The (Signless) Laplacian Spectral Radius with a Fixed Induced Subgraph of a Graph

Abstract

For a graph $G$ of size $m$, let $q(G)$ (resp. $\mu(G)$) denote the largest eigenvalue of its signless Laplacian matrix (resp. Laplacian matrix). In this paper, we investigate a complementary version of spectral Turán type problems on (signless) Laplacian matrix, concentrating on maximizing the largest (signless) Laplacian spectral radius of a graph with a fixed subgraph. For an arbitrary fixed graph $H$, we conjecture that if a graph $G$ of size $m$ contains $H$ as an induced subgraph, then $$q(G)\leq q(H_{u}^{m-m(H)}),$$ where $u$ is some vertex of $H$ and $H_u^{m-m(H)}$ is the graph obtained by attaching $m-m(H)$ pendant vertices to the vertex $u$. We confirm the conjecture for graphs of size at least $3(m(H)-\Delta(H))+3$. We also propose a similar conjecture on the Laplacian spectral radius that $$\mu(G)\leq \mu(H_{v}^{m-m(H)})$$ for a graph $G$ containing $H$ as an induced subgraph. Moreover, we validate these two conjectures when $H$ is a cycle, a path or a clique, which generalize some known results.