Maximizing the signless Laplacian spectral radius of some theta graphs

Abstract

Let $Q(G)=D(G)+A(G)$ be the signless Laplacian matrix of a simple graph $G$, where $D(G)$ and $A(G)$ are the degree diagonal matrix and the adjacency matrix of $G$, respectively. The largest eigenvalue of $Q(G)$, denoted by $q(G)$, is called the signless Laplacian spectral radius of $G$. Let $\theta (l_1,l_2,l_3)$ denote the theta graph which consists of two vertices connected by three internally disjoint paths with length $l_1$, $l_2$ and $l_3$. Let $F_n$ be the friendship graph consisting of $\frac{n-1}{2}$ triangles which intersect in exactly one common vertex for odd $n\ge 3$ and obtained by hanging an edge to the center of $F_{n-1}$ for even $n\ge 4$. Let $S_{n,k}$ denote the graph obtained by joining each vertex of $K_k$ to $n-k$ isolated vertices. Let $S_{n,k}^{+}$ denote the graph obtained by adding an edge to the two isolated vertices of $S_{n,k}$. In this paper, firstly, we show that if $G$ is $\theta (1,2,2)$-free, then $q(G)\le q(F_n)$, unless $G\cong F_n$. Secondly, we show that if $G$ is $\theta (1,2,3)$-free, then $q(G)\le q(S_{n,2})$, unless $G\cong S_{n,2}$. Finally, we show that if $G$ is $\{\theta (1,2,2),F_5\}$-free, then $q(G)\le q(S_{n,1}^{+})$, unless $G\cong S_{n,1}^{+}$.

12pages, 3 figures