On the Generalized Adjacency Spread of a Graph - Maryam Baghipur; Modjtaba Ghorbani; Shariefuddin Pirzada; Najaf Amraei (Mathematics, 2023)
For a simple finite graph G, the generalized adjacency matrix is defined as <semantics>Aα(G)=αD(G)+(1−α)A(G),α∈[0,1]</semantics>, where <semantics>A(G)</semantics> and <semantics>D(G)</semantics> are respectively the adjacency matrix and diagonal matrix of the vertex degrees. The <semantics>Aα</semantics>-spread of a graph G is defined as the difference between the largest eigenvalue and the smallest eigenvalue of the <semantics>Aα(G)</semantics>. In this paper, we answer the question posed in (Lin, Z.; Miao, L.; Guo, S. Bounds on the <semantics>Aα</semantics>-spread of a graph. Electron. J. Linear Algebra 2020, 36, 214–227). Furthermore, we show that the path graph, <semantics>Pn</semantics>, has the smallest <semantics>S(Aα)</semantics> among all trees of order n. We establish a relationship between <semantics>S(Aα)</semantics> and <semantics>S(A).</semantics> We obtain several bounds for <semantics>S(Aα)</semantics>.