New Upper Bounds on the Energy of a Graph

Abstract

The energy of a graph G, denoted by E(G), is defined as the sum of the absolute values of all eigenvalues of G. It is proved in [MATCH Commun. Math. Comput. Chem. 79 (2018) for every bipartite graph G of order n, size m and maximum degree ∆. We prove the above bound for all graphs G. We also prove new types of two bounds of Koolen and Moulton given in [Adv. Appl. Math. 26 (2001) 47-52] and [Graphs Comb. 19 (2003) 131-135].

Graph Energy Change Due to Vertex Deletion

“…It follows from Theorem 1 that there exists an orthogonal matrix P = (p ij ) such that P (A − B (1) ) and P B (1) are positive semidefinite. Note that P (A − B (1) ) and (A − B (1) )P are similar matrices since P ⊤ = P −1 and (A − B (1) )P = P ⊤ P (A − B (1) )P.…”

“…The energy of G is denoted by E(G) and defined as the sum of the absolute values of the eigenvalues of A. We refer the reader to [9,12] for further information on graph energy and [1,2,10] for recent results.…”

Note on Some New (n, m)-Type Bounds for Graph Energy

“…It is really surprising that the authors of [1] did not notice the above fact. Not only that they quote the paper [6], but (correctly) state the Koolen-Moulton bound (i.e., Eq.…”

“…Until quite recently, these were the only known (n, m)-type upper bounds for graph energy [8]. Few years ago, in the paper [1], a new such bound was reported, namely:…”

New Bounds on the Energy of a Graph

“…Many researchers presented bounds for the energy of a graph. Recently, Alawiah et al [2] proved the following upper bound for the energy of unicyclic graphs.…”