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

TL;DR

This paper establishes upper bounds on the signless Laplacian spectral radius for certain classes of graphs that are free of specific subgraphs, such as books and complete bipartite graphs, with conditions for equality.

Contribution

It provides new upper bounds on the signless Laplacian spectral radius for book-free and $K_{s,t}$-free graphs, extending spectral graph theory results.

Findings

Derived bounds depend on maximum degree and graph order.

Equality cases characterized by strongly regular graphs.

Bounds improve understanding of spectral properties in forbidden subgraph classes.

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 $1< k\leq l< \Delta < n$ and $G$ be a connected \{$B_{k+1},K_{2,l+1}$\}-free graph of order $n$ with maximum degree $\Delta$. Then $$\displaystyle q(G)\leq \frac{1}{4}[3\Delta+k-2l+1+\sqrt{(3\Delta+k-2l+1)^{2}+16l(\Delta+n-1)}.$$ with equality holds if and only if $G$ is a strongly regular graph with parameters ($\Delta$, $k$, $l$). (2) Let $s\geq t\geq 3$, and let $G$ be a connected $K_{s,t}$-free graph of order $n$ $(n\geq s+t)$. Then $$q(G)\leq n+(s-t+1)^{1/t}n^{1-1/t}+(t-1)(n-1)^{1-3/t}+t-3.$$