2024/05/23 by B. R. Rakshith, Kinkar Chandra Das, Rakshith, B. R. +3
Computer Science · #Cellular Automata and Applications #Combinatorics (math.CO) #FOS: Mathematics
paper · pdf · doi:10.48550/arxiv.2405.15547
openalex publication_date 2024/05/23 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
Let G be a simple graph on n vertices with vertex set V(G). The energy of G, denoted by, E(G) is the sum of all absolute values of the eigenvalues of the adjacency matrix A(G). It is the first eigenvalue-based topological molecular index and is related to the molecular orbital energy levels of π-electrons in conjugated hydrocarbons. Recently, the concept of energy of a graph is extended to a self-loop graph. Let S be a subset of V(G). The graph GS is obtained from the graph G by attaching a self-loop at each of the vertices of G which are in the set S. The energy of the self-loop graph GS, denoted by E(GS), is the sum of all absolute eigenvalues of the matrix A(GS). Two non-isomorphic self-loop graphs are equienergetic if their energies are equal. Akbari et al. (2023)conjectured that there exist a subset S of V(G) such that \mathcal(GS) > E(G). In this paper, we confirm this conjecture. Also, we construct pairs of equienergetic self-loop graphs of order 24n for all n ≥ 1.