{i}\|^{2}_{A}=p_{i}(\lambda_{0})$. For instance, since ${\mbox{\rm tr}}(A^{h})=n\langle A^{h},I\rangle_{A}=\sum_{i=0}^{d}m(\lambda_{i})\lambda_{i}^{h}$, the two first predistance polynomials are $p_{0}(x)=1$ and $p_{1}(x)=\frac{\lambda_{0}}{\overline{k}}x$, with $\overline{k}$ being the average degree of $\Gamma$, see also Lemma 1.2. (It is known that $\overline{k}\leq\lambda_{0}$ with equality if and only if $\Gamma$ is regular.) Moreover, the value of the highest degree polynomial $p_{d}$ at $\lambda_{0}$ can be computed from ${\mbox{\rm sp\,}}\Gamma$ as

[TABLE]

where $\phi_{i}=\prod_{j\neq i}(\lambda_{i}-\lambda_{j})$, $i=0,\ldots,d$ (see [9]).

The predistance matrices $P_{0},P_{1},\ldots,P_{d}$ are then defined by $P_{i}=p_{i}(A)$ for $i=0,1,\ldots,d$. By [1, Prop. 2.2], there exist numbers $\alpha_{i}$, $\beta_{i}$, and $\gamma_{i}$ such that $PP_{i}(=P_{i}P)=\beta_{i-1}P_{i-1}+\alpha_{i}P_{i}+\gamma_{i+1}P_{i+1}$ for $i=0,\ldots,d$, where $P=P_{1}$, $P_{-1}=P_{d+1}=0$, $\gamma_{1}=1$, and $\alpha_{0}=0$. Also, if we define the polynomial $p_{d+1}(x)=(x-\alpha_{d})p_{d}(x)-\beta_{d-1}p_{d-1}(x)$, it can be shown that $p_{d+1}(A)=0$, and the distinct eigenvalues of $\Gamma$ are precisely the zeros of $p_{d+1}$.

The above names come from the fact that, if $\Gamma$ is a distance-regular graph, then the $p_{i}$’s and $P_{i}$’s correspond to the well-known distance polynomials and distance matrices $A_{i}$, respectively. In fact, a known characterization states that $\Gamma$ is distance-regular if and only if such polynomials satisfy $p_{i}(A)=A_{i}$ for every $i=1,\ldots,D$. Moreover, in this case, $D=d$. If we do not impose that the degree of each polynomial coincide with its subindex, then it can be $D<d$ and the graph is called distance-polynomial, a concept introduced by Weichsel [11].

In fact, if $D=d$, the first author, Garriga, and Yebra [8] proved the following.

Proposition 1.1

A regular graph $\Gamma$ with diameter $D$ and $d+1$ distinct eigenvalues is distance-regular if and only if $D=d$ and its highest degree predistance polynomial satisfies $p_{d}(A)=A_{d}$.

From the predistance polynomials, we also consider their sums $q_{i}=p_{0}+\cdots+p_{i}$ for $i=0,\ldots,d$, which satisfy $1=q_{0}(\lambda_{0})<q_{1}(\lambda_{0})<\cdots<q_{d}(\lambda_{0})=|X|$, with $q_{d}=H$ being the Hoffman polynomial that characterizes the regularity of $\Gamma$ by the equality $H(A)=J$, the all-1 matrix (see [10]).

We also recall that the Laplacian matrix of $\Gamma$ is the matrix $L=K-A$, where $K={\mbox{\rm diag}}(k(x_{1}),\ldots,k(x_{n}))$, where $x_{i}\in X$ for $i=1,\ldots,n$. The Laplacian spectrum of $\Gamma$ is ${\mbox{\rm sp\,}}_{\!L}\Gamma={\mbox{\rm sp\,}}L$ $=\{\theta_{0}^{m(\theta_{0})},\theta_{1}^{m(\theta_{1})},\ldots,\theta_{d}^{m(\theta_{d})}\}$ with $\theta_{0}=0<\theta_{1}<\cdots<\theta_{d}$. In particular, since $\Gamma$ is connected, $m_{0}=1$, and the eigenvalue [math] has eigenvector $\boldsymbol{j}$, the all-1 vector. As in the case of the adjacency spectrum, we can define the Laplacian predistance polynomials $r_{0},r_{1},\ldots,r_{d}$ as the sequence of orthogonal polynomials with respect to the scalar product

[TABLE]

normalized in such a way that $\|r_{i}\|_{L}^{2}=r_{i}(0)$. The following result gives the first two Laplacian predistance polynomials.

Lemma 1.2

Let $\Gamma$ be a graph with Laplacian matrix $L=K-A$. Let $\overline{k^{2}}$ be the average of the square degrees of $\Gamma$. Then

• $(i)$

$r_{0}(x)=1$.

• $(ii)$

$r_{1}(x)=\frac{\overline{k}}{\overline{k}(\overline{k}-1)-\overline{k^{2}}}(x-\overline{k})$.

Proof. We only need to prove $(ii)$. By using the method of Gram-Schmidt, we first find a polynomial $t(x)$ orthogonal to $r_{0}=1$. That is, $t(x)=x-\frac{\langle x,r_{0}\rangle_{L}}{\|r_{0}\|_{L}^{2}}r_{0}(x)$, where

[TABLE]

Now, $r_{1}(x)=\alpha t(x)$, where $\alpha$ is a constant to be determined by the normalization condition $\|r_{1}\|_{L}=r_{1}(0)$, which gives $\alpha=\frac{t(0)}{\|t\|_{L}^{2}}$. Moreover,

[TABLE]

Then, from $r_{1}(x)=\frac{t(0)}{\|t\|^{2}_{L}}t(x)$ we get the result.

Also, as in the case of the predistance polynomials $p_{i}$’s, we have

[TABLE]

with $\psi_{i}=\prod_{j\neq i}(\theta_{i}-\theta_{j})$, $i=0,\ldots,d$ (see [1]).

The analogous of Proposition 1.1, for not necessarily regular graphs, was proved by Van Dam and the first author in [3].

Proposition 1.3

A graph $\Gamma$ with Laplacian matrix $L$, $d+1$ distinct Laplacian eigenvalues, and diameter $D$ is distance-regular if and only if $D=d$ and its highest degree Laplacian predistance polynomial satisfies $r_{d}(L)=A_{d}$.

In fact, the regularity of $\Gamma$ is already implied by the equation $r_{1}(L)=A$, as shown in the following lemma.

Lemma 1.4

Let $\Gamma$ be a graph with adjacency and Laplacian matrices $A$ and $L$, respectively, and Laplacian predistance polynomial $r_{1}$. Then, $\Gamma$ is $k$-regular if and only if $r_{1}(L)=A$.

Proof. From the Cauchy-Schwartz inequality, $\overline{k^{2}}\geq(\overline{k})^{2}$, with equality if and only if $\Gamma$ is $k$-regular. In this case, Lemma 1.2$(ii)$ becomes $r_{1}(x)=k-x$ and, hence, $r_{1}(L)=kI-(kI-A)=A$. Conversely, if $r_{1}(L)=r_{1}(K-A)=A$, by equating the coefficients of $A$ in (1.2), we get $\frac{\overline{k}}{\overline{k}(\overline{k}-1)-\overline{k^{2}}}=-1$, whence $\overline{k^{2}}=(\overline{k})^{2}$, $\Gamma$ is $k$-regular, and $K=kI$.

In this context, we also consider the sum polynomials $s_{i}=r_{0}+\cdots+r_{i}$ for $i=0,\ldots,d$, with $H_{L}=s_{d}$ being a Hoffman-like polynomial satisfying $H(L)=J$ (independently of whether $\Gamma$ is regular or not). For more details, see [3].

In our results we use the following simple result.

Lemma 1.5

Let $\Gamma=(X,E)$ be a graph with adjacency matrix $A$ and Laplacian matrix $L$. Given a vertex $x\in X$ and a polynomial $p\in{\mathbb{R}}_{h}[t]$,

• $(i)$

If $\Gamma$ is $k$-regular, then $\sum_{y\in X}p(A)_{xy}=p(k)$.

• $(ii)$

If $\Gamma$ is a general graph, then $\sum_{y\in X}p(L)_{xy}=p(0)$.

Proof. $(i)$ Since $\Gamma$ is $k$-regular, $(k,\boldsymbol{j})$ is an eigenpair of $A$ and, hence, $p(A)\boldsymbol{j}=p(k)\boldsymbol{j}$. Then, the result follows by considering the $x^{\rm th}$ component of both vectors. Case $(ii)$ is proved in the same way by considering that $(0,\boldsymbol{j})$ is an eigenpair of $L$.

2 A version of the spectral excess theorem

The spectral excess theorem, due to Fiol and Garriga [9], states that a regular (connected) graph $\Gamma$ is distance-regular if and only if its spectral excess (a number which can be computed from the spectrum of $\Gamma$) equals its average excess (the mean of the numbers of vertices at maximum distance from every vertex), see Van Dam [2], and Fiol, Gago, Garriga [7] for short proofs.

In this section we find a possible blue solution to the problem of deciding whether, from the adjacency spectrum of a (regular) graph $\Gamma$ and the harmonic (or arithmetic) mean of the numbers $(|X|-|\Gamma_{D}(x)|)_{x\in X}$, we can decide that $A_{D}$ is a polynomial in $A$. To be more precise, we provide a characterization of when $A_{D}\in{\mbox{\rm span}}\{p_{0}(A),\ldots,p_{d}(A)\}$, where the $p_{i}^{\prime}s$ are the predistance polynomials.

Before proving the main result, note that, for any $x\in X$ and any $C\in{\mbox{\rm Mat}}_{X}({\mathbb{R}})$, the Cauchy-Schwartz inequality yields

[TABLE]

That is,

[TABLE]

and equality holds if and only if all the values of $C_{xy}$ are the same for all $y\not\in\Gamma_{D}(x)$.

Theorem 2.1

Let $\Gamma=(X,R)$ be a connected $k$-regular graph with $d+1$ distinct eigenvalues, diameter $D$, and predistance polynomials $\{p_{i}\}_{i=0}^{d}$. Then,

[TABLE]

with equality if and only if $A_{D}=\sum_{i=D}^{d}p_{i}(A)$.

Proof. We just adjust the proof of [2, Lemma 1], together with Lemma 1.5 and (3). Recalling that $q_{D-1}=\sum_{i=0}^{D-1}p_{i}$, we have that

[TABLE]

and this yields

[TABLE]

Since $|X|=\sum_{i=0}^{d}p_{i}(k)=q_{D-1}(k)+\sum_{i=D}^{d}p_{i}(k)$, the inequality follows.

If equality holds, then for each $x$ the values of $q_{D-1}(A)_{xy}$ are the same, say $\alpha$, for all $y\not\in\Gamma_{D}(x)$. Moreover, since $A$ is symmetric, $q_{D-1}(A)_{xy}=q_{D-1}(A)_{yx}$ for any $x$ and $y\not\in\Gamma_{D}(x)$, so that $q_{D-1}(A)_{xy}=\alpha$ for any $x$ and $y\not\in\Gamma_{D}(x)$. Also, by Lemma 1.5$(i)$, $\sum_{y\in X}q_{D-1}(A)_{xy}=q_{D-1}(k)=n_{D-1}\alpha$, where $n_{D-1}=\sum_{i=0}^{D-1}|\Gamma_{i}(x)|$ for any $\in X$. Finally, from $\|q_{D-1}\|^{2}=q_{D-1}(k)$, we have

[TABLE]

so that $\alpha=1$. That is, $q_{D-1}(A)_{xy}=1$ for each pair of vertices $x$ and $y$ at distance less than $D-1$. Consequently, $q_{D-1}(A)=\sum_{i=0}^{D-1}A_{i}=J-A_{D}$, which yields $A_{D}=\sum_{i=D}^{d}p_{i}(A)$.

Conversely, assume that $\sum_{i=D}^{d}p_{i}(A)=A_{D}$. Then, $A_{D}\boldsymbol{j}=\sum_{i=D}^{d}p_{i}(k)\boldsymbol{j}$, and with $|\Gamma_{D}(x)|=\sum_{i=D}^{d}p_{i}(k)$, the equality follows.

As a simple consequence, notice that, if $\Gamma$ is a $k$-regular graph of diameter $2$, then $|X|-|\Gamma_{2}(x)|=1+k$ for any $x\in X$. Besides, $q_{1}(k)=p_{0}(k)+p_{1}(k)=1+k$. Thus, equality in Theorem 2.1 holds, and $\Gamma$ is distance polynomial, as already proved Weichel in [11]. Another consequence of Theorem 2.1 is the following corollary.

Corollary 2.2

Let $\Gamma=(X,R)$ be a connected $k$-regular graph on $n$ vertices, with spectrum ${\mbox{\rm sp\,}}\Gamma=$ $\{\lambda_{0}(=k)^{m(\lambda_{0})},\lambda_{1}^{m(\lambda_{1})}\ldots,\lambda_{d}^{m(\lambda_{d})}\}$, diameter $D$, and predistance polynomials $\{p_{i}\}_{i=0}^{d}$. Then the following holds.

• $(i)$

In general,

[TABLE]

with equality if and only if $A_{D}=\sum_{i=D}^{d}p_{i}(A)$.

• $(ii)$

If $A_{D}\in{\mbox{\rm span}}\{I,A,\ldots,A^{d}\}$ then

[TABLE]

• $(iii)$

If $\frac{1}{|X|}\sum_{x\in X}(|X|-|\Gamma_{D}(x)|)=|X|-\sum_{i=D}^{d}p_{i}(k)$ then $A_{D}\in{\mbox{\rm span}}\{I,A,\ldots,A^{d}\}$.

• $(iv)$

The graph $\Gamma$ is distance-regular if and only if $D=d$ and

[TABLE]

or, alternatively,

[TABLE]

Proof. $(i)$ Let $a_{1}$, $a_{2}$, …, $a_{n}$ be real numbers. Recall that the numbers

[TABLE]

are the arithmetic and harmonic mean for the numbers $a_{1}$, $a_{2}$, …, $a_{n}$, respectively, and we have $AM\geq HM$. Equalities occur if and only if $a_{1}=a_{2}=\cdots=a_{n}$. The result now follows from Theorem 2.1. The proofs in $(ii)$ and $(iii)$ are immediate from $(i)$, or from Theorem 2.1. The results in $(iv)$ correspond to different versions of the spectral excess theorem given in [2, 5] and [9], respectively. Thus, (5) is a consequence of Theorem 2.1 and Proposition 1.1, whereas (6) follows from Theorem 2.1 and $(i)$. In these two cases, we also used $|X|=n$ and (1).

3 The Laplacian approach

Theorem 3.1

Let $\Gamma=(X,R)$ be a connected graph with $d+1$ distinct eigenvalues, diameter $D$, and Laplacian predistance polynomials $\{r_{i}\}_{i=0}^{d}$. Then,

[TABLE]

with equality if and only if $A_{D}=\sum_{i=D}^{d}r_{i}(L)$. Moreover, in this case, if $D=2$, $\Gamma$ is regular.

Proof. The proof follows the same line of reasoning that in Theorem 2.1 with the polynomial $s_{D-1}$ instead of $q_{D-1}$. Thus, we have:

[TABLE]

and this yields

[TABLE]

If equality holds, then for each $x$ the values of $s_{D-1}(L)_{xy}$ are the same, say $\alpha$, for all $y\not\in\Gamma_{D}(x)$. Moreover, since $L$ is symmetric, $s_{D-1}(L)_{xy}=s_{D-1}(L)_{yx}$ for any $x$ and $y\not\in\Gamma_{D}(x)$. Also, by Lemma 1.5$(ii)$, $\sum_{y\in X}s_{D-1}(L)_{xy}=s_{D-1}(0)=n_{D-1}\alpha$, where $n_{D-1}=\sum_{i=0}^{D-1}|\Gamma_{i}(x)|$ for every $x\in X$. Finally, from $\|s_{D-1}\|^{2}=s_{D-1}(0)$, we have

[TABLE]

so that $\alpha=1$. That is, $s_{D-1}(L)_{xy}=1$ for each pair of vertices $x$ and $y$ at distance less than $D-1$. Consequently, $s_{D-1}(L)=J-A_{D}$, which yields $A_{D}=\sum_{i=D}^{d}r_{i}(L)$. In particular, if equality holds and $D=2$, we have $I+A=s_{1}(L)=r_{0}(L)+r_{1}(L)=I+r_{1}(L)$. Thus, $r_{1}(L)=A$ and, by Lemma 1.4, $\Gamma$ is regular.

Conversely, assume that $\sum_{i=D}^{d}r_{i}(L)=A_{D}$. Then, $A_{D}\boldsymbol{j}=\sum_{i=D}^{d}r_{i}(0)\boldsymbol{j}$, and with $|\Gamma_{D}(x)|=\sum_{i=D}^{d}r_{i}(k)$, and the equality follows.

From this theorem, we obtain the analogous results of Corollary 2.2$(i)$-$(iv)$. In particular, the analogous of $(iv)$ yields the following characterization of distance-regularity for a (not necessarily regular) graph.

Corollary 3.2

Let $\Gamma=(X,R)$ be a graph on $n$ vertices, with Laplacian matrix $L$, Laplacian spectrum ${\mbox{\rm sp\,}}L=\{\theta_{0}(=0)^{m(\theta_{0})},\theta_{1}^{m(\theta_{1})},\ldots,\theta_{d}^{m(\theta_{d})}\}$, diameter $D$, and Laplacian predistance polynomials $\{r_{i}\}_{i=0}^{d}$. Then, $\Gamma$ is distance-regular if and only if $D=d$ and

[TABLE]

Proof. Use Theorem 2.1, Proposition 1.3, and (2).

Also, as in Corollary 2.2$(iv)$, the above result implies the characterization given in [3] by using the arithmetic mean of the numbers $|X|-|\Gamma_{d}(x)|$.)

4 Open problems

We finish the paper by formulating some open problems which could be of interest in further studies.

Research problem 4.1

According to Corollary 2.2$(iv)$, if $d=D$ and equality in (4) holds, then $\Gamma$ is distance-regular. Classify all graphs for which equality in (4) holds.

Research problem 4.2

Let $\Gamma=(X,R)$ be a walk-regular graph (that is, for each $\ell\geq 0$, the number of closed walks of length $\ell$ from a vertex $x$ to itself is the same for each $x$) with diameter $D$ and $d+1$ distinct eigenvalues. Assume that $D<d$. Prove or disprove that

[TABLE]

More generally, prove or disprove the same when $\Gamma$ is regular.

Research problem 4.3

Let $\Gamma=(X,R)$ be a graph with diameter $D$, adjacency matrix $A$, and $d+1$ distinct eigenvalues. Let $f_{0},f_{1},\ldots,f_{d}$ be linearly independent polynomials satisfying $\sum_{i=0}^{d}f_{i}(A)=J$, where $f_{i}$, for $i=0,\ldots,d$, does not need to be of degree $i$. Find under what conditions on such polynomials we can obtain a version of the spectral excess theorem for quotient polynomial graphs. (For a definition of quotient polynomial graphs, see [6]).