Computation of Some Graph Energies of the Zero-Divisor Graph Associated with the Commutative Ring Z p 2 [ x ] / ( x 2 )

Let R be the commutative ring R = Z p 2 [ x ] /( x 2 ) with identity and Z * ( R ) be the set of all non -zero zero -divisors of R. Then, Gamma( R ) is said to be a zero -divisor graph if and only if a <middle dot> b = 0 where a, b E V (Gamma( R )) = Z * ( R ) and ( a, b ) E E (Gamma( R )) . Let lambda iota , lambda 2 , ... , lambda n be the eigenvalues of the adjacency matrix, and let mu iota , mu 2 , ... , mu n be the eigenvalues of the Laplacian matrix of Gamma( R ) . Then we discuss the energy e (Gamma( R )) = E n the Laplacian energy Le (Gamma( R )) = E n mu i - 2 m where n and i =iota | lambda i | and i =iota n m are the order and size of Gamma( R ) . c 2024 University of Kashan Press. All rights reserved.