Javarsineh, M., Fath-Tabar, G. (2017). The Main Eigenvalues of the Undirected Power Graph of a Group. Algebraic Structures and Their Applications, 4(1), 19-32.

Mehrnoosh Javarsineh; Gholam Hossein Fath-Tabar. "The Main Eigenvalues of the Undirected Power Graph of a Group". Algebraic Structures and Their Applications, 4, 1, 2017, 19-32.

Javarsineh, M., Fath-Tabar, G. (2017). 'The Main Eigenvalues of the Undirected Power Graph of a Group', Algebraic Structures and Their Applications, 4(1), pp. 19-32.

Javarsineh, M., Fath-Tabar, G. The Main Eigenvalues of the Undirected Power Graph of a Group. Algebraic Structures and Their Applications, 2017; 4(1): 19-32.

The Main Eigenvalues of the Undirected Power Graph of a Group

^{}Department of Pure Mathematics, Faculty of Mathematical Sciences, University of Kashan, Kashan 87317-53153, Iran.

Abstract

The undirected power graph of a finite group $G$, $P(G)$, is a graph with the group elements of $G$ as vertices and two vertices are adjacent if and only if one of them is a power of the other. Let $A$ be an adjacency matrix of $P(G)$. An eigenvalue $\lambda$ of $A$ is a main eigenvalue if the eigenspace $\epsilon(\lambda)$ has an eigenvector $X$ such that $X^{t}\jj\neq 0$, where $\jj$ is the all-one vector. In this paper we want to focus on the power graph of the finite cyclic group $\mathbb{Z}_{n}$ and find a condition on n where $P(\mathbb{Z}_{n})$ has exactly one main eigenvalue. Then we calculate the number of main eigenvalues of $P(\mathbb{Z}_{n})$ where $n$ has a unique prime decomposition $n = p^{r} p_2$. We also formulate a conjecture on the number of the main eigenvalues of $P(\mathbb{Z}_{n})$ for an arbitrary positive integer $n$.

[1] P.J. Cameron and Sh. Ghosh, The power graph of a finite group, Discrete Math., Vol. 311 Issue 13 (2011), pp. 1220–1222. [2] I. Chakrabarty, Sh. Ghosh and M. K. Sen, Undirected power graphs of semigroups, Semigroup Forum, Vol. 78 Issue 3 (2009), pp. 410–426. [3] D. Cvetkovi´ c, The main part of the spectrum, divisors and switching of graphs, Publ. Inst. Math. (Beograd) (N.S.), Vol. 37 (1978), pp. 31–38. [4] D. Cvetkovi´ c, P. Rowlinson and S. Simi´ c, Eigenspaces of graphs, Encyclopedia of Mathematics and its Applications, 66. Cambridge University Press, Cambridge, (1997). [5] E. Hagos, Some results on graph spectra, Linear Algebra Appl., Vol 356 (2002), pp. 103–111. [6] V. Nikiforov, Walks and the spectral radius of graphs, Linear Algebra Appl., Vol. 418 Issue 1 (2006), pp. 257–268. [7] G. R. Pourgholi, H. Yousefi-Azari and A. R. Ashrafi, The undirected power graph of a finite group, Bull. Malays. Math. Sci. Soc., Vol.38 Issue 4 (2015), pp. 1517–1525. [8] P. Rowlinson, The main eigenvalues of a graph: a survey, Appl. Anal. Discrete Math., Vol. 1 No. 2 (2007), pp. 445–471. [9] Y. Teranishi, Main eigenvalues of a graph, Linear and Multilinear Algebra, Vol. 49 Issue 4 (2001), pp. 289–303.