Spectra of comb graphs with tails

TL;DR

This paper conducts a detailed spectral analysis of comb graphs formed by paths and infinite rays, exploring their eigenvalues and spectral properties in finite and infinite cases.

Contribution

It provides a comprehensive spectral analysis of comb graphs with path and ray components, including cases with infinite rays attached.

Findings

Spectral properties of finite comb graphs with path components.

Spectral analysis of comb graphs with infinite rays.

Eigenvalue distributions in various comb graph configurations.

Abstract

Given two graphs, a backbone and a finger, a comb product is a new graph obtained by grafting a copy of the finger into each vertex of the backbone. We study the comb graphs in the case when both components are the paths of order $n$ and $k$, respectively, as well as the above comb graphs with an infinite ray attached to some of their vertices. A detailed spectral analysis is carried out in both situations.

Tables

Table 1. Table 1. Cardinality of positive discrete spectrum σ d + ​ ( H n , k ) superscript subscript 𝜎 𝑑 subscript 𝐻 𝑛 𝑘 \sigma_{d}^{+}(H_{n,k})

$(k,n)$	2	3	4	5	6	7	8	9	10	11	12	13	14	15	16	17	18	19	20
2	1	1	1	1	1	2	2	2	2	3	3	3	3	4	4	4	4	4	5
3	1	1	1	1	2	2	2	3	3	3	3	4	4	4	4	5	5	5	5
4	1	1	1	2	2	2	2	3	3	3	4	4	4	4	5	5	5	6	6
5	1	1	1	2	2	2	3	3	3	3	4	4	4	5	5	5	5	6	6
6	1	1	1	2	2	2	3	3	3	3	4	4	4	5	5	5	6	6	6

Equations

$$A(\Gamma)=\|a_{ij}\|_{i,j=1}^{\omega},\quad a_{ij}=\left\{\begin{array}[]{ll}1,&\{i,j\}\in{\mathcal{E}}(\Gamma);\\ 0,&\hbox{otherwise.}\end{array}\right.$$

$$A(\Gamma )e_k=\sum _{j\sim k}{e}_j,k=1,2,\dots ,\omega .$$

$$A({\mathbb{P}}_n⊳\Gamma )=\left[\begin{array}{cccccc}A(\Gamma )& E& & & & \\ E& A(\Gamma )& E& & & \\ & \ddots & \ddots & \ddots & & \\ & & & A(\Gamma )& E& \\ & & & E& A(\Gamma )& \end{array}\right],E=\Vert E_{ij}{\Vert }_{i,j=1}^k,$$

$$0_{11}⟶J_n=\left[\begin{array}{cccccc}0& 1& & & & \\ 1& 0& 1& & & \\ & \ddots & \ddots & \ddots & & \\ & & \ddots & 0& 1& \\ & & & 1& 0& \end{array}\right],$$

$$\mathcal{V}(\Gamma )=\mathcal{V}({\Gamma }_1)\cup \mathcal{V}({\Gamma }_2),\mathcal{E}(\Gamma )=\mathcal{E}({\Gamma }_1)\cup \mathcal{E}({\Gamma }_2)\cup \{v_1,v_2\}.$$

$$A(\Gamma )=\left[\begin{array}{cc}A({\Gamma }_1)& E\\ E^{\ast }& A({\Gamma }_2)\end{array}\right],E=\left[\begin{array}{cccc}1& 0& 0& \dots \\ 0& 0& 0& \dots \\ 0& 0& 0& \dots \\ ⋮& ⋮& ⋮& \end{array}\right].$$

$$A(\Gamma )=\left[\begin{array}{cc}A({\Gamma }_1)& E_m\\ E_m^{\ast }& A({\Gamma }_2)\end{array}\right],E_m=\left[\begin{array}{ccccc}1& 0& 0& \dots & \\ 0& 0& 0& \dots & \\ ⋮& ⋮& ⋮& & \\ 0& 0& 0& \dots & \end{array}\right]$$

$$A({\Gamma }_2)=J_{\infty }:=\left[\begin{array}{ccccc}0& 1& 0& 0& \\ 1& 0& 1& 0& \\ 0& 1& 0& 1& \\ & \ddots & \ddots & \ddots & \ddots \end{array}\right]$$

$$H_{n,k}:={\Gamma }_{n,k}+{\mathbb{P}}_{\infty },{\Gamma }_{n,k}:={\mathbb{P}}_n⊳{\mathbb{P}}_k,n,k\ge 2.$$

$$\begin{split}\bigl{|}\sigma_{d}^{+}(H_{n,k})\bigr{|}&=\Bigl{[}\frac{\omega_{k}}{\pi}\,(n+1)\Bigr{]}+\chi(a_{n,k}-1),\\ \omega_{k}&:=\arccos\,\frac{k+1}{2k}\,,\qquad a_{n,k}:=\frac{\sin n\omega_{k}}{\sin(n+1)\omega_{k}}\,,\end{split}$$

$$\sigma_{d}^{+}(H_{n,k})\subset\Bigl{[}2,\frac{5}{2}\Bigr{]},\ n,k\geq 2;\quad\sigma_{d}^{+}(H_{n,2})\subset\bigl{[}2,1+\sqrt{2}\bigr{]},\ n\geq 2.$$

$$U_m(\cos x)=\frac{\sin (m+1)x}{\sin x},U_m(t)=2^m{t}^m+\dots ,m=0,1,\dots .$$

$$V_{m}(t):=U_{m}\Bigl{(}\frac{t}{2}\Bigr{)}=t^{m}+\ldots,\qquad m=0,1,\ldots,$$

$$\begin{array}{rl}t{V}_{m+1}(t)& =V_{m+2}(t)+V_m(t),\\ V_0(t)& =1,V_1(t)=t,V_2(t)=t^2-1.\end{array}$$

$$V_{m}(t)=\prod_{j=1}^{m}\bigl{(}t-t_{j,m}\bigr{)},\qquad t_{j,m}:=2\cos\frac{j\pi}{m+1}\,.$$

$$P(\lambda ,{\Gamma }_{n,k})=V_n(v_k(\lambda ))\cdot V_{k-1}^n(\lambda ),v_k(\lambda ):=\frac{V_k(\lambda )}{V_{k-1}(\lambda )},\lambda \ne t_{j,k-1}.$$

$$\bigl{|}\sigma^{+}(\Gamma_{n,k})\bigr{|}=\Bigl{[}\frac{\omega_{k}}{\pi}\,(n+1)\Bigr{]},\qquad\omega_{k}:=\arccos\,\frac{k+1}{2k}\,.$$

$$\lambda -A({\Gamma }_{n,k})=\left[\begin{array}{cccccc}\lambda -J_n& -I_n& & & & \\ -I_n& \lambda I_n& -I_n& & & \\ & \ddots & \ddots & \ddots & & \\ & & & \lambda I_n& -I_n& \\ & & & -I_n& \lambda I_n& \end{array}\right]=\left[\begin{array}{cc}\lambda -J_n& B\\ C& D\end{array}\right].$$

$$D=I(d),d=d(\lambda )=\lambda -J_{k-1}\in {\mathcal{M}}_{k-1},$$

$$|d(\lambda )|=\left|\begin{array}{cccccc}\lambda & -1& & & & \\ -1& \lambda & -1& & & \\ & \ddots & \ddots & \ddots & & \\ & & & \lambda & -1& \\ & & & -1& \lambda & \end{array}\right|=V_{k-1}(\lambda ),$$

$$\lambda =t_{j,k-1},j=1,2,\dots ,k-1.$$

$$D^{-1}=I\bigl{(}d^{-1}\bigr{)}=\|(\lambda-J_{k-1})^{-1}_{ij}\cdot I_{n}\|_{i,j=1}^{k-1},$$

$$G_m(\lambda )=(\lambda -J_m{)}_{11}^{-1}=\frac{V_{m-1}(\lambda )}{V_m(\lambda )}$$

$$P(\lambda,\Gamma_{n,k}):=|\lambda-A(\Gamma_{n,k})|=|\bigl{(}\lambda-G_{k-1}(\lambda)\bigr{)}\,I_{n}-J_{n}|\cdot V_{k-1}^{n}(\lambda).$$

$$\lambda -G_{k-1}(\lambda )=\lambda -\frac{V_{k-2}(\lambda )}{V_{k-1}(\lambda )}=\frac{V_k(\lambda )}{V_{k-1}(\lambda )}=v_k(\lambda ),$$

$$P(-\lambda ,{\Gamma }_{n,k})=(-1{)}^{kn}P(\lambda ,{\Gamma }_{n,k}),$$

$$P(t_{j,k-1},{\Gamma }_{n,k})=V_k^n(t_{j,k-1})\ne 0.$$

$$V_n(v_k(\lambda ))=0\iff v_k(\lambda )=t_{j,n}=2\cos \frac{j\pi }{n+1},j=1,\dots ,n.$$

$$v_k(t)=t+\sum _{j=1}^{k-1}\frac{d_{j,k}}{t_{j,k-1}-t},d_{j,k}>0.$$

$$L_{m,k}:=\bigl{(}t_{m,k-1},t_{m-1,k-1}\bigr{)},\qquad m=1,\ldots,k,$$

Full text

1. Introduction

We begin with some rudiments of the graph theory. For the sake of simplicity we restrict ourselves with simple, connected, undirected, finite or infinite (countable) unweighted graphs, although the method works for weighted multigraphs and graphs with loops as well. We will enumerate the vertex set ${\mathcal{V}}(\Gamma)$ with positive integers ${\mathbb{N}}=\{1,2,\ldots\}$, $\{v\}_{v\in{\mathcal{V}}}=\{j\}_{j=1}^{\omega}$, $\omega\leq\infty$. The symbol $i\sim j$ means that the vertices $i$ and $j$ are incident, i.e., $\{i,j\}$ belongs to the edge set ${\mathcal{E}}(\Gamma)$.

The degree (valency) of a vertex $v\in{\mathcal{V}}(\Gamma)$ is a number $\nu(v)$ of edges emanating from $v$. A graph $\Gamma$ is said to be locally finite, if $\nu(v)<\infty$ for all $v\in{\mathcal{V}}(\Gamma)$, and uniformly locally finite, if $\sup_{{\mathcal{V}}}\nu(v)<\infty$.

The spectral graph theory studies spectra and spectral properties of certain matrices related to graphs (more precisely, operators generated by such matrices in the standard basis $\{e_{k}\}_{k\in{\mathbb{N}}}$ and acting on the corresponding Hilbert spaces ${\mathbb{C}}^{n}$ or $\ell^{2}=\ell^{2}({\mathbb{N}})$). One of the most notable of them is the adjacency matrix $A(\Gamma)$

[TABLE]

The corresponding adjacency operator (sometimes called the shift operator) will be denoted by the same symbol. It acts as

[TABLE]

Clearly, $A(\Gamma)\,$ is a symmetric, densely-defined linear operator, whose domain is the set of all finite linear combinations of the basis vectors. The operator $A(\Gamma)$ is bounded and self-adjoint on $\ell^{2}=\ell^{2}({\mathbb{N}})$ if and only if the graph $\Gamma$ is uniformly locally finite.

In what follows, under the spectrum $\sigma(\Gamma)$ of a graph $\Gamma$ we always mean the spectrum $\sigma(A(\Gamma)\,)$ of its adjacency operator. The set of isolated eigenvalues of finite multiplicity is usually referred to as the discrete spectrum $\sigma_{d}(\Gamma)$.

We proceed with the definition of a comb product (comb graph), see, e.g., [8, Section 8.5].

Let \Gamma^{(1)}=\bigl{(}{\mathcal{V}}^{(1)},{\mathcal{E}}^{(1)}\bigr{)}, \Gamma^{(2)}=\bigl{(}{\mathcal{V}}^{(2)},{\mathcal{E}}^{(2)}\bigr{)} be two graphs, the first one is referred to as a backbone, and the second as a finger, and assume that the second graph is given a distinguished vertex [math]. Consider the Cartesian product ${\mathcal{V}}={\mathcal{V}}^{(1)}\times{\mathcal{V}}^{(2)}$, which is a set of vertices for a new graph ${\mathcal{V}}$. We say that $(x_{1},x_{2})\sim(x_{1}^{\prime},x_{2}^{\prime})$, so $\{(x_{1},x_{2}),(x_{1}^{\prime},x_{2}^{\prime})\}\in{\mathcal{E}}$, if one of the following is satisfied:

(i). $x_{1}\sim x_{1}^{\prime}$, $x_{2}=x_{2}^{\prime}=0$;

(ii). $x_{1}=x_{1}^{\prime}$, $x_{2}\sim x_{2}^{\prime}$.

Then $\Gamma=({\mathcal{V}},{\mathcal{E}})$ becomes a graph, known as the comb graph or the comb product of $\Gamma^{(1)}$ and $\Gamma^{(2)}$ with a contact vertex $0\in{\mathcal{V}}^{(2)}$. The standard notation here is $\Gamma=\Gamma^{(1)}\rhd\Gamma^{(2)}$. The comb graph is (uniformly) locally finite whenever so are its components $\Gamma^{(j)}$, $j=1,2$. In fact, the comb graph is obtained by grafting a copy of $\Gamma^{(2)}$ at a vertex [math] into each vertex of $\Gamma^{(1)}$. If $\Gamma^{(1)}={\mathbb{P}}_{n}$, the path of order $n$, the comb product ${\mathbb{P}}_{n}\rhd\Gamma$ is given on Fig.1.

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\Gamma$$\Gamma$$\Gamma$$\Gamma$$\Gamma

1$$2$$3$$n-1$$n

n+1$$n+2$$n+3$$2n-1$$2n$$2n+1$$2n+2$$2n+3$$3n-1$$3n

Fig. 1

Remark 1.1**.**

Although enumeration of the vertex set of a graph does not affect its spectral properties, some particular enumerations can be more convenient for computational purposes. Let us explain this on the above comb graph.

The first way to label the vertices looks as follows. We use the numbers $\{1,2,\ldots,k\}$ to enumerate the vertices of the first copy of $\Gamma$, the next $k$ numbers $\{k+1,k+2,\ldots,2k\}$ to enumerate the vertices of the second copy of $\Gamma$ etc. The adjacency matrix now takes the form

[TABLE]

where $E_{11}=1$, and $E_{ij}=0$ for the rest of the entries. The matrix structure is transparent but not good enough for computation.

In what follows we stick to another way of enumeration, see Fig.1. Precisely, the numbers $\{1,2,\ldots,n\}$ label the vertices of the backbone ${\mathbb{P}}_{n}$, then come $n$ identical vertices of the “first level” labeled with $\{n+1,n+2,\ldots,2n\}$, then $n$ identical vertices of the “second level” etc. The adjacency matrix $A({\mathbb{P}}_{n}\rhd\Gamma)$ can be obtained by using the “inflation” procedure: take the adjacency matrix $A(\Gamma)$ and replace the first zero entry $a_{11}=0$ with the $n\times n$ matrix, known as a discrete Laplacian of order $n$,

[TABLE]

the rest of the zero entries with $n\times n$ zero matrices $0_{n}$, and all unit entries $1$ with $n\times n$ unit matrices $I_{n}$, see (2.7).

For a formal description of the adjacency matrix of a comb product see [8, Lemma 8.35].

To define a main object under consideration – a comb graph with tail – we recall first the operation of coupling by means of the bridge, well known for finite graphs, see, e.g., [4, Theorem 2.12].

Let $\Gamma_{j}$, $j=1,2$, be two graphs with no common vertices, with the vertex sets and edge sets ${\mathcal{V}}(\Gamma_{j})$ and ${\mathcal{E}}(\Gamma_{j})$, respectively, and let $v_{j}\in{\mathcal{V}}(\Gamma_{j})$. A graph $\Gamma=\Gamma_{1}+\Gamma_{2}$ is called a coupling by means of the bridge $\{v_{1},v_{2}\}$ if

[TABLE]

So we join $\Gamma_{2}$ to $\Gamma_{1}$ by the new edge between $v_{2}$ and $v_{1}$, see Fig.2.

$\Gamma_{1}$

$\Gamma_{2}$

v_{1}$$v_{2}

Fig. 2

In general, the adjacency matrix $A(\Gamma_{1}+\Gamma_{2})$ takes the form of a block operator matrix

[TABLE]

If the graph $\Gamma_{1}$ is finite, ${\mathcal{V}}(\Gamma_{1})=\{1,2,\ldots,m\}$, and ${\mathcal{V}}(\Gamma_{2})=\{j\}_{j=m+1}^{\omega}$, we can with no loss of generality put $v_{1}=1$, $v_{2}=m+1$, so the adjacency matrix $A(\Gamma)$ can be written as a block operator matrix

[TABLE]

has $m$ rows. If $\Gamma_{2}={\mathbb{P}}_{\infty}$, the one-sided infinite path, we can view the coupling $\Gamma=\Gamma_{1}+{\mathbb{P}}_{\infty}$ as a finite graph with the tail. Now

[TABLE]

is an infinite Jacobi matrix called a discrete Laplacian.

The spectral theory of infinite graphs with one or several rays attached to certain finite graphs was initiated in [11, 12, 14] wherein a number of particular examples of graphs was examined. In [6] this collection was enlarged considerably. The canonical form of the adjacency matrix $A(\Gamma_{1}+{\mathbb{P}}_{\infty})$ suggested there enabled one to compute the spectrum of such graph by using the spectral theory of Jacobi matrices of finite rank. In [7] we study the spectra of the graphs with tails by means of the Schur complement.

The problem we address in the paper concerns the structure of the spectrum of the basic comb graph with the tail, see Fig.5,

[TABLE]

Here is our main result.

Theorem 1.2**.**

For the comb graph $H_{n,k}$ the following hold.

(i).  The spectrum $\sigma(H_{n,k})=[-2,2]\cup\sigma_{d}(H_{n,k})$, the disjoint union, the discrete spectrum is finite and symmetric with respect to the origin.

(ii).  Let $\sigma_{d}^{+}(H_{n,k}):=\sigma_{d}(H_{n,k})\cap(2,\infty)$ be the positive part of the discrete spectrum. Its cardinality equals

[TABLE]

$\chi$* is the Heaviside function, $[x]$ is the integer part of a real number $x$.*

(iii).

[TABLE]

Both intervals are the smallest ones with such property.

(iv).  The multiplicity of each eigenvalue $\nu\in\sigma_{d}(H_{n,k})$ is at most $2$, and the number of eigenvalues of multiplicity $2$ is at most $4$.

Let us mention a recent manuscript [1], wherein some partial results concerning the spectrum of the graph $H_{n,2}$ are obtained.

We proceed as follows. In Section 2 we study the spectrum of the basic comb graph $\Gamma_{n,k}$ (location and upper bounds). The comb graph with the tail $H_{n,k}$ (1.7) is examined in Section 3, where the main result is proved. We include some auxiliary results from linear algebra, number theory, and perturbation theory in a short Appendix at the end of the paper.

2. Basic comb graph of height $k$

The spectral analysis of the comb graph $\Gamma_{n,k}$, see Fig.3, is performed in this section. We take an endpoint of ${\mathbb{P}}_{k}$ as the contact vertex [math].

1$$2$$3$$n-1$$n$$n+1$$n+2$$2n$$(k-1)n+1$$kn

$\Gamma_{n,k}$

Fig. 3

The Chebyshev polynomials of the second kind enter the expression for the characteristic polynomial and the spectrum of $\Gamma_{n,k}$. They are defined by the relations

[TABLE]

The corresponding monic polynomials

[TABLE]

satisfy the three-term recurrence relation

[TABLE]

The set of the roots of $V_{m}$ comes from (2.1)

[TABLE]

It is easy to see that $V_{m}(-t)=(-1)^{m}V_{m}(t)$, so the set $\{t_{j,m}\}_{j=1}^{m}$ is symmetric with respect to the origin.

The results in Theorem 2.1 below are not completely new. For example, formula (2.5) in (i) follows from the general formula for the characteristic polynomial of a comb product [5], see also [15, Theorem 1]. Yet, our strategy is somewhat different and, what is more to the point, some elements of the construction are applied later on to the study of infinite graphs with tails.

Theorem 2.1**.**

For the comb graph $\Gamma_{n,k}$ the following hold.

(i).  The characteristic polynomial of $\Gamma_{n,k}$ is

[TABLE]

The eigenvalues of $\Gamma_{n,k}$ are simple, and the spectrum $\sigma(\Gamma_{n,k})$ is symmetric with respect to the origin.

(ii).  Let $\sigma^{+}(\Gamma_{n,k}):=\sigma(\Gamma_{n,k})\cap(2,\infty)$. Its cardinality equals

[TABLE]

(iii).  \sigma(\Gamma_{n,k})\subset\bigl{[}-\frac{5}{2},\frac{5}{2}\bigr{]} for all $n,k\geq 2$, and this is the smallest interval with such property.

Proof.

(i). The inflation procedure described in Introduction, applied to the graph $\Gamma_{n,k}$, leads to the following expression as a block matrix with $k^{2}$ blocks, each one belongs to the class ${\mathcal{M}}_{n}$ of $n\times n$ matrices

[TABLE]

Now $D=D(\lambda)$ is the block matrix of scalar type (see Appendix), with the symbol

[TABLE]

(for $k=2$ we have $d=\lambda 1$). So, $|D|=|d|^{n}$, see (4.2).

By the recurrence relations (2.3), the value $|d|=\det d$ agrees with the monic Chebyshev polynomial of the second kind

[TABLE]

$|D(\lambda)|=V_{k-1}^{n}(\lambda)$. So, $|D(\lambda)|=0$ if and only if

[TABLE]

Next, when $\lambda\not=t_{j,k-1}$, we have, by (4.2),

[TABLE]

and so $BD^{-1}C=(\lambda-J_{k-1})^{-1}_{11}\cdot I_{n}$, the scalar matrix from ${\mathcal{M}}_{n}$. The scalar factor

[TABLE]

is called the Green’s function. Due to the second relation in (4.1), we have

[TABLE]

It follows from (2.3) that

[TABLE]

so (2.5) is proved.

Since $w_{k}(-\lambda)=-w_{k}(\lambda)$, we have

[TABLE]

so $\sigma(\Gamma_{n,k})$ is symmetric with respect to the origin.

Note that each point $t_{j,k-1}$, $j=1,2,\ldots,k-1$, is regular for the right side of (2.5), so we can pass to the limit for $\lambda\to t_{j,k-1}$ to obtain

[TABLE]

Thereby, $\lambda\in\sigma(\Gamma_{n,k})$ if and only if any one of the following $n$ relations holds

[TABLE]

Since the zeros of $V_{k-1}$ and $V_{k}$ interlace, the function $v_{k}$ admits the following representation

[TABLE]

It is clear from (2.11) that $v_{k}$ is monotone increasing from $-\infty$ to $+\infty$ on each interval

[TABLE]

where we put $t_{0,k-1}:=+\infty$ and $t_{k,k-1}:=-\infty$ (for $k=2$ there are only two unbounded intervals $L_{1}=(0,+\infty)$ and $L_{2}=(-\infty,0)$). So, for each $j$ there is exactly one root of the $j$-th relation (2.10) on each interval $L_{m,k}$. So all the roots of (2.10) $\{\lambda_{j,m}=\lambda_{j,m}(n,k)\}_{j=1}^{n}$ (the eigenvalues of $\Gamma_{n,k}$) are simple and can be arranged in the groups

[TABLE]

(ii). The eigenvalues of $\Gamma_{n,k}$ on the interval $(2,\infty)$ certainly belong to the first group $\{\lambda_{j,1}\}$. Choose $p=p_{n,k}$ from the conditions

[TABLE]

or

[TABLE]

(note that, by Corollary 4.2, the number $\omega_{k}/\pi$ is irrational, and so both inequalities in (2.14) are strict). Hence, $\lambda_{p,1}>2>\lambda_{p+1,1}$, and (2.6) follows.

(iii). By (2.10), the maximal eigenvalue $\lambda_{1,1}=\lambda_{1,1}(n,k)$ satisfies

[TABLE]

Denote by $\lambda_{1}=\lambda_{1}(k)$ the maximal root of the equation $v_{k}(\cdot)=2$. Clearly, $\lambda_{1,1}<\lambda_{1}$, and also

[TABLE]

implies $\lambda_{1}>2$. Take $y_{1}=y_{1}(k)>0$ from the relation $\lambda_{1}=2\cosh y_{1}$, so, by (2.1),

[TABLE]

It is easy to check that $s_{k}(t)>e^{t}$ for all $t>0$, and so

[TABLE]

as claimed.

The thorough analysis provides the announced extremal property of the interval $[-5/2,5/2]$. Indeed, let $\sigma(\Gamma_{n,k})\subset[-a,a]$. It follows from the construction that $\{\lambda_{1,1}(\cdot,k)\}$ is the monotone increasing sequence, and

[TABLE]

so $\lambda_{1}(k)\in[-a,a]$, i.e., $\lambda_{1}(k)\leq a$ for all $k$.

Next, by elementary calculus, it is not hard to see that each function $s_{k}$ (2.15) is monotone increasing, whereas the sequence $\{s_{k}\}$ is monotone decreasing, that is, $s_{k+1}<s_{k}$, and so the sequence $\{y_{1}(k)\}$ is monotone increasing. On the other hand, $y_{1}(k)<\log 2$, which means that the limits exist

[TABLE]

Now, let us write the definition

[TABLE]

and tend $k\to\infty$. We see that actually $y_{0}=\log 2$. But $2\cosh y_{1}(k)\leq a$ implies, as $k\to\infty$,

[TABLE]

as claimed. The proof is complete. ∎

Note that formula (2.5) holds (for obvious reasons) for $k=1$ and $n=1$, since

[TABLE]

Remark 2.2**.**

Since the maximal valency of the vertices of $\Gamma_{n,k}$ equals $3$, there is an obvious inclusion for the spectrum, which is a consequence of Gershgorin’s theorem, see [9, Corollary 6.1.3],

[TABLE]

Example 2.3**.**

Consider the comb graph $\Gamma_{n,2}$, see Fig.4. Its characteristic polynomial is

[TABLE]

The calculation of the bounds for $\sigma(\Gamma_{n,2})$ can be made more precise than for the general case (iii) above.

Indeed, the maximal eigenvalue $\lambda_{1,1}(n,2)$ is the maximal root of the quadratic equation

[TABLE]

We see that $\lambda_{1,1}(n,2)<\lambda_{1}(2)=1+\sqrt{2}$ for all $n\geq 2$, and so

[TABLE]

In fact,

[TABLE]

cf. [3, Theorem 3.1.3] with $\Gamma_{2,2}={\mathbb{P}}_{4}$, $\Gamma_{3,2}=E_{6}$, and the maximal eigenvalue (the index of the graph) is bigger than $2$ for $n\geq 4$.

\Gamma_{n,2}$$1$$2$$3$$n-1$$n$$n+1$$n+2$$2n

Fig. 4

3. Comb graphs with tail

We study here the coupling $H_{n,k}=\Gamma_{n,k}+{\mathbb{P}}_{\infty}$ and prove Theorem 1.2.

$v$

$H_{n,k}$

Fig. 5

The adjacency matrix $A(H_{n,k})$ takes the form

[TABLE]

so the finiteness of the discrete spectrum follows.

Our argument relies heavily upon the result from [7, Theorem 1 and relation (1.6)]. It states that $\lambda\in\sigma_{d}(H_{n,k})$ if and only if

[TABLE]

It is clear that $\Gamma_{n,k}\backslash\{v\}=\Gamma_{n-1,k}\cup{\mathbb{P}}_{k-1}$, the disjoint union, and so

[TABLE]

We have, by (2.5) and (2.8),

[TABLE]

Since $V_{k-1}\not=0$ on ${\mathbb{R}}\backslash[-2,2]$, we can cancel $V_{k-1}^{n}$ out to obtain

[TABLE]

Since $R_{n,k}(-\lambda)=(-1)^{n}R_{n,k}(\lambda)$, the discrete spectrum is symmetric with respect to the origin. We restrict ourselves with its positive part $\sigma_{d}^{+}(H_{n,k})$.

Write (3.3) as

[TABLE]

The function $v_{k}$ (2.5) is regular on $(2,\infty)$ and grows monotonically from $k+1/k$ to $+\infty$. The zeros of the denominator in (3.4) are exactly the eigenvalues of $\Gamma_{n,k}$, see (2.10). So, the only eigenvalues $\lambda_{j,1}(n,k)$ in the first group can arise as the singularities of the rational function on the left side (3.4) on the half-line $(2,\infty)$

[TABLE]

Take $p=p_{n,k}$ from (2.14), so that $\lambda_{p,1}>2>\lambda_{p+1,1}$. The function $v_{n,k}$ is monotone decreasing from $+\infty$ to $-\infty$ on each interval $(\lambda_{j+1,1},\lambda_{j,1})$, $j=1,\ldots,p-1$, and from $+\infty$ to [math] on $(\lambda_{1,1},\infty)$, whereas $v$ is monotone increasing from $1$ to $+\infty$ on $(2,\infty)$. Hence, there is exactly one root of (3.4) on each such interval, that is, $p$ roots altogether. The existence of the root on $(2,\lambda_{p,1})$ depends on the value

[TABLE]

Specifically, there is one root as soon as $a_{n,k}>1$, and there is no such root otherwise (note that, by Corollary 4.2, $a_{n,k}\not=1$). Thereby, relation (1.8) is established.

Denote by $\nu_{j}=\nu_{j}(n,k)$ the different eigenvalues in $\sigma_{d}^{+}(H_{n,k})$, arranged in non-increasing order (we will discuss the multiplicity of the eigenvalues later on). So, $\nu_{1}=\nu_{1}(n,k)$ is the maximal such eigenvalue, that is, the root of (3.4) on $(\lambda_{1,1},\infty)$. Clearly, $\nu_{1}>\lambda_{1,1}$. As in the proof of Theorem 2.1, (iii), denote by $\lambda_{1}=\lambda_{1}(k)$ the maximal root of the equation $v_{k}(\cdot)=2$. Let us compare the values of both sides in (3.4) at $\lambda=\lambda_{1}$:

[TABLE]

so $\nu_{1}<\lambda_{1}$. But, as we have already shown in (2.16), $\lambda_{1}<5/2$, which means that $\sigma_{d}^{+}(H_{n,k})\subset[2,5/2]$, as needed.

Moreover, assume that $\sigma_{d}^{+}(H_{n,k})\subset[2,a]$ for all $n,k\geq 2$. Then, by (2.17),

[TABLE]

The rest is the same as in the proof of Theorem 2.1, (iii).

The second inclusion in (1.9) and the extremal property of the corresponding interval follows from (3.6) with $k=2$, since $\lambda_{1}(2)=1+\sqrt{2}$, see (2.20) in Example 2.3.

Finally, let us turn to the multiplicity issue. Put $A:=A(\Gamma_{n,k})\bigoplus A({\mathbb{P}}_{\infty})$, $B:=H_{n,k}$. In view of (3.1), $B$ is the rank $2$ perturbation of $A$. Take the interval $I:=(2,3)$, then both $A$ and $B$ have a pure discrete spectrum on $I$. Furthermore, $\sigma_{d}(A)$ on $I$ consists of $p=p_{n,k}$ simple eigenvalues, so $\pi_{A}(I)=p$. Hence, by Theorem 4.3, the spectral multiplicity of $B$ on $I$ is at most $p+2$. But, we know that the number of different eigenvalues of $B$ on $I$ is $p$ or $p+1$, and there are no common eigenvalues for $A$ and $B$. Then there are at most two eigenvalues of $B$, having multiplicity $2$ (there are no eigenvalues of $B$ of multiplicity $3$). The argument for the negative part of $\sigma_{d}(B)$ is identical.

The proof is complete.

Remark 3.1**.**

Let $\Gamma$ be a finite graph of order $m$, $H=\Gamma+{\mathbb{P}}_{\infty}$. According to [6, Theorem 1.2], the adjacency operator $A(H)$ is unitarily equivalent to the orthogonal sum

[TABLE]

where $F(H)$ is an operator on a finite dimensional subspace of dimension at most $m-1$, $J(H)$ is a Jacobi operator of finite rank at most $m$, generated by a Jacobi (tridiagonal) matrix

[TABLE]

in an appropriate orthonormal basis. $J$ is said to have a finite rank if

[TABLE]

and the minimal such $j_{0}$ is called the rank of $J$. $F(H)$ and $J(H)$ are referred to as the finite and Jacobi components of $H$, respectively.

The structure of $\sigma(J(H))$ is quite transparent: the absolutely continuous spectrum on $[-2,2]$ along with a finite number of simple eigenvalues off $[-2,2]$, which come from the roots of a certain algebraic equation (the Jost equation, see [6, Section 5]). The eigenvalues of $F(H)$ can appear anywhere on the real line including $[-2,2]$. It is not clear how to find this later set (the “hidden spectrum”) without computing the finite component $F(H)$ explicitly, the problem, which seems to be hard enough. For example, if $H=H_{3,2}$, the finite component is missing (no hidden spectrum), and the Jacobi component is given by $J(\{b_{j}\},\{a_{j}\})$ with

[TABLE]

Some information on the hidden spectrum can be gathered from [1, Theorem 4.3], which claims that $A(H_{n,k})$ has a trivial kernel, i.e., [math] is not an eigenvalue of $A(H_{n,k})$.

In the table below we display the number of the positive eigenvalues for the graph $H_{n,k}$ for $n=2,3,\ldots,20$ and $k=2,3,\ldots,6$ obtained by using the computer experiment.

4. Appendix

4.1. Transformation of block matrices

We collect here some information about operations on block matrices, see, e.g., [9, Sections 0.7, 0.8]. We denote by ${\mathcal{M}}_{m}$ the set of all square matrices of order $m$.

Let

[TABLE]

be a block matrix, $A$ and $D$ the square matrices (in general, of the different order). Then for the determinant $|\Delta|=\det(\Delta)$ the following relations hold

[TABLE]

The matrices $D-CA^{-1}B$ and $A-BD^{-1}C$ are usually referred to as the Schur complements.

Let $A=\|A_{ij}\|_{i,j=1}^{m}$, $A_{ij}\in{\mathcal{M}}_{n}$, be a block matrix. We call $A$ a block matrix of scalar type if each block is a scalar matrix, i.e., $A_{ij}=a_{ij}I_{n}$, $I_{n}$ is the identity matrix of order $n$, $a_{ij}\in{\mathbb{C}}$. We call the matrix $a=\|a_{ij}\|_{i,j=1}^{m}\in{\mathcal{M}}_{m}$ a symbol of $A$, and $A$ the inflation of $a$, $A=I(a)$.

The operations on block matrices of scalar type are easily reduced to the corresponding operations on their symbols

[TABLE]

If $\sigma(a)=\{\lambda_{j}\}_{j=1}^{p}$ be the set of all different eigenvalues of $a$, and the algebraic multiplicity of $\lambda_{j}$ equals $\kappa_{j}$, then $\sigma(I(a))=\{\lambda_{j}\}_{j=1}^{p}$, with the algebraic multiplicity $n\kappa_{j}$.

4.2. Rational values of the arccosine functions

The arithmetic properties of trigonometric functions has been a recurring topic in the number theory. In particular, all the rational values of the function $\pi^{-1}\,\arccos t$, $t\in{\mathbb{Q}}$, the set of the rational numbers, can be determined explicitly. This is a special case of a result of D. Lehmer [13]. For completeness sake we provide an elementary argument, borrowed from [10].

Proposition 4.1**.**

The only rational values of the function $\cos(\pi r)$, $r\in{\mathbb{Q}}$ are [math], $\pm\frac{1}{2}$, $\pm 1$.

Proof.

Put $g_{j}=g_{j}(r):=2\cos(2^{j}\,\pi r)$, $j=0,1,\ldots$. The addition formula for cosine implies

[TABLE]

By the hypothesis, $g_{0}\in{\mathbb{Q}}$, and so are $g_{j}$ for all $j\geq 0$. Write the irreducible fraction $g_{0}=a_{0}/b_{0}$. Then

[TABLE]

Clearly, the latter fraction is irreducible, since the assumption ${\rm gcd}(a_{0},b_{0})=~{}\!1$ implies ${\rm gcd}(a_{0}^{2},b_{0}^{2})=1$, and so ${\rm gcd}(a_{0}^{2}-2b_{0}^{2},b_{0}^{2})=1$. Hence, the irreducible representation for $g_{j}$ is

[TABLE]

If $|b_{0}|\geq 2$, the denominators $b_{j}$ tend to infinity.

On the other hand, as it follows from periodicity of cosine, and $r\in{\mathbb{Q}}$, the sequence $\{g_{j}\}_{j\geq 0}$ takes the finite number of values, that is inconsistent with $\beta_{j}\to\infty$. So, $b_{0}=\pm 1$ and $a_{0}=0,\,\pm 1,\,\pm 2$, as claimed. ∎

Corollary 4.2**.**

(i). The numbers

[TABLE]

are irrational.

(ii). Let $a_{n,k}$ be the numbers defined in $\eqref{numpds}$. Then $a_{n,k}\not=1$ for all $n,k=2,3,\ldots$.

Proof.

The first statement is an immediate consequence of Proposition 4.1. As for the second one, assume that

[TABLE]

or

[TABLE]

The first factor above is nonzero, so \cos\Bigl{(}n+\frac{1}{2}\Bigr{)}\,\omega_{k}=0, that is

[TABLE]

which contradicts (i). ∎

4.3. Finite rank perturbations of self-adjoint operators

Given a self-adjoint operator $T=T^{*}$ on the Hilbert space ${\mathcal{H}}$, denote by $E_{T}$ its resolution of the identity. For an interval $I\subset{\mathbb{R}}$ the operator $E_{T}(I)$ is the projection on the spectral subspace $E_{T}(I)\,{\mathcal{H}}$. The dimension $\pi_{T}(I)=\dim E_{T}(I)\,{\mathcal{H}}$ of this subspace is called a spectral multiplicity of $T$ on $I$.

We say that $T$ has a discrete spectrum on $I$ if $\sigma(T)\cap I\subset\sigma_{d}(T)$. In this case $I$ contains only isolated eigenvalues of finite multiplicity, and the spectral multiplicity equals the sum of multiplicities of all eigenvalues in $I$.

The following result is well known, see, e.g., [2, Theorem 9.3.3].

Theorem 4.3**.**

Let $A$, $B$ be self-adjoint operators on the Hilbert space ${\mathcal{H}}$, so that

[TABLE]

If the spectrum of $A$ is discrete on a bounded interval $I$, then so is the spectrum of $B$, and $|\pi_{B}(I)-\pi_{A}(I)|\leq r$.