TY - JOUR

T1 - On Eigenvalues and Energy of Geometric–Arithmetic Matrix of Graphs

AU - Pirzada, S.

AU - Rather, Bilal A.

AU - Aouchiche, M.

N1 - Funding Information:
The research of S. Pirzada is supported by SERB-DST, New Delhi under the research project number CRG/2020/000109.
Publisher Copyright:
© 2022, The Author(s), under exclusive licence to Springer Nature Switzerland AG.

PY - 2022/6

Y1 - 2022/6

N2 - Let G be a graph with vertex set { v1, v2, … , vn} and let di denote the degree of vertex vi. The geometric–arithmetic matrix GA(G) of G is indexed by the vertices of G, whose (i, j) -th entry is 2didjdi+dj if the vertices i and j are adjacent and 0 otherwise. The multi-set of eigenvalues of GA(G) is known as the geometric–arithmetic spectrum of G. In this article, we obtain geometric–arithmetic spectra of various families of graphs and characterize the connected graphs with two and three distinct GA eigenvalues. We obtain several upper and lower bounds for the geometric arithmetic energy and characterize the graphs attaining such bounds.

AB - Let G be a graph with vertex set { v1, v2, … , vn} and let di denote the degree of vertex vi. The geometric–arithmetic matrix GA(G) of G is indexed by the vertices of G, whose (i, j) -th entry is 2didjdi+dj if the vertices i and j are adjacent and 0 otherwise. The multi-set of eigenvalues of GA(G) is known as the geometric–arithmetic spectrum of G. In this article, we obtain geometric–arithmetic spectra of various families of graphs and characterize the connected graphs with two and three distinct GA eigenvalues. We obtain several upper and lower bounds for the geometric arithmetic energy and characterize the graphs attaining such bounds.

KW - Adjacency matrix

KW - energy

KW - geometric–arithmetic eigenvalues

KW - geometric–arithmetic index

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U2 - 10.1007/s00009-022-02035-0

DO - 10.1007/s00009-022-02035-0

M3 - Article

AN - SCOPUS:85128916855

SN - 1660-5446

VL - 19

JO - Mediterranean Journal of Mathematics

JF - Mediterranean Journal of Mathematics

IS - 3

M1 - 115

ER -