The maximum spectral radius of wheel-free graphs

Abstract

A wheel graph is a graph formed by connecting a single vertex to all vertices of a cycle. A graph is called wheel-free if it does not contain any wheel graph as a subgraph. In 2010, Nikiforov proposed a Brualdi–Solheid–Turán type problem: what is the maximum spectral radius of a graph of order $n$ that does not contain subgraphs of particular kind. In this paper, we study the Brualdi–Solheid–Turán type problem for wheel-free graphs, and we determine the maximum (signless Laplacian) spectral radius of a wheel-free graph of order $n$. Furthermore, we characterize the extremal graphs.

Introduction

Let $G$ be an undirected simple graph with vertex set $V\left(G\right)$ and edge set $E\left(G\right)$ (denote by $e\left(G\right)=\left|E\left(G\right)\right|$). For any $v\in V\left(G\right)$, let $N_k\left(v\right)$ denote the set of vertices at distance $k$ from $v$ in $G$. In particular, the vertex subset $N\left(v\right)=N_1\left(v\right)$ is called the neighborhood of $v$, and $d_v=\left|N\left(v\right)\right|$ is called the degree of $v$. The adjacency matrix of $G$ is defined as $A\left(G\right)={\left(a_{u,v}\right)}_{u,v\in V\left(G\right)}$, where $a_{u,v}=1$ if $uv\in E\left(G\right)$, and $a_{u,v}=0$ otherwise. Let $D\left(G\right)=\mathrm{diag}(d_v:v\in V\left(G\right))$ denote the diagonal matrix of vertex degrees of $G$. Then $Q\left(G\right)=D\left(G\right)+A\left(G\right)$ is called the signless Laplacian matrix of $G$. The (adjacency) spectral radius ${\rho }_A\left(G\right)$ and the signless Laplacian spectral radius ${\rho }_Q\left(G\right)$ of $G$ are the largest eigenvalues of $A\left(G\right)$ and $Q\left(G\right)$, respectively. In addition, for any $n\times n$ matrix $M$ with only real eigenvalues, we always arrange its eigenvalues in a non-increasing order: ${\lambda }_1\left(M\right)\ge {\lambda }_2\left(M\right)\ge \cdots \ge {\lambda }_n\left(M\right)$.

For any $S,T\subseteq V\left(G\right)$ with $S\cap T=0̸$, let $E(S,T)$ denote the set of edges between $S$ and $T$ in $G$ (denote by $e(S,T)=\left|E(S,T)\right|$), and let $G\left[S\right]$ denote the subgraph of $G$ induced by $S$. For any $e\in E\left(G\right)$, let $G-e$ denote the graph obtained by deleting $e$ from $G$. Given two graphs $G$ and $H$, let $G\nabla H$ denote the graph obtained from the disjoint union $G\cup H$ by adding all edges between $G$ and $H$. For any nonnegative integer $k$, let $kG$ denote the disjoint union of $k$ copies of $G$. As usual, we denote by $K_n$, $P_n$, $C_n$ and $W_n=K_1\nabla C_{n-1}$ the complete graph, the path, the cycle and the wheel graph on $n$ vertices, respectively, and $K_{s,t}=s{K}_1\nabla t{K}_1$ the complete bipartite graph with two parts of size $s$ and $t$. Also, we denote by $B_k$ the book graph with $k$-pages consisting of $k$ triangles sharing one edge, and $F_k$ the friendship graph on $2k+1$ vertices consisting of $k$ triangles which intersect in exactly one common vertex.

Let $\mathcal{H}$ be a family of graphs. A graph $G$ is called $\mathcal{H}$-free if it does not contain any graph of $\mathcal{H}$ as a subgraph. The Turán number of $\mathcal{H}$, denoted by $ex(n,\mathcal{H})$, is the maximum number of edges in an $\mathcal{H}$-free graph of order $n$. Let $Ex(n,\mathcal{H})$ denote the set of $\mathcal{H}$-free graphs of order $n$ with $ex(n,\mathcal{H})$ edges. To determine $ex(n,\mathcal{H})$ and characterize the graphs in $Ex(n,\mathcal{H})$ for various kinds of $\mathcal{H}$ is a basic problem in extremal graph theory (see [6], [17], [19] for surveys). In particular, for $\mathcal{H}=\left\{W_{2k}\right\}$, the Simonovits’s theorem (see [20, Theorem 1, p. 285]) implies that $ex(n,\left\{W_{2k}\right\})=ex(n,\left\{K_4\right\})=\lfloor \frac{n^2}{3}\rfloor$ and $Ex(n,\left\{W_{2k}\right\})=\left\{T_{n,3}\right\}$ for sufficiently large $n$, where $T_{n,3}$ is the complete $3$-partite graph of order $n$ with part sizes as equal as possible. In 2013, Dzido [7] improved this result to $n\ge 6k-10$ for $k\ge 3$. For $\mathcal{H}=\left\{W_{2k+1}\right\}$, Dzido and Jastrzȩbski [8] proved that $ex(n,\left\{W_5\right\})=\lfloor \frac{n^2}{4}+\frac{n}{2}\rfloor$, $ex(n,\left\{W_7\right\})=\lfloor \frac{n^2}{4}+\frac{n}{2}+1\rfloor$, and $ex(n,\left\{W_{2k+1}\right\})\ge \lfloor \frac{n^2}{4}+\frac{n}{2}\rfloor$ for all values of $n$ and $k$. Very recently, Yuan [24] established that $ex(n,\left\{W_{2k+1}\right\})=max\{n_0{n}_1+\lfloor \frac{(k-1)n_0}{2}\rfloor +2:n_0+n_1=n\}$ for $k\ge 3$ and sufficiently large $n$.

In spectral graph theory, the well-known Brualdi–Solheid problem (see [3]) asks for the maximum spectral radius of a graph belonging to a specified class of graphs and the characterization of the extremal graphs. Up to now, this problem has been studied for many classes of graphs, and readers are referred to [22] for systematic results. As the blending of the Brualdi–Solheid problem and the general Turán type problem, Nikiforov [16] proposed a Brualdi–Solheid-Turán type problem:

Problem 1

What is the maximum spectral radius of an $\mathcal{H}$-free graph of order $n$?

In the past few decades, much attention has been paid to Problem 1 for various families of graphs $\mathcal{H}$ such as $\left\{K_s\right\}$ [13], [23], $\left\{K_{s,t}\right\}$ [1], [13], [15], $\{B_{k+1},K_{2,l+1}\}$ [18], $\left\{F_k\right\}$ [5], $\left\{P_s\right\}$ [16], $\left\{C_{2k+1}\right\}$ [14], $\{C_3,C_4\}$ [12], $\left\{C_4\right\}$ [13], [27], $\{C_5,C_6\}$ [25], $\{W_5,C_6\}$ [28], $\left\{C_6\right\}$ [26], $\left\{{\cup }_{i=1}^k{P}_{s_i}\right\}$ [4], $\{C_l:l\ge 2k+1\}$ and $\{C_l:l\ge 2k+2\}$ [10]. For more results on extremal spectral graph theory, we refer the reader to [17].

Motivated by the general Turán type problem, it is natural to consider the Brualdi–Solheid–Turán type problem for $\left\{W_{\ell }\right\}$-free graphs, where $\ell$ is any fixed integer. However, it seems difficult to determine the maximum spectral radius of a $\left\{W_{\ell }\right\}$-free graph of order $n$. In this paper, we consider a closely related problem:

Problem 2

What is the maximal spectral radius of a wheel-free (i.e., $\{W_{\ell }:\ell \ge 4\}$-free) graph of order $n$?

Let $H_n$ be defined as

$$H_n=\left\{\begin{array}{cc}\frac{n-1}{4}{K}_2\nabla \frac{n+1}{2}{K}_1\hfill & \text{if }n\equiv 1\mathrm{mod}4,\hfill \\ \frac{n+1}{4}{K}_2\nabla \frac{n-1}{2}{K}_1\hfill & \text{if }n\equiv 3\mathrm{mod}4,\hfill \\ \frac{n}{4}{K}_2\nabla \frac{n}{2}{K}_1\hfill & \text{if }n\equiv 0\mathrm{mod}4,\hfill \\ (\frac{n-2}{4}{K}_2\cup K_1)\nabla \frac{n}{2}{K}_1\hfill & \text{if }n\equiv 2\mathrm{mod}4,\hfill \end{array}\right.$$

and let $F$ denote the graph in Fig. 1. Note that $F$ is the complement of the cycle $C_7$ on $7$ vertices. Clearly, both $H_n$ and $F$ are wheel-free. As an answer to Problem 2, we prove that

Theorem 1

Let $G$ be a wheel-free graph of order $n\ge 4$. Then

$${\rho }_A\left(G\right)\le {\rho }_A\left(H_n\right),$$

with equality holding if and only if $G=H_n$ for $n\ne 7$ and $G=H_7$ or $F$ for $n=7$.

Furthermore, we consider the same problem for the signless Laplacian spectral radius of wheel-free graphs. Surprisingly, the following result shows that the extremal graphs are not the same as that of Theorem 1.

Theorem 2

Let $G$ be a wheel-free graph of order $n\ge 4$. Then

$${\rho }_Q\left(G\right)\le {\rho }_Q(K_2\nabla (n-2)K_1)=\frac{n+2+\sqrt{{(n+2)}^2-16}}{2},$$

with equality holding if and only if $G=K_2\nabla (n-2)K_1$.