Maxima of the Q-index: Forbidden a fan

Abstract

The complete split graph, denoted by $S_{n,k}$, is the graph obtained by joining a copy of $K_k$ to $n-k$ isolated vertices. As a special threshold graph, it plays an important role in graph theory and spectral graph theory. Some known results tell us that if $q(G)>q(S_{n,k})$ then G can contain many kinds of subgraphs, such as paths, linear forests, friendship graphs, flowers, and so on (where $q(G)$ is the Q-index of G). Motivated by these results, we are interested in finding larger subgraphs in G. In this paper, we show that if $q(G)>q(S_{n,k})$ then G contains a fan $H_{2k}$, where $H_{2k}$ denotes the graph obtained by joining a vertex to a path of order 2k. This implies several previous results, and a conjecture proposed by Li and Peng.

Introduction

The terminology fan, posed by Dirac [9] in 1960, refers to the union of paths from a vertex to a vertex set such that any two of them share only the previous vertex. The famous Fan-Lemma, due to Dirac, extends Menger's theorem to the fan-type family of paths. In 1995, Erdős et al. [11] characterized the extremal graphs with maximum number of edges forbidding a friendship graph, denoted by $F_k$. In this paper, we consider a new fan $H_k$, which is the graph obtained by joining a vertex to a copy of $P_k$.

A graph is said to be H-free, if it does not contain a subgraph isomorphic to H. The Turán number of H, denoted by $\text{ex}(n,H)$, is the maximum number of edges in an H-free graph of order n. Determining $\text{ex}(n,H)$ for a given H is well-known as Turán-type problem. A spectral version of Turán-type problem, proposed by Nikiforov [22], asks what is the maximum adjacent spectral radius of all H-free graphs of order n? Much of the classic extremal graph theory has been recast to adjacent spectral radius with an astonishing preservation of details; however, not all extremal results about adjacent spectral radius are similar to corresponding edge extremal results. We refer readers to related surveys ([6], [18], [21]).

Let $Q(G)$ be the signless Laplacian matrix of a graph G and $q(G)$ be its spectral radius (or Q-index for short). In 2009, Hansen and Lucas [15] posed the following conjecture.

Conjecture 1.1

Let G be a $K_{r+1}$-free graph of order n. Then $q(G)\le 2(1-\frac{1}{r})n$.

In 2013, Abreu and Nikiforov [1] confirmed Conjecture 1.1. He, Jin and Zhang [16] further obtained that $q(G)\le q(T_{n,r})$, with equality if and only if G is the r-partite Turán graph $T_{n,r}$ for $r\ge 3$ and G is a complete bipartite graph for $r=2$. Then, Freitas, Nikiforov and Patuzzi [12] proposed the following problem.

Problem 1.1

What is the maximum Q-index of all H-free graphs of order n?

Problem 1.1 was considered for some kinds of subgraphs H (see $k{P}_2$ [25]; paths [23]; $C_4$ and $C_5$ [12]; odd cycles [26]; even cycles [24]; $K_{s,t}$ [13]; linear forests [4]; friendship graphs [27]; flowers [5]). Some known results imply that the extremal graphs for Problem 1.1 are usually different from the extremal graphs for corresponding adjacent version (for example, friendship graphs, see [7]).

Let $S_{n,k}$ be the complete split graph obtained by joining a copy of $K_k$ to $n-k$ isolated vertices. The following two theorems solve Problem 1.1 for odd cycles and friendship graphs, respectively.

Theorem 1.1

(Yuan [26]) Let $k\ge 3$ and $n\ge 110k^2$. If G is a $C_{2k+1}$-free graph of order n, then $q(G)\le q(S_{n,k})$, with equality if and only if $G\cong S_{n,k}$.

Theorem 1.2

(Zhao, Huang and Guo [27]) Let $k\ge 2$ and $n\ge 3k^2-k-2$. If G is an $F_k$-free graph of order n, then $q(G)\le q(S_{n,k})$, with equality if and only if $G\cong S_{n,k}$.

Let $F_{k,r}$ be the graph consisting of k cliques of order r which intersect in exactly one common vertex. Recently, Desai et al. [8] proposed the following conjecture.

Conjecture 1.2

For integers $k\ge 1$ and $r\ge 3$, there exists an integer $n_0(k,r)$ such that if $n\ge n_0(k,r)$ and G is an $F_{k,r}$-free graph of order n, then $q(G)\le q(S_{n,k(r-2)})$, with equality if and only if $G\cong S_{n,k(r-2)}$.

Clearly, when $r=3$, this conjecture reduces to Theorem 1.2. Let $F_{a_1,\dots ,a_s}$ be a flower, i.e., the graph obtained from s odd cycles of lengths $2a_1+1,\dots ,2a_s+1$ respectively which intersect in exactly one common vertex. If $a_1=\cdots =a_s=t$, then we write $F_{a_1,\dots ,a_s}$ for $C_{s,2t+1}$. Li and Peng [19] proposed the following conjecture.

Conjecture 1.3

Let $s\ge 2$, $t\ge 1$ and $\ell =2t+1$. Then there exists an integer $n_0(s,t)$ such that if $n\ge n_0(s,t)$ and G is a $C_{s,\ell }$-free graph of order n, then $q(G)\le q(S_{n,st})$, with equality if and only if $G\cong S_{n,st}$.

Chen, Liu and Zhang [5] showed the following result, which confirms Conjecture 1.3.

Theorem 1.3

(Chen, Liu and Zhang [5]) Let $a_1\ge \cdots \ge a_s\ge 1$ be s integers, $k={\sum }_{i=1}^s{a}_i$ and $n\ge k^2-12k+9$. If G is an $F_{a_1,\dots ,a_s}$-free graph of order n, then $q(G)\le q(S_{n,k})$, with equality if and only if G is a complete bipartite graph for $k=1$ and $G\cong S_{n,k}$ otherwise.

In this paper, we consider a larger family of graphs. The main result is as follows.

Theorem 1.4

Let $k\ge 2$ and $n\ge 36k^6$. If G is an $H_{2k}$-free graph of order n, then $q(G)\le q(S_{n,k})$, with equality if and only if $G\cong S_{n,k}$.

Note that $C_{2k+1}$, $F_k$ and $F_{a_1,\dots ,a_s}$ (where ${\sum }_{i=1}^s{a}_i=k$) are three spanning subgraphs of $H_{2k}$. Therefore, Theorem 1.4 implies Theorem 1.1, Theorem 1.2, Theorem 1.3 for $k\ge 2$ and $n\ge 36k^6$.