The general spectral radius and majorization theorem of $t$-cone graphs with given degree sequences

Abstract

The general spectral radius of a graph $G$, denoted by $\Theta (G,\alpha )$, is the maximal eigenvalue of $M_{\alpha }(G)=A(G)+\alpha D(G)$ $(\alpha \ge 0)$, where $A(G)$ and $D(G)$ are the adjacency matrix and the diagonal matrix of vertex degrees of $G$, respectively. A graph $G$ is called ${\Theta }_{\alpha }$-maximal in a class of connected simple graphs $\mathcal{G}$ if $\Theta (G,\alpha )$ is maximal among all graphs of $\mathcal{G}$. A $t$-cone $c$-cyclic graph is the join of a complete graph $K_t$ and a $c$-cyclic connected simple graph. Let $\pi =(d_1,d_2,\dots ,d_n)$ and ${\pi }^{\prime }=(d_1^{\prime },d_2^{\prime },\dots ,d_n^{\prime })$ be two non-increasing degree sequences of $t$-cone $c$-cyclic graphs with $n$ vertices. We say $\pi$ is strictly majorized by ${\pi }^{\prime }$, denoted by $\pi ⊲{\pi }^{\prime }$, if $\pi \ne {\pi }^{\prime }$, ${\sum }_{i=1}^n{d}_i={\sum }_{i=1}^n{d}_i^{\prime }$, and ${\sum }_{i=1}^k{d}_i\le {\sum }_{i=1}^k{d}_i^{\prime }$ for $k=1,2,\dots ,n-1$. Denote by $\Gamma (\pi ,t;c)$ the class of $t$-cone $c$-cyclic graphs with $\pi$ as its degree sequence. In this paper, we determine some properties of ${\Theta }_{\alpha }$-maximal graphs of $\Gamma (\pi ,t;c)$ and characterize the unique ${\Theta }_{\alpha }$-maximal graph of $\Gamma (\pi ,t;0)$ \big(resp. $\Gamma (\pi ,t;1)$ and $\Gamma (\pi ,t;2)$\big). Moreover, we prove that if $\pi ⊲{\pi }^{\prime }$, $G$ and $G^{\prime }$ are the ${\Theta }_{\alpha }$-maximal graphs of $\Gamma (\pi ,t;c)$ and $\Gamma ({\pi }^{\prime },t;c)$ respectively, then $\Theta (G,\alpha )<\Theta (G^{\prime },\alpha )$ for $c\in \{0,1\}$, and we also consider the similar result for $c=2$.