A general method to obtain the spectrum and local spectra of a graph from its regular partitions

TL;DR

This paper introduces a comprehensive method to derive the entire spectrum and local spectra of a graph using quotient matrices from regular partitions, applicable to various classes like walk-regular and distance-regular graphs.

Contribution

It presents a new general approach to obtain all spectral information of a graph from its regular partitions, extending previous partial results.

Findings

Method successfully computes full spectra of various graph classes.

Applicable to walk-regular, distance-regular, and distance-biregular graphs.

Provides explicit eigenvalues and multiplicities from quotient matrices.

Abstract

It is well known that, in general, part of the spectrum of a graph can be obtained from the adjacency matrix of its quotient graph given by a regular partition. In this paper, we propose a method to obtain all the spectrum, and also the local spectra, of a graph $\Gamma$ from the quotient matrices of some of its regular partitions. As examples, it is shown how to find the eigenvalues and (local) multiplicities of walk-regular, distance-regular, and distance-biregular graphs.

Figures

Tables

Table 1. Table 1: Local multiplicities, from a vertex u ∈ V 1 𝑢 subscript 𝑉 1 u\in V_{1} , of the Hadamard distance-biregular graph H ​ b ​ ( n ) 𝐻 𝑏 𝑛 Hb(n) .

$dist(u,v)$, $u\in V_1$	0	1	2	3	4
$m_{uv}\left({\theta }_0\right)$	$\frac{1}{16n-4}$	$\frac{\sqrt{2}}{8\sqrt{4n^2-n}}$	$\frac{1}{16n-4}$	$\frac{\sqrt{2}}{8\sqrt{4n^2-n}}$	$\frac{1}{16n-4}$
$m_{uv}\left({\theta }_1\right)$	$\frac{1}{4}$	$\frac{\sqrt{2}}{8\sqrt{n}}$	$0$	$-\frac{\sqrt{2}}{8\sqrt{n}}$	$-\frac{1}{4}$
$m_{uv}\left({\theta }_2\right)$	$\frac{2n-1}{4n-1}$	$0$	$-\frac{1}{8n-2}$	$0$	$\frac{2n-1}{4n-1}$
$m_{uv}\left({\theta }_3\right)$	$\frac{1}{4}$	$-\frac{\sqrt{2}}{8\sqrt{n}}$	$0$	$\frac{\sqrt{2}}{8\sqrt{n}}$	$-\frac{1}{4}$
$m_{uv}\left({\theta }_4\right)$	$\frac{1}{16n-4}$	$-\frac{\sqrt{2}}{8\sqrt{4n^2-n}}$	$\frac{1}{16n-4}$	$-\frac{\sqrt{2}}{8\sqrt{4n^2-n}}$	$\frac{1}{16n-4}$
${\sum }_{i=0}^4{m}_{uv}\left({\theta }_i\right)$	$1$	$0$	$0$	$0$	$0$

Table 2. Table 2: Local multiplicities, from a vertex u ∈ V 2 𝑢 subscript 𝑉 2 u\in V_{2} , of the Hadamard distance-biregular graph H ​ b ​ ( n ) 𝐻 𝑏 𝑛 Hb(n) .

$dist(u,v)$, $u\in V_2$	0	1	2	3
$m_{uv}\left({\theta }_0\right)$	$\frac{1}{8n}$	$\frac{\sqrt{2}}{8\sqrt{4n^2-n}}$	$\frac{1}{8n}$	$\frac{\sqrt{2}}{8\sqrt{4n^2-n}}$
$m_{uv}\left({\theta }_1\right)$	$\frac{4n-1}{8n}$	$\frac{\sqrt{2}}{8\sqrt{n}}$	$-\frac{1}{8n}$	$-\frac{\sqrt{2}}{8\sqrt{n}}$
$m_{uv}\left({\theta }_2\right)$	$0$	$0$	$0$	$0$
$m_{uv}\left({\theta }_3\right)$	$\frac{4n-1}{8n}$	$-\frac{\sqrt{2}}{8\sqrt{n}}$	$-\frac{1}{8n}$	$\frac{\sqrt{2}}{8\sqrt{n}}$
$m_{uv}\left({\theta }_4\right)$	$\frac{1}{8n}$	$-\frac{\sqrt{2}}{8\sqrt{4n^2-n}}$	$\frac{1}{8n}$	$-\frac{\sqrt{2}}{8\sqrt{4n^2-n}}$
${\sum }_{i=0}^3{m}_{uv}\left({\theta }_i\right)$	$1$	$0$	$0$	$0$

Equations

$$\mathop{\rm sp}\nolimits\Gamma=\mathop{\rm sp}\nolimits{\mbox{\boldmath$A$}}=\{\theta_{0}^{m(\theta_{0})},\theta_{1}^{m(\theta_{1})},\dots,\theta_{d}^{m(\theta_{d})}\},$$

$$L_i=\frac{1}{{\varphi }_i}\prod _{\stackrel{j=0}{j\ne i}}^d(x-{\theta }_j)$$

$${\mbox{\boldmath$E$}}_{i}=L_{i}({\mbox{\boldmath$A$}})=\frac{1}{\phi_{i}}\prod_{\stackrel{{\scriptstyle j=0}}{{j\neq i}}}^{d}({\mbox{\boldmath$A$}}-\theta_{j}{\mbox{\boldmath$I$}}),$$

$$m_{u}(\theta_{i})=\|{\mbox{\boldmath$E$}}_{i}{\mbox{\boldmath$e$}}_{u}\|^{2}=\langle{\mbox{\boldmath$E$}}_{i}{\mbox{\boldmath$e$}}_{u},{\mbox{\boldmath$e$}}_{u}\rangle=({\mbox{\boldmath$E$}}_{i})_{uu},\qquad u\in V,\ i=0,1,\dots,d.$$

$$\sum _{i=0}^d{m}_u({\theta }_i)$$

$$\sum _{u\in V}{m}_u({\theta }_i)$$

$${\mathrm{sp}}_u\Gamma =\{{\mu }_0^{m_u({\mu }_0)},{\mu }_1^{m_u({\mu }_1)},\dots ,{\mu }_{d_u}^{m_u({\mu }_{d_u})}\}.$$

$$m_{uv}(\theta_{i})=\langle{\mbox{\boldmath$E$}}_{i}{\mbox{\boldmath$e$}}_{u},{\mbox{\boldmath$E$}}_{i}{\mbox{\boldmath$e$}}_{v}\rangle=\langle{\mbox{\boldmath$E$}}_{i}{\mbox{\boldmath$e$}}_{u},{\mbox{\boldmath$e$}}_{v}\rangle=({\mbox{\boldmath$E$}}_{i})_{uv},\qquad u,v\in V,\ i=0,1,\dots,d.$$

$$a_{uv}^{({\ell})}=({\mbox{\boldmath$A$}}^{\ell})_{uv}=\sum_{i=0}^{d}m_{uv}(\theta_{i})\theta_{i}^{\ell},\qquad\ell=0,1,\dots$$

$$({\mbox{\boldmath$A$}}^{\ell})_{uv}=a_{j}^{(\ell)}=\frac{1}{|V_{j}|}({\mbox{\boldmath$B$}}^{\ell})_{1j}.$$

$$({\mbox{\boldmath$A$}}^{\ell+1})_{uv}=\sum_{h=1}^{m}\sum_{w\in\Gamma(v)\cap V_{h}}({\mbox{\boldmath$A$}}^{\ell})_{uw}=\sum_{h=1}^{m}b_{jh}a_{h}^{(\ell)}=a^{(\ell+1)}_{j},$$

$$\overline{{\mbox{\boldmath$E$}}}_{i}=\overline{L}_{i}({\mbox{\boldmath$B$}})=\frac{1}{\displaystyle\prod_{\stackrel{{\scriptstyle j=0}}{{j\neq i}}}^{e}(\tau_{i}-\tau_{j})}\prod_{\stackrel{{\scriptstyle j=0}}{{j\neq i}}}^{e}({\mbox{\boldmath$B$}}-\tau_{j}{\mbox{\boldmath$I$}}).$$

$$m_{uv}({\theta }_i)$$

$$\displaystyle=\frac{1}{|V_{j}|}\sum_{r=0}^{d}\zeta_{r}({\mbox{\boldmath$B$}}^{r})_{1j}=\frac{1}{|V_{j}|}(L_{i}({\mbox{\boldmath$B$}}))_{1j},$$

$$a_{uv}^{(\ell)}=\frac{1}{|V_{j}|}\sum_{i=0}^{e}\tau_{i}^{\ell}({\mbox{\boldmath$E$}}_{i})_{1j},\qquad\ell=0,\ldots,d,$$

$$\sum _{i=0}^d{\theta }_i^{\ell }{m}_{uv}({\theta }_i)=a_{uv}^{(\ell )},\ell =0,\dots ,d.$$

$${\mathrm{sp}}_u\Gamma =\{{\tau }_0^{m_u({\tau }_0)},{\tau }_1^{m_u({\tau }_1)},\dots ,{\tau }_e^{m_u({\tau }_e)}\},$$

$${\mbox{\boldmath$B$}}=\left(\begin{array}[]{cc}0&n\\ 1&k\end{array}\right)$$

$${\mathrm{sp}}_u\Delta =\left\{\frac{1}{2}{\left(k+\sqrt{k^2+4n}\right)}^{m_u({\theta }_0)},\frac{1}{2}{\left(k-\sqrt{k^2+4n}\right)}^{m_u({\theta }_1)}\right\},$$

$$m_{uv}(\tau_{i})=\frac{1}{|V_{j}|}\frac{v_{1i}u_{ij}}{\langle{\mbox{\boldmath$u$}}_{i},{\mbox{\boldmath$v$}}_{i}\rangle},\qquad j=1,\ldots,m.$$

$$m_{uv}(\tau_{i})=\frac{1}{|V_{j}|}(\mbox{\boldmath$Q$}^{-1})_{ij}(\mbox{\boldmath$Q$})_{1i},\qquad j=1,\ldots,m.$$

$$\mathrm{sp}\Gamma =\{{\theta }_0^{m({\theta }_0)},{\theta }_1^{m({\theta }_1)},\dots ,{\theta }_d^{m({\theta }_d)}\},$$

$$\textstyle m(\theta_{i})=n(\overline{{\mbox{\boldmath$E$}}}_{i})_{11}=\frac{n}{\prod_{\stackrel{{\scriptstyle j=0}}{{j\neq i}}}^{d}(\theta_{i}-\theta_{j})}\left(\prod_{\stackrel{{\scriptstyle j=0}}{{j\neq i}}}^{d}({\mbox{\boldmath$B$}}-\theta_{j}{\mbox{\boldmath$I$}})\right)_{11}.$$

$$\scriptstyle\displaystyle m(\theta_{0})=\frac{12}{(\theta_{0}-\theta_{1})(\theta_{0}-\theta_{2})(\theta_{0}-\theta_{3})}\left(({\mbox{\boldmath$\scriptstyle B$}}-\theta_{1}{\mbox{\boldmath$\scriptstyle I$}})({\mbox{\boldmath$\scriptstyle B$}}-\theta_{2}{\mbox{\boldmath$\scriptstyle I$}})({\mbox{\boldmath$\scriptstyle B$}}-\theta_{3}{\mbox{\boldmath$\scriptstyle I$}})\right)_{11}=1,$$

$$\scriptstyle\displaystyle m(\theta_{1})=\frac{12}{(\theta_{1}-\theta_{0})(\theta_{1}-\theta_{2})(\theta_{1}-\theta_{3})}\left(({\mbox{\boldmath$\scriptstyle B$}}-\theta_{0}{\mbox{\boldmath$\scriptstyle I$}})({\mbox{\boldmath$\scriptstyle B$}}-\theta_{2}{\mbox{\boldmath$\scriptstyle I$}})({\mbox{\boldmath$\scriptstyle B$}}-\theta_{3}{\mbox{\boldmath$\scriptstyle I$}})\right)_{11}=3,$$

$$\scriptstyle\displaystyle m(\theta_{2})=\frac{12}{(\theta_{2}-\theta_{0})(\theta_{2}-\theta_{1})(\theta_{2}-\theta_{3})}\left(({\mbox{\boldmath$\scriptstyle B$}}-\theta_{0}{\mbox{\boldmath$\scriptstyle I$}})({\mbox{\boldmath$\scriptstyle B$}}-\theta_{1}{\mbox{\boldmath$\scriptstyle I$}})({\mbox{\boldmath$\scriptstyle B$}}-\theta_{3}{\mbox{\boldmath$\scriptstyle I$}})\right)_{11}=3,$$

$$\scriptstyle\displaystyle m(\theta_{3})=\frac{12}{(\theta_{3}-\theta_{0})(\theta_{3}-\theta_{1})(\theta_{3}-\theta_{2})}\left(({\mbox{\boldmath$\scriptstyle B$}}-\theta_{0}{\mbox{\boldmath$\scriptstyle I$}})({\mbox{\boldmath$\scriptstyle B$}}-\theta_{1}{\mbox{\boldmath$\scriptstyle I$}})({\mbox{\boldmath$\scriptstyle B$}}-\theta_{2}{\mbox{\boldmath$\scriptstyle I$}})\right)_{11}=5.$$

$$m(\theta_{i})=n(\overline{{\mbox{\boldmath$E$}}}_{i})_{11}=\frac{n}{\langle{\mbox{\boldmath$u$}}_{i},{\mbox{\boldmath$v$}}_{i}\rangle},\qquad j=1,\ldots,m,$$

$${\mbox{\boldmath$B$}}_{1}=\left(\begin{array}[]{ccccc}0&2n&0&0&0\\ 1&0&4n-2&0&0\\ 0&n&0&n&0\\ 0&0&4n-2&0&1\\ 0&0&0&2n&0\end{array}\right),\qquad{\mbox{\boldmath$B$}}_{2}=\left(\begin{array}[]{cccc}0&4n-1&0&0\\ 1&0&2n-1&0\\ 0&2n-1&0&2n\\ 0&0&2n&0\end{array}\right),$$

$$m({\theta }_i)=\sum _{u\in V_1}{m}_u({\theta }_i)+\sum _{v\in V_2}{m}_v({\theta }_i)=(8n-2)\cdot m_u({\theta }_i)+4n\cdot m_v({\theta }_i).$$

$$\mathrm{sp}Hb(n)=\{{\sqrt{8n^2-2n}}^1,{\sqrt{2n}}^{4n-1},0^{4n-2},-{\sqrt{2n}}^{4n-1},-{\sqrt{8n^2-2n}}^1\}.$$

Full text

Keywords: Graph; Adjacency matrix; Spectrum; Eigenvalues; Local multiplicities; Walk-regular graph.

2010 Mathematics Subject Classification: 05E30, 05C50.

††

The first author has also received funding from the European Union’s Horizon 2020 research and innovation programme under the Marie Skłodowska-Curie grant agreement No 734922.

1 Preliminaries

First, let us recall some basic concepts and define our generic notation for graphs.

1.1 Some notions on graphs and their spectra

Throughout this paper, $\Gamma=(V,E)$ denotes a simple and connected graph with order $n=|V|$, size $m=|E|$, and adjacency matrix $A$. The distance between two vertices $u$ and $v$ is denoted by $\mathop{\rm dist}\nolimits(u,v)$, so that the eccentricity of a vertex $u$ is $\mathop{\rm ecc}\nolimits(u)=\max_{v\in V}\mathop{\rm dist}\nolimits(u,v)$, and the diameter of the graph is $D=\max_{u\in V}\mathop{\rm ecc}\nolimits(u)$. The set of vertices at distance $i$, from a given vertex $u\in V$ is denoted by $\Gamma_{i}(u)$, for $i=0,1,\dots,D$, and we write $\Gamma(u)=\Gamma_{1}(u)$ for short. The degree of a vertex $u$ is denoted by $\delta(u)=|\Gamma_{1}(u)|$. The distance-$i$ graph $\Gamma_{i}$ is the graph with vertex set $V$, and where two vertices $u$ and $v$ are adjacent if and only if $\mathop{\rm dist}\nolimits(u,v)=i$ in $\Gamma$. Its adjacency matrix {\mbox{\boldmathA}}_{i} is usually referred to as the distance-$i$ matrix of $\Gamma$. The spectrum of a graph $\Gamma$ of its adjacency matrix {\mbox{\boldmathA}}(={\mbox{\boldmathA}}_{1}) is denoted by

[TABLE]

where the different eigenvalues of $\Gamma$, whose set is denoted by $\mathop{\rm ev}\nolimits\Gamma$, are in decreasing order, $\theta_{0}>\theta_{1}>\cdots>\theta_{d}$, and the superscripts stand for their multiplicities $m(\theta_{i})$ for $i=0,\ldots,d$. In particular, note that $m(\theta_{0})=1$, since $\Gamma$ is connected, and $m(\theta_{0})+m(\theta_{1})+\cdots+m(\theta_{d})=n$. Alternatively, if we include repetitions, the eigenvalues of $\Gamma$ are denoted as $\lambda_{1}\geq\lambda_{2}\geq\cdots\geq\lambda_{n}.$

1.2 Projections and local spectra

For any graph with eigenvalue $\theta_{i}$ having multiplicity $m(\theta_{i})$, its corresponding $($principal$)$ idempotent can be computed as {\mbox{\boldmathE}}_{i}=\mbox{\boldmathV}_{i}\mbox{\boldmathV}_{i}^{\top}, where \mbox{\boldmathV}_{i} is the $n\times m(\theta_{i})$ matrix whose columns form an orthonormal basis of the eigenspace {\cal E}_{i}=\mathop{\rm Ker}\nolimits({\mbox{\boldmathA}}-\theta_{i}{\mbox{\boldmathI}}). For instance, when $\Gamma$ is a $\delta$-regular graph on $n$ vertices, its largest eigenvalue $\theta_{0}=\delta$ has eigenvector $j$, the all-$1$ (column) vector, and corresponding idempotent {\mbox{\boldmathE}}_{0}=\frac{1}{n}{\mbox{\boldmathj}}{\mbox{\boldmathj}}^{\top}=\frac{1}{n}{\mbox{\boldmathJ}}, where $J$ is the all-$1$ matrix.

Alternatively, for every $i=0,1,\dots,d$, the orthogonal projection of $\mathbb{R}^{n}$ onto the eigenspace ${\cal E}_{i}$ is given by the Lagrange interpolating polynomial

[TABLE]

of degree $d$, where $\phi_{i}=\prod_{j=0,j\neq i}^{d}(\theta_{i}-\theta_{j})$. These polynomials satisfy $L_{i}(\theta_{i})=1$ and $L_{i}(\theta_{j})=0$ for $j\neq i$. The idempotents are, then,

[TABLE]

and they are known to satisfy the following properties (see, for instance, Godsil [9, p. 28]):

• $(a)$

{\mbox{\boldmathE}}_{i}{\mbox{\boldmathE}}_{j}=\delta_{ij}{\mbox{\boldmathE}}_{i};

• $(b)$

{\mbox{\boldmathA}}{\mbox{\boldmathE}}_{i}=\theta_{i}{\mbox{\boldmathE}}_{i};

• $(c)$

p({\mbox{\boldmathA}})=\displaystyle\sum_{i=0}^{d}p(\theta_{i}){\mbox{\boldmathE}}_{i}, for any polynomial $p\in\mathbb{R}[x]$.

In particular, when $p(x)=x$ in $(c)$, we have the so-called spectral decomposition theorem: {\mbox{\boldmathA}}=\sum_{i=0}^{d}\theta_{i}{\mbox{\boldmathE}}_{i}. The $(u$-$)$local multiplicities of the eigenvalue $\theta_{i}$, introduced by Fiol and Garriga in [7], were defined as the square norm of the projection of {\mbox{\boldmathe}}_{u} onto the eigenspace ${\cal E}_{i}$, where {\mbox{\boldmathe}}_{u} is the unitary characteristic vector of a vertex $u\in V$. That is,

[TABLE]

Notice that, in fact, $m_{u}(\theta_{i})=\cos^{2}\beta_{ui}$, where $\beta_{ui}$ is the angle between {\mbox{\boldmathe}}_{u} and {\mbox{\boldmathE}}_{i}{\mbox{\boldmathe}}_{u}. The values $\cos\beta_{ui}$, for $u\in V$ and $i=0,\ldots,d$, were formally introduced by Cvetković as the ‘angles’ of $\Gamma$ (see, for instance, Cvetković and Doob [2]).

The local multiplicities can be seen as a generalization of the (standard) multiplicities when the graph is ‘seen’ from the ‘base vertex’ $u$. Indeed, they satisfy the following properties (see Fiol and Garriga [7]):

[TABLE]

If $\mu_{0}(=\theta_{0})>\mu_{1}>\cdots>\mu_{d_{u}}$ represent the eigenvalues of $\Gamma$ with non-null $u$-local multiplicity, we define the $u$-local spectrum of $\Gamma$ as

[TABLE]

By analogy with the local multiplicities, which correspond to the diagonal entries of the idempotents, Fiol, Garriga, and Yebra [8] defined the crossed $(uv$-$)$local multiplicities of the eigenvalue $\theta_{i}$, denoted by $m_{uv}(\theta_{i})$, as

[TABLE]

(Thus, in particular, $m_{uu}(\theta_{i})=m_{u}(\theta_{i})$.) These parameters allow us to compute the number of walks of length $\ell$ between two vertices $u,v$ in the following way:

[TABLE]

Conversely, the values $a_{uv}^{(\ell)}$, for $\ell=0,1,\ldots,d$, determine the crossed local multiplicities $m_{uv}(\theta_{i})$. (Indeed, notice that the coefficients of the system in (4) are the entries of a Vandermonde matrix).

2 Regular partitions and local spectra

Let $\Gamma=(V,E)$ be a graph with adjacency matrix $A$. A partition $\pi=(V_{1},\ldots,V_{m})$ of its vertex set $V$ is called regular (or equitable) whenever, for any $i,j=1,\ldots,m$, the intersection numbers $b_{ij}(u)=|\Gamma(u)\cap V_{j}|$, where $u\in V_{i}$, do not depend on the vertex $u$ but only on the subsets (usually called classes or cells) $V_{i}$ and $V_{j}$. In this case, such numbers are simply written as $b_{ij}$, and the $m\times m$ matrix {\mbox{\boldmathB}}=(b_{ij}) is referred to as the quotient matrix of $A$ with respect to $\pi$. This is also represented by the quotient (weighted) graph $\pi(\Gamma)$ (associated to the partition $\pi$), with vertices representing the cells, and there is an edge with weight $b_{ij}$ between vertex $V_{i}$ and vertex $V_{j}$ if and only if $b_{ij}\neq 0$. Of course, if $b_{ii}>0$, for some $i=1,\ldots,m$, the quotient graph $\pi(\Gamma)$ has loops.

The characteristic matrix of (any) partition $\pi$ is the $n\times m$ matrix \mbox{\boldmathS}=(s_{ui}) whose $i$-th column is the characteristic vector of $V_{i}$, that is, $s_{ui}=1$ if $u\in V_{i}$, and $s_{ui}=0$ otherwise. In terms of this matrix, we have the following characterization of regular partitions (see Godsil [9]).

Lemma 2.1** ([9]).**

Let $\Gamma=(V,E)$ be a graph with adjacency matrix $A$, and vertex partition $\pi$ with characteristic matrix $S$. Then, $\pi$ is regular if and only if there exists an $m\times m$ matrix $C$ such that \mbox{\boldmathS}{\mbox{\boldmathC}}={\mbox{\boldmathA}}\mbox{\boldmathS}. Moreover, {\mbox{\boldmathC}}={\mbox{\boldmathB}}, the quotient matrix of $A$ with respect to $\pi$.

Using the above lemma, it can be proved that \mathop{\rm sp}\nolimits{\mbox{\boldmathB}}\subseteq\mathop{\rm sp}\nolimits{\mbox{\boldmathA}}. Moreover, we have the following result by the authors [3].

Lemma 2.2** ([3]).**

Let $\Gamma$ be a graph with adjacency matrix $A$. Let $\pi=(V_{1},\ldots,V_{m})$ be a regular partition of $\Gamma$, with quotient matrix $B$. Then, the number of $\ell$-walks from any vertex $u\in V_{i}$ to all vertices of $V_{j}$ is the $ij$-entry $b_{ij}^{(\ell)}$ of {\mbox{\boldmathB}}^{\ell}.

By using the last lemma, now we have the next result.

Lemma 2.3**.**

Let $\Gamma$ be a graph with adjacency matrix $A$, and $\pi=(V_{1},\ldots,V_{m})$ a regular partition of $\Gamma$ with quotient matrix $B$. If $V_{1}=\{u\}$, then the number of $\ell$-walks from vertex $u$ to a vertex $v\in V_{j}$, for $j=1,\ldots,m$, only depends on $j$:

[TABLE]

Proof.

To prove that the number of $\ell$-walks between $u$ and $v\in V_{j}$ is a constant, we use induction on $\ell$. The result is clearly true for $\ell=0$, since {\mbox{\boldmathB}}^{0}={\mbox{\boldmathI}}, and for $\ell=1$ because of the definition of $B$. Suppose that the result holds for some $\ell>1$. Then, the set of walks of length $\ell+1$ from $u$ to $v\in V_{j}$ is obtained from the set of $\ell$-walks from $u$ to vertices $w\in V_{h}$ adjacent to $v$. Then, the number of these walks is

[TABLE]

as claimed. ∎

Let $B$ be a quotient (diagonalizable) $m\times m$ matrix as above, with \mathop{\rm sp}\nolimits{\mbox{\boldmathB}}=\{\tau_{0}^{m(\tau_{0})},\tau_{1}^{m(\tau_{1})}, $\ldots,\tau_{e}^{m(\tau_{e})}\}$, and let {\mbox{\boldmathD}}=\mathop{\rm diag}\nolimits(\tau_{0},\tau_{1},\ldots,\tau_{e}). Let $Q$ be the $m\times m$ matrix that diagonalizes $B$, that is, \mbox{\boldmathQ}^{-1}{\mbox{\boldmathB}}\mbox{\boldmathQ}={\mbox{\boldmathD}}. For $i=0,\ldots,e$, let \mbox{\boldmathV}_{i} be the $m\times m(\tau_{i})$ matrix formed by the columns of $Q$ corresponding to the right $\tau_{i}$-eigenvectors of $B$. Let \mbox{\boldmathU}_{i} be the $m(\tau_{i})\times m$ matrix formed by the corresponding rows of \mbox{\boldmathQ}^{-1}, which are the left $\tau_{i}$-eigenvectors of $B$. Then, the $i$-th idempotent of $B$ is \overline{{\mbox{\boldmathE}}}_{i}=\mbox{\boldmathV}_{i}\mbox{\boldmathU}_{i}. Moreover, \overline{{\mbox{\boldmathE}}}_{i} can be computed as in the case of the (symmetric) adjacency matrix by using the Lagrange interpolating polynomial $\overline{L}_{i}$ satisfying $\overline{L}_{i}(\tau_{j})=\delta_{ij}$:

[TABLE]

The above results yield a simple method to compute the local spectra of a vertex $u$ in a given regular partition or, more generally, the crossed multiplicities between $u$ and any other vertex $v$. Besides, with the union of the local spectra (applying (3)) of the different classes of vertices according to their corresponding regular partitions (that is, we ‘hung’ the quotient graph from every one of the different classes of vertices), we obtain all the spectrum of the original graph. Thus, the main result is the following.

Theorem 2.4**.**

Let $\Gamma$ be a graph with adjacency matrix $A$ and set of different eigenvalues $\mathop{\rm ev}\nolimits\Gamma=\{\theta_{0},\theta_{1},\ldots,\theta_{d}\}$. Let $\pi=(V_{1},\ldots,V_{m})$ be a regular partition of $\Gamma$, with $V_{1}=\{u\}$. Let $B$ be the quotient matrix of $\pi$, with set of different eigenvalues \mathop{\rm ev}\nolimits{\mbox{\boldmathB}}=\{\tau_{0},\tau_{1},\ldots,\tau_{e}\}\subseteq\mathop{\rm ev}\nolimits\Gamma. Let $L_{i}$ and $\overline{L}_{i}$ be the Lagrange interpolating polynomials satisfying $L_{i}(\theta_{j})=\delta_{ij}$ for $i,j=0,\ldots,d$, and $\overline{L}_{i}(\tau_{j})=\delta_{ij}$ for $i,j=0,\ldots,e$, respectively. Let {\mbox{\boldmathE}}_{i}=L_{i}({\mbox{\boldmathA}}) and \overline{{\mbox{\boldmathE}}}_{i}=\overline{L}_{i}({\mbox{\boldmathB}}) be the corresponding idempotents. Then, for every vertex $v\in V_{j}$, the crossed $uv$-local multiplicity of $\theta_{i}$ is

[TABLE]

or, alternatively,

[TABLE]

Proof.

Let $L_{i}(x)=\sum_{r=0}^{d}\zeta_{r}x^{r}$. Then, for every $v\in V_{j}$ and $i=0,1,\ldots,d$, and using Lemma 2.3, we have

[TABLE]

which proves (6). To prove (9), note first that, by the spectral decomposition theorem, {\mbox{\boldmathB}}^{r}=\sum_{i=0}^{e}\tau_{i}^{r}\overline{{\mbox{\boldmathE}}}_{i}. Then, by Lemma 2.3, the numbers of $\ell$-walks from $u$ to $v\in V_{j}$ are

[TABLE]

which, as already commented, determine the local multiplicities because of the system of equations

[TABLE]

But a (the) solution of this system is obtained when the multiplicities $m_{uv}(\theta_{i})$, for $i=0,\ldots,d$, are given by (9), as claimed. ∎

In particular, notice that this result allows us to compute the $u$-local spectrum of $\Gamma$ as

[TABLE]

where \tau_{i}\in\mathop{\rm ev}\nolimits{\mbox{\boldmathB}} and m_{u}(\tau_{i})=(\overline{{\mbox{\boldmathE}}}_{i})_{11}, for $i=0,\ldots,e$.

Let us show an example.

Example 1**.**

Let $\Delta=\Gamma+u$ be the cone of a $k$-regular graph on $n$ vertices, where the ‘new’ vertex $u$ is joined to all vertices of $\Gamma$. Then, $\Delta$ has a regular partition with quotient matrix

[TABLE]

and eigenvalues $\theta_{0}=\frac{1}{2}(k+\sqrt{k^{2}+4n})$ and $\theta_{1}=\frac{1}{2}(k-\sqrt{k^{2}+4n})$. (Notice that the first expression can be rewritten as $k=\theta_{0}-\frac{\theta_{0}}{n}$, in agreement with the results of Dalfó, Fiol, and Garriga [4].) Thus, the idempotents of $B$ turn out to be \overline{{\mbox{\boldmathE}}}_{0}=({\mbox{\boldmathB}}-\theta_{1}{\mbox{\boldmathI}})/(\theta_{0}-\theta_{1}) and \overline{{\mbox{\boldmathE}}}_{1}=({\mbox{\boldmathB}}-\theta_{0}{\mbox{\boldmathI}})/(\theta_{1}-\theta_{0}). Consequently, Theorem 2.4 implies that the local $u$-spectrum of $\Delta$ is

[TABLE]

where $m_{u}(\theta_{0})=\frac{1}{2}(1-k/\sqrt{k^{2}+4n})$ and $m_{u}(\theta_{1})=\frac{1}{2}(1+k/\sqrt{k^{2}+4n})$.

Another simple consequence of our main result is obtained for simple eigenvalues of $B$.

Corollary 2.5**.**

Let $\Gamma$ be a graph with adjacency matrix $A$ and a set of different eigenvalues $\mathop{\rm ev}\nolimits\Gamma=\{\theta_{0},\theta_{1},\ldots,\theta_{d}\}$, $\pi=(V_{1},\ldots,V_{m})$ a regular partition of $\Gamma$ with $V_{1}=\{u\}$, $B$ the quotient matrix of $\pi$ with a set of different eigenvalues \mathop{\rm ev}\nolimits{\mbox{\boldmathB}}=\{\tau_{0},\tau_{1},\ldots,\tau_{e}\}\subseteq\mathop{\rm ev}\nolimits\Gamma, and {\mbox{\boldmathE}}_{i} and \overline{{\mbox{\boldmathE}}}_{i} the corresponding idempotents. Suppose that, for some $i$, \theta_{i}\in\mathop{\rm ev}\nolimits{\mbox{\boldmathA}}\cap\mathop{\rm ev}\nolimits{\mbox{\boldmathB}} has multiplicity $1$. Let {\mbox{\boldmathu}}_{i}=(u_{i1},\ldots,u_{im}) and {\mbox{\boldmathv}}_{i}=(v_{1i},\ldots,v_{im})^{\top} be the left and right eigenvectors of $B$, respectively, corresponding to the eigenvalue $\theta_{i}$. Then, for every vertex $v\in V_{j}$, the crossed $uv$-local multiplicity of $\theta_{i}$ in $\Gamma$ is

[TABLE]

Proof.

Let $Q$ be a matrix that diagonalizes $B$. If, for some constants $\alpha$ and $\beta$, we have that \alpha{\mbox{\boldmathu}}_{i} and \beta{\mbox{\boldmathv}}_{i} are the corresponding row of \mbox{\boldmathQ}^{-1} and column of $Q$, respectively, then (\mbox{\boldmathQ}^{-1}\mbox{\boldmathQ})_{ii}=1 implies that \alpha\beta=\langle{\mbox{\boldmathu}}_{i},{\mbox{\boldmathv}}_{i}\rangle^{-1}. Thus, (\overline{{\mbox{\boldmathE}}}_{i})_{1j}=(\alpha{\mbox{\boldmathv}}_{i}\cdot\beta{\mbox{\boldmathu}}_{i})_{1j}=\frac{v_{1i}u_{ij}}{\langle{\mbox{\boldmathu}}_{i},{\mbox{\boldmathv}}_{i}\rangle} (where ‘$\cdot$’ stands for the matrix product), and the result follows from (9). ∎

Alternatively, if {\mbox{\boldmathu}}_{i} and {\mbox{\boldmathv}}_{i} are already taken from the corresponding row and column of \mbox{\boldmathQ}^{-1} and $Q$, respectively, then $\alpha=\beta=1$, and (11) can be simply written as

[TABLE]

3 The local spectra of some families of graphs

In this last section, we show the application of our method to obtain the local spectra and the complete spectrum of different well-known families of graphs.

3.1 Walk-regular graphs

Let $\Gamma$ be a graph with spectrum as above. If the number of closed walks of length ${\ell}$ rooted at vertex $u$, that is, $a_{uu}^{({\ell})}=\sum_{i=0}^{d}m_{u}(\theta_{i})\theta_{i}^{\ell}$ only depends on ${\ell}$, for each $\ell\geq 0$, then $\Gamma$ is called walk-regular (a concept introduced by Godsil and McKay [10]). In this case, we write $a_{uu}^{({\ell})}=a^{({\ell})}$. Note that, since $a_{uu}^{(2)}=\delta(u)$, the degree of vertex $u$, a walk-regular graph is necessarily regular. Moreover, we say that $\Gamma$ is spectrum-regular if, for any $i=0,1,\ldots,d$, the $u$-local multiplicity of $\theta_{i}$ does not depend on the vertex $u$. By (4) and the subsequent comment, it follows that spectrum-regularity and walk-regularity are equivalent concepts. Equation (4) also shows that the existence of the constants $a^{(0)},a^{(1)},\ldots,a^{(d)}$ suffices to assure walk-regularity. It is well known that any distance-regular graph, as well as any vertex-transitive graph, is walk-regular, but the converse is not true.

Proposition 3.1**.**

Let $\Gamma$ be a walk-regular graph with $n$ vertices, having a regular partition $\pi=(V_{1},\ldots,V_{m})$ with $V_{1}=\{u\}$ and quotient matrix $B$. Then, the spectrum of $\Gamma$ is

[TABLE]

where, for every $i=0,\ldots,d$, $\theta_{i}$ also is an eigenvalue of $B$, with multiplicity

[TABLE]

Proof.

Since $\sum_{u\in V}m_{uu}(\theta_{i})=m(\theta_{i})$, the (standard) multiplicity $m(\theta_{i})$ ‘splits’ equitably among the $n$ vertices, giving $m_{u}(\theta_{i})=m(\theta_{i})/n$. Therefore, the different eigenvalues of $B$ coincide with those of $B$, and Theorem 2.4 yields (13). ∎

Let us see an example.

Example 2**.**

Consider the walk-regular graph $\Gamma$, that is not distance-regular, given by Godsil [9]. This graph and its quotient $\pi(\Gamma)$ are represented in Figure 1. The spectrum of $\pi(\Gamma)$ is $\mathop{\rm sp}\nolimits\pi(\Gamma)=\{\theta_{0}^{m(\tau_{0})},\theta_{1}^{m(\tau_{1})},\theta_{2}^{m(\tau_{2})},\theta_{3}^{m(\tau_{3})}\}=\{4^{1},2^{2},0^{1},-2^{3},\}$. Since $\Gamma$ is walk-regular, the spectrum of its quotient from a regular partition has all the different eigenvalues of the spectrum of $\Gamma$. Now we compute the multiplicities $m(\theta_{i})$, for $i=0,1,2,3$.

[TABLE]

This gives that the spectrum of $\Gamma$ is $\mathop{\rm sp}\nolimits\Gamma=\{4^{1},2^{3},0^{3},-2^{5},\}$, as it is known to be. Note that, in this example, we need to ‘hang’ the graph $\Gamma$ from only one of its vertices.

3.2 Distance-regular graphs

In particular, when $\Gamma$ is distance-regular, the distance-partition with respect to any vertex is regular with the same quotient matrix $B$ (see, for instance, Biggs [1] or Fiol [6]). Moreover, since $B$ is tridiagonal, all its eigenvalues are simple and (13), together with (11), leads to the known formula (see Biggs [1])

[TABLE]

where the eigenvectors {\mbox{\boldmathu}}_{i} and {\mbox{\boldmathv}}_{i} have been chosen to have the first entry $1$.

3.3 Distance-biregular graphs

Distance-biregular graphs are defined in a similar way as distance-regular graphs. They are connected bipartite graphs in which each of the two classes of vertices has its own intersection array. It was proved by Godsil and Shawe-Taylor [11] that all vertices in the same bipartition class have the same intersection array. Delorme [5] gave the basic properties and some new examples of distance-biregular graphs.

Let us give an example.

Example 3**.**

A Hadamard matrix $H$ (with entries $\pm 1$ and mutually orthogonal rows) with size $4n$ gives a bipartite distance-biregular graph $Hb(n)$ on $12n-2$ vertices (see, for instance, Wallis [12, p. 426]). Its stable sets $V_{1}$ and $V_{2}$ have $4n$ and $8n-2$ vertices, respectively. For instance, $Hb(1)$ is the subdivided complete graph $K_{4}$. The quotient graphs corresponding to the regular distance-partitions with respect to vertices in $V_{1}$ and $V_{2}$ are shown in Figure 2. Thus, the respective quotient matrices are

[TABLE]

with (simple) eigenvalues \mathop{\rm ev}\nolimits{\mbox{\boldmathB}}_{1}=\{\sqrt{8n^{2}-1},\sqrt{2n},0,-\sqrt{2n},-\sqrt{8n^{2}-1}\} and \mathop{\rm ev}\nolimits{\mbox{\boldmathB}}_{2}=\mathop{\rm ev}\nolimits{\mbox{\boldmathB}}_{1}\setminus\{0\}. Then, according to Theorem 2.4, we can compute all the local (crossed) multiplicities of vertices in each stable set from the idempotents of {\mbox{\boldmathB}}_{1} and {\mbox{\boldmathB}}_{2}. The results obtained are shown in Table 1 (for $u\in V_{1}$) and Table 2 (for $u\in V_{2}$), where the last row in both tables corresponds to the sums in (4) for $\ell=0$ (or, for the case $u=v$, to (2)). Moreover, from the columns of local multiplicities ($\mathop{\rm dist}\nolimits(u,v)=0$), we can find the (global) multiplicities by using (3), which in our case becomes

[TABLE]

Then, the complete spectrum of the Hadamard distance-biregular graph turns out to be

[TABLE]