Laplacian Energy and Eccentricity Based Energy for Cross Monic Zero Divisor Graph Associated With Commutative Rings

For a commutative ring, a cross monic zero divisor graph is discussed, whose vertices are nonzero zero divisors of the commutative ring, then the two vertices x and y are adjacent if and only if xy = 0. Sum of absolute auxiliary eigenvalues of Laplacian matrix of a simple graph is known as Laplacian energy of a simple graph. Eccentricity sum matrix of a graph is a positive matrix, whose entries are sum of the eccentricities of two vertices and 0 for diagonals of the eccentricity sum matrix. Then the sum of absolute eigenvalues of eccentricity sum matrix is known as eccentricity sum energy. Laplacian sum eccentricity energy is the absolute difference between the eigenvalues of Laplacian sum eccentricity matrix and twice the edge cardinality by vertex cardinality. Maximum eccentricity energy is the sum of absolute eigenvalues of maximum eccentricity matrix. For a simple graph, the entries of average degree eccentricity matrix, if the two vertices are adjacent, are average sum of degree and eccentricity of their two vertices, otherwise 0. Then the sum of absolute eigenvalues of average degree eccentricity matrix is known as average degree eccentricity energy. This paper discusses the degree squence, vertex cardinality, and edge cardinality of the cross monic zero divisor graph for commutative rings. For a certain family of cross monic zero divisor graphs for commutative ring, moveover, Laplacian energy, eccentricity sum energy, Laplacian sum eccentricity energy, maximum eccentricity energy, and average degree-eccentricity energy are also briefly covered.