Let F be a finite field, n∈N and CMn(F) denote the unitary Cayley graph of the matrix algebra Mn(F). In this paper, we study the first and second subconstituents of CMn(F). We determine the spectra ...of the subconstituents of CM2(F) by using the character table of GL2(F) and elementary linear algebra, and conclude their hyperenergeticity and Ramanujan property. Moreover, we compute the clique number, the chromatic number and the independence number of those subconstituents.
In this paper, we first study zero divisor graphs over finite chain rings. We determine their rank, determinant, and eigenvalues using reduction graphs. Moreover, we extend the work to zero divisor ...graphs over finite commutative principal ideal rings using a combinatorial method, finding the number of positive eigenvalues and the number of negative eigenvalues, and finding upper and lower bounds for the largest eigenvalue. Finally, we characterize all finite commutative principal ideal rings such that their zero divisor graphs are complete and compute the Wiener index of the zero divisor graphs over finite commutative principal ideal rings.
Let R be a finite commutative ring and n a positive integer. In this paper, we study the unitary Cayley graph CMn(R) of the matrix ring over R. If F is a field, we use the additive characters of ...Mn(F) to determine three eigenvalues of CMn(F) and use them to analyze strong regularity and hyperenergetic graphs. We find conditions on R and n such that CMn(R) is strongly regular. Without explicitly having the spectrum of the graph, we can show that CMn(R) is hyperenergetic and characterize R and n such that CMn(R) is Ramanujan. Moreover, we compute the clique number, the chromatic number and the independence number of the graph.
