On Energy of Graphs with Self-Loops
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arXiv
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| Format: | Preprint |
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2024
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| _version_ | 1866929357520371712 |
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| author | Rakshith, B. R. Das, Kinkar Chandra Manjunatha, B. J. |
| author_facet | Rakshith, B. R. Das, Kinkar Chandra Manjunatha, B. J. |
| contents | Let G be a simple graph on n vertices with vertex set V(G). The energy of G, denoted by, $\mathcal{E}(G)$ 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 $G_S$ 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 $G_S$, denoted by $\mathcal{E}(G_S)$, is the sum of all absolute eigenvalues of the matrix $A(G_S)$. 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 $\mathcal(G_S) > \mathcal{E}(G)$. In this paper, we confirm this conjecture. Also, we construct pairs of equienergetic self-loop graphs of order 24n for all n \ge 1. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2405_15547 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | On Energy of Graphs with Self-Loops Rakshith, B. R. Das, Kinkar Chandra Manjunatha, B. J. Combinatorics Let G be a simple graph on n vertices with vertex set V(G). The energy of G, denoted by, $\mathcal{E}(G)$ 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 $G_S$ 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 $G_S$, denoted by $\mathcal{E}(G_S)$, is the sum of all absolute eigenvalues of the matrix $A(G_S)$. 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 $\mathcal(G_S) > \mathcal{E}(G)$. In this paper, we confirm this conjecture. Also, we construct pairs of equienergetic self-loop graphs of order 24n for all n \ge 1. |
| title | On Energy of Graphs with Self-Loops |
| topic | Combinatorics |
| url | https://arxiv.org/abs/2405.15547 |