Applications on color (distance) signless laplacian energy of annihilator monic prime graph of commutative rings

In this study, we define the structure formation of the annihilator monic prime graph of commutative rings, whose distinct vertices X and J satisfies a condition ann(XJ) not equal ann(X)boolean OR ann(J), graph is denoted by AMPG(Z(n)[x]/f(x)). The chromatic number of the annihilator monic prime graph is determined to establish the color (signless laplacian) energy of the graph. Also, we introduce color based energy of the annihilator monic prime graph, namely color distance signless Laplacian energy. Diag(Tr) +D(G) +A(chi), where Diag(Tr) denotes the diagonal matrix of the vertex transmissions in the graph, D(G) is the distance matrix of vertices whose diagonals are zero, the entries of the matrix A(chi) are 1 if eta(i) and eta(j) are adjacent, -1 if eta(i) and eta(j) are non-adjacent with different colors, otherwise 0. The eigenvalues of color distance signless Laplacian matrix will be denoted by partial differential partial derivative(chi+)(1)<= partial derivative(chi+)(2) <= partial derivative(chi+)(3) <= ... <= partial derivative(chi+)(n), then DSL chi+ of annihilator monic prime graph are define as E(DSL chi+ (Z(n)[x]/< f(x)>)) -Sigma(n)(i) vertical bar partial derivative(chi+)(i) -t(G)vertical bar where t(G) is the transmission of the graph. Based on the eigenvalues, we can study some more properties of graphs. Applications of color based energy are also discuss in this paper.