On the Generalized Adjacency Spread of a Graph.

For a simple finite graph G, the generalized adjacency matrix is defined as A α (G) = α D (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 A α -spread of a graph G is defined as the difference between the largest eigenvalue and the smallest eigenvalue of the A α (G) . In this paper, we answer the question posed in (Lin, Z.; Miao, L.; Guo, S. Bounds on the A α -spread of a graph. Electron. J. Linear Algebra2020, 36, 214–227). Furthermore, we show that the path graph, P n , has the smallest S (A α) among all trees of order n. We establish a relationship between S (A α) and S (A). We obtain several bounds for S (A α) . [ABSTRACT FROM AUTHOR]