On the Extended Adjacency Eigenvalues of Graphs and Applications

Hilal A. Ganie () and Amal Alsaluli ()
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Hilal A. Ganie: Department of Mathematics, Government Degree College Uri, Srinagar 190001, India
Amal Alsaluli: Department of Mathematics, College of Science, University of Bisha, Bisha 61922, Saudi Arabia

Mathematics, 2025, vol. 13, issue 22, 1-20

Abstract: Let A e x ( G ) be the extended adjacency matrix of G . The eigenvalues of A e x ( G ) are called extended adjacency eigenvalues of G . The sum of the absolute values of eigenvalues of the A e x -matrix is called the extended adjacency energy E e x ( G ) of G . In this paper, we obtain the A e x -spectrum of the joined union of regular graphs in terms of their adjacency spectrum and the eigenvalues of an auxiliary matrix. Consequently, we derive the A e x -spectrum of the join of two regular graphs, the lexicographic product of regular graphs, and the A e x -spectrum of various families of graphs. Further, as applications of our results, we construct infinite classes of infinite families of extended adjacency equienergetic graphs. We show that the A e x -energy of the join of two regular graphs is greater than or equal to their energy. We also determine the A e x -eigenvalues of the power graph of finite abelian groups.

Keywords: graphs; eigenvalues; extended adjacency eigenvalues; extended adjacency energy; power graph; equienergetic graphs (search for similar items in EconPapers)
JEL-codes: C (search for similar items in EconPapers)
Date: 2025
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