On the Generalized Adjacency Spread of a Graph

Abstract

For a simple finite graph G, the generalized adjacency matrix is defined as Aα(G)=αD(G)+(1−α)A(G),α∈[0,1], where A(G) and D(G) are respectively the adjacency matrix and diagonal matrix of the vertex degrees. The Aα-spread of a graph G is defined as the difference between the largest eigenvalue and the smallest eigenvalue of the Aα(G). In this paper, we answer the question posed in (Lin, Z.; Miao, L.; Guo, S. Bounds on the Aα-spread of a graph. Electron. J. Linear Algebra 2020, 36, 214–227). Furthermore, we show that the path graph, Pn, has the smallest S(Aα) among all trees of order n. We establish a relationship between S(Aα) and S(A). We obtain several bounds for S(Aα).

Keywords

• generalized adjacency matrix
• spread
• eigenvalue