ANTIADJACENCY MATRICES FOR SOME STRONG PRODUCTS OF GRAPHS

Let G be an undirected graphs with no multiple edges. There are many ways to represent a graph, and one of them is in a matrix form, by constructing an antiadjacency matrix. Given a connected graph G with  vertex set $V$ consisting of n members, an antiadjacency matrix of the graph G is a matrix B of order n \times n such that if there is an edge that connects vertex v_i to vertex v_j (v_i \sim v_j ) then the element of i^{th} row and b^{th} column of B is 0, otherwise 1. In this paper we investigate some properties of antiadjacency matrices for some strong product of two graphs. Our results are general forms of the antiadjacency matrix of the strong product of path graphs P_m with P_n for m, n\ge 3, and cycle graphs C_m with C_m for m \ge 3

ON THE GIRTH, INDEPENDENCE NUMBER, AND WIENER INDEX OF COPRIME GRAPH OF DIHEDRAL GROUP

The coprime graph of a finite group , denoted by , is a graph with vertex set  such that two distinct vertices  and  are adjacent if and only if their orders are coprime, i.e.,  where |x| is the order of x. In this paper, we complete the form of the coprime graph of a dihedral group that was given by previous research and it has been proved that  if , for some  and  if . Moreover, we prove that if  is even, then the independence number of  is , where  is the greatest odd divisor of  and if  is odd, then the independence number of  is . Furthermore, the Wiener index of coprime graph of dihedral group has been stated here

SOME CARTESIAN PRODUCTS OF A PATH AND PRISM RELATED GRAPHS THAT ARE EDGE ODD GRACEFUL

Let $G$ be a connected undirected simple graph of size $q$ and let $k$ be the maximum number of its order and its size. Let $f$ be a bijective edge labeling which codomain is the set of odd integers from 1 up to $2q-1$. Then $f$ is called an edge odd graceful on $G$ if the weights of all vertices are distinct, where the weight of a vertex $v$ is defined as the sum $mod(2k)$ of all labels of edges incident to $v$. Any graph that admits an edge odd graceful labeling is called an edge odd graceful graph. In this paper, some new graph classes that are edge odd graceful are presented, namely some cartesian products of path of length two and some circular related graphs

PENGKONTRUKSIAN SOLUSI EKSPLISIT PERSAMAAN MATRIKS X²AX=AXA

The solution space is explicitly constructed for all 2x2 complex matrices using Basis Groebner techniques. When A is a 2 x 2 matrix, the equation X²AX=AXA is equivalent to a system of four polynomial equations. The solution space is then the variety defined by the polynomials involved. The ideal of the underlying polynomial ring generated by the defining polynomials plays an important role in solving the system. In the procedure for solving these equations, Grobner bases are used to transform the polynomial system into a simpler one, which makes it possible to classify all the solutions. In addition to classify all solution for 2 x 2 matrices, certain explicit solutions are produced in arbitrary dimensions when A is nonsingular. In higher dimensions, Toeplitz matrices are used to construct the solution of the matrix equation

Generalized Arithmetic Staircase Graphs and Their Total Edge Irregularity Strengths

Let (Formula presented.) be a simple undirected graph with finite vertex set (Formula presented.) and edge set (Formula presented.). A total (Formula presented.) -labeling (Formula presented.) is called a total edge irregular labeling on (Formula presented.) if for any two different edges (Formula presented.) and (Formula presented.) in (Formula presented.) the numbers (Formula presented.) and (Formula presented.) are distinct. The smallest positive integer n such that (Formula presented.) can be labeled by a total edge irregular labeling is called the total edge irregularity strength of the graph (Formula presented.). In this paper, we provide the total edge irregularity strength of some asymmetric graphs and some symmetric graphs, namely generalized arithmetic staircase graphs and generalized double-staircase graphs, as the generalized forms of some existing staircase graphs. Moreover, we give the construction of the corresponding total edge irregular labelings