The Wiener Index of an Undirected Power Graph

• 作者: Volkan Aşkin Şerife Büyükköse
• 来源期刊: 线性代数与矩阵理论研究进展(英文)
• 学科: 数学
• 关键词: Wiener Index Edge-Wiener Index An Undirected Power Graph Line Graph
• 年，卷（期）: 2021,（1）
• 页码范围: 21-29
• 页数: 9页
• 分类号: O15

摘要

The undirected power graph <i>P</i>(<i>Z_n</i>) of a finite group <i>Z_n</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_n</i>)) of an undirected power graph <i>P</i>(<i>Z_n</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_n</i>). Similarly, the edge-Wiener index <i>W_e</i>(<i>P</i>(<i>Z_n</i>)) of <i>P</i>(<i>Z_n</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_n</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_pq</i>) and <i>P</i>(<i>Z_p</i>). Firstly, we obtain new results on the wiener index and edge-wiener index of power graph <i>P</i>(<i>Z_n</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_n</i>.