On the Generalized Adjacency Spread of a Graph

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

Automated summary

This paper introduces the concepts of the generalized adjacency matrix, A(a)-spread of a graph, and the smallest S(A(a)) of the path graph. It answers the question raised in a previous paper and establishes a relationship between S(A(a)) and S(A). It also obtains several bounds for S(A(a)).