Upper bounds on the Q-spectral radius of book-free and/or $K_{s,t}$-free graphs

Abstract

In this paper, we prove two results about the signless Laplacian spectral radius $q(G)$ of a graph $G$ of order $n$ with maximum degree $\Delta$. Let $B_n=K_2+\overline{K_n}$ denote a book, i.e., the graph $B_n$ consists of $n$ triangles sharing an edge. (1) Let $1<k\le l<\Delta <n$ and $G$ be a connected \{$B_{k+1},K_{2,l+1}$\}-free graph of order $n$ with maximum degree $\Delta$. Then $$q(G)\le \frac{1}{4}[3\Delta +k-2l+1+\sqrt{(3\Delta +k-2l+1{)}^2+16l(\Delta +n-1)}.$$ with equality holds if and only if $G$ is a strongly regular graph with parameters ($\Delta$, $k$, $l$). (2) Let $s\ge t\ge 3$, and let $G$ be a connected $K_{s,t}$-free graph of order $n$ $(n\ge s+t)$. Then $$q(G)\le n+(s-t+1{)}^{1/t}{n}^{1-1/t}+(t-1)(n-1{)}^{1-3/t}+t-3.$$

8 pages