The signless Laplacian spectral radius of subgraphs of regular graphs

Abstract

Let $q(H)$ be the signless Laplacian spectral radius of a graph $H$. In this paper, we prove that \\1. Let $H$ be a proper subgraph of a $\Delta$-regular graph $G$ with $n$ vertices and diameter $D$. Then $$2\Delta -q(H)>\frac{1}{n(D-\frac{1}{4})}.$$ \\2. Let $H$ be a proper subgraph of a $k$-connected $\Delta$-regular graph $G$ with $n$ vertices, where $k\ge 2$. Then $$2\Delta -q(H)>\frac{2(k-1{)}^2}{2(n-\Delta )(n-\Delta +2k-4)+(n+1)(k-1{)}^2}.$$ Finally, we compare the two bounds. We obtain that when $k>2\sqrt{\frac{(n-\Delta )(n+\Delta -4)}{n(4D-3)-2}}+1$, the second bound is always better than the first. On the other hand, when $k<\frac{2(n-\Delta )}{\sqrt{n(4D-3)-2}}+1$, the first bound is always better than the second.

9 pages, 2 figures, 1 table