A perfect r-code in a graph is a subset of the graph’s vertices with the property that each vertex in the graph is within distance r of exactly one vertex in the subset. We prove that the n-fold strong product of simple graphs has a perfect r-code if and only if each factor has a perfect r-code.

The partition of indeterminacy function the neutrosophic set into contradiction part and ignorance represent quadripartitioned single valued set. In this work, new concept bipolar graph is established, operations on it are studied. Cartesian product, cross lexicographic strong product composition investigated. proposed concepts illustrated with examples.

A new operation on graphs is introduced and some of its properties are studied. We call it hierarchical product, because of the strong (connectedness) hierarchy of the vertices in the resulting graphs. In fact, the obtained graphs turn out to be subgraphs of the cartesian product of the corresponding factors. Some well-known properties of the cartesian product, such as a reduced mean distance a...

Recently the author introduced a semidefinite upper bound on the square of the stability number of a graph, the inverse theta number, which is proved here to be multiplicative with respect to the strong graph product, hence to be an upper bound for the square of the Shannon capacity of the graph. We also describe a heuristic algorithm for the stable set problem based on semidefinite programming...

Let $P(G,lambda)$ be the chromatic polynomial of a graph $G$. A graph $G$ ischromatically unique if for any graph $H$, $P(H, lambda) = P(G,lambda)$ implies $H$ is isomorphic to $G$. In this paper, we determine the chromaticity of all Tur'{a}n graphs with at most three edges deleted. As a by product, we found many families of chromatically unique graphs and chromatic equivalence classes of graph...

5 We give an optimal degree condition for a tripartite graph to have 6 a spanning subgraph consisting of complete graphs of order 3. This 7 result is used to give an upper bound of 2∆ for the strong chromatic 8 number of n vertex graphs with ∆ ≥ n/6. 9

For a connected graph G containing no bridges, let D(G) be the family of strong orientations of G; and for any D ∈ D(G), we denote by d(D) the diameter of D. The orientation number −→ d (G) of G is defined by −→ d (G) = min{d(D)|D ∈ D(G)}. Let G(p, q;m) denote the family of simple graphs obtained from the disjoint union of two complete graphs K p and Kq by adding m edges linking them in an arbi...

The basis number of a graph G is defined to be the least integer d such that there is a basis B of the cycle space of G such that each edge of G is contained in at most d members of B. In this paper we give an upper bound of the basis number of the strong product of a graph with a bipartite graph and we show that this upper bound is the best possible.