2024/06/17 by Ming-Hua Li, Yue Liu, Li, Minghua +1
Computer Science · Mathematics · #Cellular Automata and Applications #Stochastic processes and statistical mechanics #Mathematical Dynamics and Fractals
paper · pdf · doi:10.48550/arxiv.2406.11412
Let GS be a graph with n vertices obtained from a simple graph G by attaching one self-loop at each vertex in S ⊆ V(G). The energy of GS is defined by Gutman et al. as E(GS)=∑i=1n| λi -\fracσn |, where λ1,…,λn are the adjacency eigenvalues of GS and σ is the number of self-loops of GS. In this paper, several upper and lower bounds of E(GS) regarding λ1 and λn are obtained. Especially, the upper bound E(GS) ≤ √n(2m+σ-\fracσ2n) (∗) given by Gutman et al. is improved to the following bound E(GS)≤ √n(2m+σ-\fracσ2n)-(n)/(2)( |λ1-\fracσn |- |λn-\fracσn |)2, where | λ1-\fracσn| ≥ … ≥ | λn-\fracσn|. Moreover, all graphs are characterized when the equality holds in Gutmans' bound (∗) by using this new bound.