The signless Laplacian spectral radius of book-free graphs

Abstract

Given a graph H, a graph is called H-free if it does not contain H as a subgraph. For a positive integer k, a book $B_{k+1}$ is a graph consisting of $k+1$ triangles sharing a common edge. In this paper, let G be a $B_{k+1}$-free graph of order $n\ge 49k^2-22k+4$. Then the signless Laplacian spectral radius $q\left(G\right)\le \frac{n+2k+\sqrt{{(n-2k)}^2+8k^2}}{2}$ with equality if and only if $G={\bar{K}}_k\vee H$, where k or $n-k$ is even, and H is a $K_3$-free k-regular graph of order $n-k$. Furthermore, if k and $n-k$ are both odd, the extremal graphs with the maximum signless Laplacian spectral radius are also characterized.