On the distance spectral gap and construction of D-equienergetic graphs
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arXiv
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| Natura: | Preprint |
| Pubblicazione: |
2025
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| _version_ | 1866910822124486656 |
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| author | T, Haritha A. V, Chithra |
| author_facet | T, Haritha A. V, Chithra |
| contents | Let $D(G)$ denote the distance matrix of a connected graph $G$ with $n$ vertices. The distance spectral gap of a graph $G$ is defined as $δ_{D^G} = ρ_1 - ρ_2$, where $ρ_1$ and $ρ_2$ represent the largest and second largest eigenvalues of $D(G)$, respectively. For a $k$-transmission regular graph $G$, the second smallest eigenvalue of the distance Laplacian matrix equals the distance spectral gap of $G$. In this article, we obtain some upper and lower bounds for the distance spectral gap of a graph in terms of the sum of squares of its distance eigenvalues. Additionally, we provide some bounds for the distance eigenvalues and distance energy of graphs. Furthermore, we construct new families of non $D$-cospectral $D$-equienergetic graphs with diameters of $3$ and $4$. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2502_07398 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | On the distance spectral gap and construction of D-equienergetic graphs T, Haritha A. V, Chithra Combinatorics 05C50, 05C76, 05C12 Let $D(G)$ denote the distance matrix of a connected graph $G$ with $n$ vertices. The distance spectral gap of a graph $G$ is defined as $δ_{D^G} = ρ_1 - ρ_2$, where $ρ_1$ and $ρ_2$ represent the largest and second largest eigenvalues of $D(G)$, respectively. For a $k$-transmission regular graph $G$, the second smallest eigenvalue of the distance Laplacian matrix equals the distance spectral gap of $G$. In this article, we obtain some upper and lower bounds for the distance spectral gap of a graph in terms of the sum of squares of its distance eigenvalues. Additionally, we provide some bounds for the distance eigenvalues and distance energy of graphs. Furthermore, we construct new families of non $D$-cospectral $D$-equienergetic graphs with diameters of $3$ and $4$. |
| title | On the distance spectral gap and construction of D-equienergetic graphs |
| topic | Combinatorics 05C50, 05C76, 05C12 |
| url | https://arxiv.org/abs/2502.07398 |