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

摘要

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. The results are the following: (1) Let $1< kleq l< Delta < n$ and $G$ be a connected {$B_{k+1},K_{2,l+1}$}-free graph of order $n$ with maximum degree $Delta$. Then $$displaystyle q(G)leq rac{1}{4}[3Delta+k-2l+1+sqrt{(3Delta+k-2l+1)^{2}+16l(Delta+n-1)}$$ with equality if and only if $G$ is a strongly regular graph with parameters ($Delta$, $k$, $l$). (2) Let $sgeq tgeq 3$, and let $G$ be a connected $K_{s,t}$-free graph of order $n$ $(ngeq s+t)$. Then $$q(G)leq n+(s-t+1)^{1/t}n^{1-1/t}+(t-1)(n-1)^{1-3/t}+t-3.$$.