Some New Results on Energy of Graphs with Self Loops
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arXiv
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| Auteurs principaux: | , |
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| Format: | Preprint |
| Publié: |
2026
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| _version_ | 1866913048679153664 |
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| author | Popat, Kalpesh M. Shingala, Kunal R. |
| author_facet | Popat, Kalpesh M. Shingala, Kunal R. |
| contents | The graph $G_σ$ is obtained from graph $G$ by attaching self loops on $σ$ vertices. The energy $ E(G_σ)$ of the graph $G_σ$ with order $n$ and eigenvalues $λ_1,λ_2,\dots,λ_n$ is defined as $ E(G_σ)= \displaystyle \sum_{i=1}^n\left|λ_i-\dfracσ{n}\right| $. It has been proved that if $σ=0\; or\; n$ then $ E(G)=E(G_σ) $. The obvious question arise: Are there any graph such that $E(G)=E(G_σ)$ and 0$<σ<n$? We have found an affirmative answer of this question and contributed a graph family which satisfies this property. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2604_18651 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | Some New Results on Energy of Graphs with Self Loops Popat, Kalpesh M. Shingala, Kunal R. Combinatorics Spectral Theory 05C50, 05C76 The graph $G_σ$ is obtained from graph $G$ by attaching self loops on $σ$ vertices. The energy $ E(G_σ)$ of the graph $G_σ$ with order $n$ and eigenvalues $λ_1,λ_2,\dots,λ_n$ is defined as $ E(G_σ)= \displaystyle \sum_{i=1}^n\left|λ_i-\dfracσ{n}\right| $. It has been proved that if $σ=0\; or\; n$ then $ E(G)=E(G_σ) $. The obvious question arise: Are there any graph such that $E(G)=E(G_σ)$ and 0$<σ<n$? We have found an affirmative answer of this question and contributed a graph family which satisfies this property. |
| title | Some New Results on Energy of Graphs with Self Loops |
| topic | Combinatorics Spectral Theory 05C50, 05C76 |
| url | https://arxiv.org/abs/2604.18651 |