On the α-spectral radius of graphs

Applicable Analysis and Discrete Mathematics

article

Haiyan Guo¹  Bo Zhou¹ 

[1] School of Mathematical Sciences, South China Normal University

关键词: α-spectral radius;    adjacency matrix;    tree;    unicyclic graph;    irregular graph;   

DOI  :  10.2298/AADM180210022G

学科分类：社会科学、人文和艺术（综合）

来源: Univerzitet u Beogradu * Elektrotehnicki Fakultet / University of Belgrade, Faculty of Electrical Engineering

【 摘 要 】

For 0 ≤ α ≤ 1, Nikiforov proposed to study the spectral properties of thefamily of matrices Aα(G) = αD(G) + (1−α)A(G) of a graph G, where D(G)is the degree diagonal matrix and A(G) is the adjacency matrix of G. Theα-spectral radius of G is the largest eigenvalue of Aα(G). For a graph withtwo pendant paths at a vertex or at two adjacent vertices, we prove resultsconcerning the behavior of the α-spectral radius under relocation of a pendantedge in a pendant path. We give upper bounds for the α-spectral radius forunicyclic graphs G with maximum degree ∆ ≥ 2, connected irregular graphswith given maximum degree and some other graph parameters, and graphswith given domination number, respectively. We determine the unique treewith the second largest α-spectral radius among trees, and the unique treewith the largest α-spectral radius among trees with given diameter. We alsodetermine the unique graphs so that the difference between the maximumdegree and the α-spectral radius is maximum among trees, unicyclic graphsand non-bipartite graphs, respectively.

【 授权许可】

Unknown