Distance powers of unitary Cayley graphs

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• 作者：Xiaogang Liu ; ^{xiaogliu.yzhang@gmail.com" class="auth_mail" title="E-mail the corresponding author} ; ^{xiaogliu@nwpu.edu.cn" class="auth_mail" title="E-mail the corresponding author} ; Binlong Li ^{binlongli@nwpu.edu.cn" class="auth_mail" title="E-mail the corresponding author}
• 关键词：Unitary Cayley graph ; Distance power ; Energy of a graph ; Hyperenergetic graph ; Equienergetic graphs ; Ramanujan graph
• 刊名：Applied Mathematics and Computation
• 出版年：2016
• 出版时间：20 October 2016
• 年：2016
• 卷：289
• 期：Complete
• 页码：272-280
• 全文大小：438 K

文摘

Let G be a graph and let diam(G) denote the diameter of G. The distance power G^N of G is the undirected graph with vertex set V(G), in which x and y are adjacent if their distance d(x, y) in G belongs to N, where N   is a non-empty subset of {1,2,…,diam(G)}$\{1,2,\dots ,\mathrm{diam}(G\left)\right\}$. The unitary Cayley graph   is the graph having the vertex set Z_n${\mathbb{Z}}_n$ and the edge set $\{(a,b):a,b\in {\mathbb{Z}}_n,\gcd (a-b,n)=1\}$. In this paper, we determine the energies of distance powers of unitary Cayley graphs, and classify all Ramanujan distance powers of unitary Cayley graphs. By the energies of distance powers of unitary Cayley graphs, we construct infinitely many pairs of non-cospectral equienergetic graphs. Moreover, we characterize all hyperenergetic distance powers of unitary Cayley graphs.