Spectral Properties of Unitary Cayley Graphs of Finite Commutative Rings

Let $R$ be a finite commutative ring. The unitary Cayley graph of $R$, denoted $G_R$, is the graph with vertex set $R$ and edge set $\left\{\{a,b\}:a,b\in R, a-b\in R^\times\right\}$, where $R^\times$ is the set of units of $R$. An $r$-regular graph is Ramanujan if the absolute value of every eigenvalue of it other than $\pm r$ is at most $2\sqrt{r-1}$. In this paper we give a necessary and sufficient condition for $G_R$ to be Ramanujan, and a necessary and sufficient condition for the complement of $G_R$ to be Ramanujan. We also determine the energy of the line graph of $G_R$, and compute the spectral moments of $G_R$ and its line graph.

[1]  Gordon F. Royle,et al.  Algebraic Graph Theory , 2001, Graduate texts in mathematics.

[2]  Elena Fuchs,et al.  Longest Induced Cycles on Cayley Graphs , 2004 .

[3]  Dariush Kiani,et al.  On the Unitary Cayley Graph of a Ring , 2012, Electron. J. Comb..

[4]  David Burton.  Elementary Number Theory , 1976 .

[5]  Torsten Sander,et al.  Integral circulant graphs of prime power order with maximal energy , 2011 .

[6]  Aleksandar Ilic,et al.  The energy of unitary cayley graphs , 2009 .

[7]  Domingos M. Cardoso,et al.  Energy of line graphs , 2010 .

[8]  M. Murty.  Ramanujan Graphs , 1965 .

[9]  Oscar Rojo.  Line graph eigenvalues and line energy of caterpillars , 2011 .

[10]  R. Balakrishnan.  The energy of a graph , 2004 .

[11]  Michael Francis Atiyah,et al.  Introduction to commutative algebra , 1969 .

[12]  Giuliana P. Davidoff,et al.  Elementary number theory, group theory, and Ramanujan graphs , 2003 .

[13]  Andrew Droll.  A Classification of Ramanujan Unitary Cayley Graphs , 2010, Electron. J. Comb..

[14]  N. Linial,et al.  Expander Graphs and their Applications , 2006 .

[15]  Ivan Gutman,et al.  Spectra and energies of iterated line graphs of regular graphs , 2005, Appl. Math. Lett..

[16]  Elena D. Fuchs.  Longest Induced Cycles in Circulant Graphs , 2005, Electron. J. Comb..

[17]  N. Ganesan.  Properties of rings with a finite number of zero divisors , 1964 .

[18]  Yongtang Shi,et al.  Note on the energy of regular graphs , 2009 .

[19]  Reza Akhtar,et al.  On the Unitary Cayley Graph of a Finite Ring , 2009, Electron. J. Comb..

[20]  Juan Rada,et al.  Upper and lower bounds for the energy of bipartite graphs , 2004 .

[21]  Peter Sarnak,et al.  Elementary Number Theory, Group Theory, and Ramanujan Graphs: Graph Theory , 2003 .

[22]  H. O. Foulkes.  Abstract Algebra , 1967, Nature.

[23]  Rajat Tandon.  Algebra and number theory : proceedings of the silver jubilee conference, University of Hyderabad , 2005 .

[24]  D. West.  Introduction to Graph Theory , 1995 .

[25]  M. Ram Murty.  Ramanujan graphs and zeta functions , 2005 .

[26]  Dariush Kiani,et al.  Energy of unitary Cayley graphs and gcd-graphs , 2011 .

[27]  D. Cvetkovic,et al.  An Introduction to the Theory of Graph Spectra: Introduction , 2009 .

[28]  Ivan Gutman.  HYPERENERGETIC MOLECULAR GRAPHS , 1999 .

[29]  P ? ? ? ? ? ? ? % ? ? ? ? , 1991 .

[30]  I. Gutman.  The Energy of a Graph: Old and New Results , 2001 .

[31]  Torsten Sander,et al.  Some Properties of Unitary Cayley Graphs , 2007, Electron. J. Comb..

[32]  Oscar Rojo,et al.  Line graph of combinations of generalized Bethe trees: Eigenvalues and energy , 2011 .