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

The general spectral radius of a graph $G$, denoted by $\mathrm{\Theta }(G,\alpha )$, 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 ${\mathrm{\Theta }}_{\alpha }$-maximal in a class of connected simple graphs $\mathcal{G}$ if $\mathrm{\Theta }(G,\alpha )$ 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 ${\pi }^{\mathrm{\prime }}=(d_1^{\mathrm{\prime }},d_2^{\mathrm{\prime }},\dots ,d_n^{\mathrm{\prime }})$ be two non-increasing degree sequences of $t$-cone $c$-cyclic graphs with $n$ vertices. We say $\pi$ is strictly majorized by ${\pi }^{\mathrm{\prime }}$, denoted by $\pi ⊲{\pi }^{\mathrm{\prime }}$, if $\pi \mathrm{\ne }{\pi }^{\mathrm{\prime }}$, ${\sum }_{i=1}^n{d}_i={\sum }_{i=1}^n{d}_i^{\mathrm{\prime }}$, and $\sum_{i=1}^k d_i\leq \sum_{i=1}^k d_i’$for$k=1,2,\ldots,n-1$\mathrm{.}Denoteby$\Gamma(\pi,t;c)$theclassof$t$-cone$c$-cyclicgraphswith$\pi$ as its degree sequence. In this paper, we determine some properties of ${\mathrm{\Theta }}_{\alpha }$-maximal graphs of $\mathrm{\Gamma }(\pi ,t;c)$ and characterize the unique ${\mathrm{\Theta }}_{\alpha }$-maximal graph of $\mathrm{\Gamma }(\pi ,t;0)$ \big(resp. $\mathrm{\Gamma }(\pi ,t;1)$ and $\mathrm{\Gamma }(\pi ,t;2)$\big). Moreover, we prove that if $\pi ⊲{\pi }^{\mathrm{\prime }}$, $G$ and $G^{\mathrm{\prime }}$ are the ${\mathrm{\Theta }}_{\alpha }$-maximal graphs of $\mathrm{\Gamma }(\pi ,t;c)$ and $\mathrm{\Gamma }({\pi }^{\mathrm{\prime }},t;c)$ respectively, then $\mathrm{\Theta }(G,\alpha )<\mathrm{\Theta }(G^{\mathrm{\prime }},\alpha )$ for $c\in \big{0,1\big}$,andwealsoconsiderthesimilarresultfor$c=2$.