A characterization of extremal non-transmission-regular graphs by the distance (signless Laplacian) spectral radius
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| Format: | Preprint |
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2024
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| _version_ | 1866909090071969792 |
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| author | Lan, Jingfen Liu, Lele |
| author_facet | Lan, Jingfen Liu, Lele |
| contents | Let $G$ be a simple connected graph of order $n$ and $\partial(G)$ is the spectral radius of the distance matrix $D(G)$ of $G$. The transmission $D_i$ of vertex $i$ is the $i$-th row sum of $D(G)$. Denote by $D_{\max}(G)$ the maximum of transmissions over all vertices of $G$, and $\partial^Q(G)$ is the spectral radius of the distance signless Laplacian matrix $D(G)+\mbox{diag}(D_1,D_2,\ldots,D_n)$. In this paper, we present a sharp lower bound of $2D_{\max}(G)-\partial^Q(G)$ among all $n$-vertex connected graphs, and characterize the extremal graphs. Furthermore, we give the minimum values of respective $D_{\max}(G)-\partial(G)$ and $2D_{\max}(G)-\partial^Q(G)$ on trees and characterize the extremal trees. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2402_00416 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | A characterization of extremal non-transmission-regular graphs by the distance (signless Laplacian) spectral radius Lan, Jingfen Liu, Lele Combinatorics Spectral Theory 05C50 Let $G$ be a simple connected graph of order $n$ and $\partial(G)$ is the spectral radius of the distance matrix $D(G)$ of $G$. The transmission $D_i$ of vertex $i$ is the $i$-th row sum of $D(G)$. Denote by $D_{\max}(G)$ the maximum of transmissions over all vertices of $G$, and $\partial^Q(G)$ is the spectral radius of the distance signless Laplacian matrix $D(G)+\mbox{diag}(D_1,D_2,\ldots,D_n)$. In this paper, we present a sharp lower bound of $2D_{\max}(G)-\partial^Q(G)$ among all $n$-vertex connected graphs, and characterize the extremal graphs. Furthermore, we give the minimum values of respective $D_{\max}(G)-\partial(G)$ and $2D_{\max}(G)-\partial^Q(G)$ on trees and characterize the extremal trees. |
| title | A characterization of extremal non-transmission-regular graphs by the distance (signless Laplacian) spectral radius |
| topic | Combinatorics Spectral Theory 05C50 |
| url | https://arxiv.org/abs/2402.00416 |