On Bounds of Energy of a Graph with Self-Loops
Abujafar Mandal, Shaikh Mohammed Abu Nayeem · match Communications in Mathematical and in Computer Chemistry · 2024
The energy of a graph GS with n vertices and σ self-loops is defined as ε(GS) = n i=1 |λi -σ n |, where the eigenvalues of the adjacency matrix of GS are λ1, λ2, . . ., λn.In this article, we have established some upper and lower bounds for the energy of such a graph.Those new bounds involve parameters like number of vertices (n), number of edges (m), number of self-loops (σ), maximum vertex degree (∆), and minimum vertex degree (δ).We show that for 1 ≤ σ < n, the quantity |λi -σ n | is always greater than 0, and using that fact we establish a lower bound.We have compared and concluded that the new bounds are either better than the existing bounds or incomparable to a few bounds obtained by some researchers recently.