The

Cartesian product of these two NSSs is [O.sub.1] x [O.sub.2] [member of] SNS ([X.sub.1] x [X.sub.2]) defined as

Cartesian product This terminal [y.sup.*.sub.j] is called a bud by Labelle [10].

Then, the

Cartesian product of [[psi].sub.K] and [[OMEGA].sub.L] is obtained as follows;

In this section, we introduce the

cartesian product of Q-NSGs and discuss some of its properties.

We consider the

cartesian product space [OMEGA] = [[OMEGA].sub.1] x [[OMEGA].sub.2] equipped with the product sigma algebra.

In this article, we study the

Cartesian product and composition of two m-polar fuzzy graphs and compute the degrees of the vertices in these graphs.

Figures 3(a), 3(b), and 3(c) illustrate two directed graphs and their

Cartesian product, respectively.

Keywords: efficient open domination,

Cartesian product, vertex labeling, total domination

The

Cartesian product of two SIVNGs [G.sub.1] and [G.sub.2] is denoted by [G.sub.1] x [G.sub.2] = [[A.sub.1] x [A.sub.2], [B.sub.1] x [B.sub.2]) and is defined as follows:

We are interested in the

cartesian product of codes in the Hamming graphs H(n, q) and H(n', q').

Let X denote the

Cartesian product of the [X.sub.i]'s, namely ,

Then the

Cartesian product G [??] H has the vertex set V(G) x V(H) and vertices ([u.sub.r], [v.sub.i]) and ([u.sub.s], [v.sub.k]) are adjacent if and only if [[u.sub.r] = [u.sub.s] [member of] V(G) and [v.sub.i][v.sub.k] [member of] E(H)] or [[v.sub.i] = [v.sub.k] [member of] V(H) and [u.sub.r][u.sub.s] [member of] E(G)], where r, s = 1, 2, ..., [absolute value of G] and i, k = 1, 2, ..., [absolute value of H].

The

cartesian product of two fuzzy sets [6] was introduce by Bhattacharya and Mukherjee in 1985.

A

Cartesian product [G.sub.1] x [G.sub.2] of graphs [G.sub.1] and [G.sub.2] with disjoint point sets [V.sub.1] and [V.sub.2] and edge sets [X.sub.1] and [X.sub.2] is the graph with point set [V.sub.1] x [V.sub.2] and u = ([u.sub.1], [u.sub.2]) adjacent with [nu] =([[nu].sub.1], [[nu].sub.2]) whenever [u.sub.1] = [[nu].sub.1] and [u.sub.2] adjacent with [[nu].sub.2] or [u.sub.2] = [[nu].sub.2] and [u.sub.1] adjacent with [[nu].sub.1].