For simplicity we follow the rules: x, y, y1, y2, z, a will be arbitrary, f , g, h, h′, f1, f2 will denote functions, i will denote a natural number, X, Y , Z, V1, V2 will denote sets, P will denote a permutation of X, D, D1, D2, D3 will denote non-empty sets, d1 will denote an element of D1, d2 will denote an element of D2, and d3 will denote an element of D3. We now state a number of proposit...

Graph bundles generalize the notion of covering graphs and products of graphs. The chromatic numbers of product bundles with respect to the Cartesian, strong and tensor product whose base and fiber are cycles are determined.

A subset $S$ of vertex set $V(D)$ is an {em indpendent dominating set} of $D$ if $S$ is both an independent and a dominating set of $D$. The {em indpendent domination number}, $i(D)$ is the cardinality of the smallest independent dominating set of $D$. In this paper we calculate the independent domination number of the { em cartesian product} of two {em directed paths} $P_m$ and $P_n$ for arbi...

In this paper the concept of inverse domintion in some operations on bipolar fuzzy graphs was introduced and investigated bound γ8 are obtained like union, join,Cartesian product, strong product composition.