禁用{B_(k+1),K_(2,l+1)}的图α谱半径极值问题

Extremal α-spectral results of graphs without{B_(k+1),K_(2,l+1)}

令K_(s,t)是完全二部图,K_(n)是完全图,其中s,t和n是正整数.令B_(4,l)是由l个共享一条边的K_(4)构成的图,B_(l)是由B_(4,l)的所有生成子图构成的集合.本文研究了禁用{B_(k+1),K_(2,l+1)}的图的最大α-谱半径问题.利用B_(k+1)和K_(2,l+1)的结构特点以及基本不等式,在具有n个顶点、最大度为Δ且禁用{B_(k+1),K_(2,l+1)}的连通图中,获得了α-谱半径的上界,且刻画了达到上界的极值图.相应地,在具有n个顶点、最大度为Δ且禁用B_(k+1)或K_(2,l+1)的连通图中,得到了α-谱半径的上界.

Let K_(s,t)be a complete bipartite graph and K_(n)be a complete graph,where s and n are positive intergers.Let B_(4,l)be a graph consisting of l copies of K_(4)which share a common edge and B_(l)be the set of all spanning subgraphs of B_(4,l).This article investigates the maximumα-spectral radius for graphs without{B_(k+1),K_(2,l+1)}.By using the structural characteristics of the forbidden subgraphs B_(k+1)and K_(2,l+1)and some scaling methods,among the set of connected graphs with n vertices and maximum degreeΔwithout{B_(k+1),K_(2,l+1)},we obtain the upper bound for theirα-spectral radii and characterize the graph which attains the tight bound.Accordingly,among the set of connected graphs with n vertices and maximum degreeΔwithout B_(k+1)or K_(2,l+1),the upper bounds for theirα-spectral radii are also deduced.