On the Generalized Adjacency Spread of a Graph

Maryam Baghipur, Modjtaba Ghorbani (), Shariefuddin Pirzada and Najaf Amraei
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Maryam Baghipur: Department of Mathematics, Faculty of Science, Shahid Rajaee Teacher Training University, Tehran 16785-163, Iran
Modjtaba Ghorbani: Department of Mathematics, Faculty of Science, Shahid Rajaee Teacher Training University, Tehran 16785-163, Iran
Shariefuddin Pirzada: Department of Mathematics, University of Kashmir, Srinagar 192101, India
Najaf Amraei: Department of Mathematics, Faculty of Science, Shahid Rajaee Teacher Training University, Tehran 16785-163, Iran

Mathematics, 2023, vol. 11, issue 6, 1-9

Abstract: 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 Algebra 2020 , 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 ) .

Keywords: generalized adjacency matrix; spread; eigenvalue (search for similar items in EconPapers)
JEL-codes: C (search for similar items in EconPapers)
Date: 2023
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