Exploring new upper and lower bounds for the $A_α$-energy of graphs
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arXiv
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| Autores principales: | , |
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| Formato: | Preprint |
| Publicado: |
2026
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| _version_ | 1866911543323525120 |
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| author | Basunia, Mainak Panigrahi, Pratima |
| author_facet | Basunia, Mainak Panigrahi, Pratima |
| contents | Let $G$ be a graph on $n$ vertices and $m$ edges. For $α\in [0,1]$, the $A_α$-matrix of $G$ is defined as $A_α(G) = αD(G) + (1- α) A(G)$, where $A(G)$ is the adjacency matrix and $D(G)$ is the degree diagonal matrix of $G$. If $ρ_1 \geq ρ_2 \ldots \geq ρ_n$ are the eigenvalues of $A_α(G)$, the $A_α$-energy of $G$ is defined as $E_{A_α}(G) = \sum_{i=1}^{n} |ρ_i -\frac{2αm}{n}|$. In this paper, we present novel upper and lower bounds for $E_{A_α}(G)$ in terms of standard graph invariants, showing that each bound is sharp and identifying the specific graphs attaining them. For selected bounds, we provide brief comparative analysis with existing results, observing improved estimates. Furthermore, we establish new relations between $E_{A_α}(G)$ and other well known graph energies, including adjacency, Laplacian, as well as the adjacency energy of the line graph. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2603_23920 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | Exploring new upper and lower bounds for the $A_α$-energy of graphs Basunia, Mainak Panigrahi, Pratima Combinatorics 05C50, 05C05, 15A18 Let $G$ be a graph on $n$ vertices and $m$ edges. For $α\in [0,1]$, the $A_α$-matrix of $G$ is defined as $A_α(G) = αD(G) + (1- α) A(G)$, where $A(G)$ is the adjacency matrix and $D(G)$ is the degree diagonal matrix of $G$. If $ρ_1 \geq ρ_2 \ldots \geq ρ_n$ are the eigenvalues of $A_α(G)$, the $A_α$-energy of $G$ is defined as $E_{A_α}(G) = \sum_{i=1}^{n} |ρ_i -\frac{2αm}{n}|$. In this paper, we present novel upper and lower bounds for $E_{A_α}(G)$ in terms of standard graph invariants, showing that each bound is sharp and identifying the specific graphs attaining them. For selected bounds, we provide brief comparative analysis with existing results, observing improved estimates. Furthermore, we establish new relations between $E_{A_α}(G)$ and other well known graph energies, including adjacency, Laplacian, as well as the adjacency energy of the line graph. |
| title | Exploring new upper and lower bounds for the $A_α$-energy of graphs |
| topic | Combinatorics 05C50, 05C05, 15A18 |
| url | https://arxiv.org/abs/2603.23920 |