The Signless Dirichlet Spectral Radius of Unicyclic Graphs

Abstract:

      Let $G$ be a simple connected graph with pendant vertex set $\partial V$ and nonpendant vertex set $V_0$. The signless Laplacian matrix of $G$ is denoted by $Q(G)$. The signless Dirichlet eigenvalue is a real number $\lambda$ such that there exists a function $f \neq 0$ on $V(G)$ such that $Q(G)f(u)=\lambda f(u)$ for $u \in V_0$ and $f(u)=0$ for $u \in \partial V$. The signless Dirichlet spectral radius $\lambda(G)$ is the largest signless Dirichlet eigenvalue. In this paper, the unicyclic graphs with the largest signless Dirichlet spectral radius among all unicyclic graphs with a given degree sequence are characterized.