The undirected power graph <i>P</i>(<i>Z<sub>n</sub></i>) of a finite group <i>Z<sub>n</sub></i> is the graph with vertex set G and two distinct vertices u a...The undirected power graph <i>P</i>(<i>Z<sub>n</sub></i>) of a finite group <i>Z<sub>n</sub></i> is the graph with vertex set G and two distinct vertices u and v are adjacent if and only if <i>u</i> ≠ <i>v</i> and <img src="Edit_3b1df203-9ff2-4c13-93d1-4bba568eae54.png" width="40" height="20" alt="" /> or <img src="Edit_094c8f88-deb6-4f41-825a-ba91c0306ae8.png" width="40" height="20" alt="" />. The Wiener index <i>W</i>(<i>P</i>(<i>Z<sub>n</sub></i>)) of an undirected power graph <i>P</i>(<i>Z<sub>n</sub></i>) is defined to be sum <img src="Edit_348337df-b9c2-480d-9713-ec299a6fcd4e.png" width="110" height="25" alt="" /> of distances between all unordered pair of vertices in <i>P</i>(<i>Z<sub>n</sub></i>). Similarly, the edge-Wiener index <i>W<sub>e</sub></i>(<i>P</i>(<i>Z<sub>n</sub></i>)) of <i>P</i>(<i>Z<sub>n</sub></i>) is defined to be the sum <img src="Edit_e9b89765-f71e-4865-a0c5-c688710ff0c6.png" width="60" height="25" alt="" /> of distances between all unordered pairs of edges in <i>P</i>(<i>Z<sub>n</sub></i>). In this paper, we concentrate on the wiener index of a power graph <img src="Edit_dff0cd99-eb11-4123-a437-78cbbd8ebf96.png" width="40" height="20" alt="" />, <i>P</i>(<i>Z<sub>pq</sub></i>) and <i>P</i>(<i>Z<sub>p</sub></i>). Firstly, we obtain new results on the wiener index and edge-wiener index of power graph <i>P</i>(<i>Z<sub>n</sub></i>), using <i>m,n</i> and Euler function. Also, we obtain an equivalence between the edge-wiener index and wiener index of a power graph of <i>Z<sub>n</sub></i>.展开更多
文摘The undirected power graph <i>P</i>(<i>Z<sub>n</sub></i>) of a finite group <i>Z<sub>n</sub></i> is the graph with vertex set G and two distinct vertices u and v are adjacent if and only if <i>u</i> ≠ <i>v</i> and <img src="Edit_3b1df203-9ff2-4c13-93d1-4bba568eae54.png" width="40" height="20" alt="" /> or <img src="Edit_094c8f88-deb6-4f41-825a-ba91c0306ae8.png" width="40" height="20" alt="" />. The Wiener index <i>W</i>(<i>P</i>(<i>Z<sub>n</sub></i>)) of an undirected power graph <i>P</i>(<i>Z<sub>n</sub></i>) is defined to be sum <img src="Edit_348337df-b9c2-480d-9713-ec299a6fcd4e.png" width="110" height="25" alt="" /> of distances between all unordered pair of vertices in <i>P</i>(<i>Z<sub>n</sub></i>). Similarly, the edge-Wiener index <i>W<sub>e</sub></i>(<i>P</i>(<i>Z<sub>n</sub></i>)) of <i>P</i>(<i>Z<sub>n</sub></i>) is defined to be the sum <img src="Edit_e9b89765-f71e-4865-a0c5-c688710ff0c6.png" width="60" height="25" alt="" /> of distances between all unordered pairs of edges in <i>P</i>(<i>Z<sub>n</sub></i>). In this paper, we concentrate on the wiener index of a power graph <img src="Edit_dff0cd99-eb11-4123-a437-78cbbd8ebf96.png" width="40" height="20" alt="" />, <i>P</i>(<i>Z<sub>pq</sub></i>) and <i>P</i>(<i>Z<sub>p</sub></i>). Firstly, we obtain new results on the wiener index and edge-wiener index of power graph <i>P</i>(<i>Z<sub>n</sub></i>), using <i>m,n</i> and Euler function. Also, we obtain an equivalence between the edge-wiener index and wiener index of a power graph of <i>Z<sub>n</sub></i>.
