%0 Journal Article
%T The Wiener Index of an Undirected Power Graph
%A Volkan A£¿kin
%A £¿erife B¨¹y¨¹kk£¿se
%J Advances in Linear Algebra & Matrix Theory
%P 21-29
%@ 2165-3348
%D 2021
%I Scientific Research Publishing
%R 10.4236/alamt.2021.111003
%X 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> ¡Ù <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>.
%K Wiener Index
%K Edge-Wiener Index
%K An Undirected Power Graph
%K Line Graph
%U http://www.scirp.org/journal/PaperInformation.aspx?PaperID=107706