In the last decade, several scholars proposed an unifying approach to study the spectral theories of the adjacency, Laplacian and signless Laplacian of graphs. The most general graph matrix is the universal adjacency matrix $U=\alpha A+\text{β}D+\gamma J+\delta I$, where A, D, J, and I are the adjacency matrix of G, the degree matrix of G, the all-ones matrix, the identity matrix, respectively. Here, we consider $M_{\text{β}}=A+\text{β}D$, with $\text{β}\ge 0$, and we study the graphs belonging to some given class Γ maximizing the corresponding spectral radius ${\rho }_{\text{β}}$. In particular, we consider connected graphs with prescribed c-cyclic degree sequence, $c\in \{0,1,2\}$, and the multicone graphs defined over them, where the multicone graph is the join of a clique with a given graph. The aim of this paper is to provide the best possible generalization of results to the spectral radius of $M_{\text{β}}$ (and the graph matrices related to it) of several well-known results for multicone graphs over connected graphs with prescribed c-cyclic degree sequence, where $c\in \{0,1,2\}$.

在过去的十年中，几位学者提出了一种统一的方法来研究图的邻接，拉普拉斯算子和无符号拉普拉斯算子的频谱理论。最通用的图矩阵是通用邻接矩阵 $ü=\alpha \mathrm{一种}+\text{β}d+\gamma Ĵ+\delta \mathrm{一世}$其中甲，d，Ĵ，和我是的邻接矩阵G ^，的程度矩阵ģ分别全1矩阵，单位矩阵，。在这里，我们考虑${\mathrm{中号}}_{\text{β}}=\mathrm{一种}+\text{β}d$， 和 $\text{β}\ge 0$，我们研究属于某个给定类Γ的图，从而最大化相应的光谱半径 ${\rho }_{\text{β}}$。特别地，我们考虑具有规定c循环度序列的连通图，$C\in \{0，\mathrm{1个}，\mathrm{2个}\}$以及在其上定义的多锥图，其中多锥图是集团与给定图的连接。本文的目的是对结果的谱半径提供最佳的概括。${\mathrm{中号}}_{\text{β}}$带有已知c环次数序列的连通图上的多锥图的几个众所周知的结果（以及与之相关的图矩阵），其中$C\in \{0，\mathrm{1个}，\mathrm{2个}\}$。