Resistance distance and Kirchhoff index in central vertex join and central edge join of two graphs
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arXiv
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| Format: | Preprint |
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2024
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| _version_ | 1866914746252394496 |
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| author | T, Haritha A V, Chithra |
| author_facet | T, Haritha A V, Chithra |
| contents | The central graph $C(G)$ of a graph $G$ is the graph obtained by inserting a new vertex into each edge of $G$ exactly once and joining all the non-adjacent vertices in $G$. Let $G_1$ and $G_2$ be two vertex disjoint graphs. The central vertex join of $G_1$ and $G_2$ is the graph $ G_1\dot{\vee} G_2$, is obtained from $C(G_1)$ and $G_2$ by joining each vertex of $G_1$ with every vertex of $G_2$. The central edge join of $G_1$ and $G_2$ is the graph $ G_1\veebar G_2$, is obtained from $C(G_1)$ and $G_2$ by joining each vertex corresponding to the edges of $G_1$ with every vertex of $G_2$. In this article, we obtain formulae for the resistance distance and Kirchhoff index of $G_1\dot{\vee} G_2$ and $ G_1\veebar G_2$. In addition, we provide the resistance distance, Kirchhoff index, and Kemeny's constant of the central graph of a graph. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2404_06317 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Resistance distance and Kirchhoff index in central vertex join and central edge join of two graphs T, Haritha A V, Chithra Combinatorics 05C50, 05C76, 60J20 The central graph $C(G)$ of a graph $G$ is the graph obtained by inserting a new vertex into each edge of $G$ exactly once and joining all the non-adjacent vertices in $G$. Let $G_1$ and $G_2$ be two vertex disjoint graphs. The central vertex join of $G_1$ and $G_2$ is the graph $ G_1\dot{\vee} G_2$, is obtained from $C(G_1)$ and $G_2$ by joining each vertex of $G_1$ with every vertex of $G_2$. The central edge join of $G_1$ and $G_2$ is the graph $ G_1\veebar G_2$, is obtained from $C(G_1)$ and $G_2$ by joining each vertex corresponding to the edges of $G_1$ with every vertex of $G_2$. In this article, we obtain formulae for the resistance distance and Kirchhoff index of $G_1\dot{\vee} G_2$ and $ G_1\veebar G_2$. In addition, we provide the resistance distance, Kirchhoff index, and Kemeny's constant of the central graph of a graph. |
| title | Resistance distance and Kirchhoff index in central vertex join and central edge join of two graphs |
| topic | Combinatorics 05C50, 05C76, 60J20 |
| url | https://arxiv.org/abs/2404.06317 |