For a nontrivial connected graph G = (V (G), E(G)), a set S ⊆ V (G) is called an edge geodetic set of G if every edge of G is contained in a geodesic joining some pair of vertices in S. The edge geodetic number g1(G) of G is the minimum order of its edge geodetic sets. Bounds for the edge geodetic number of Cartesian product graphs are proved and improved upper bounds are determined for a speci...

The Padmakar-Ivan (PI) index of a graph G is the sum over all edges uv of G of the number of edges which are not equidistant from u and v. In this paper, the notion of vertex PI index of a graph is introduced. We apply this notion to compute an exact expression for the PI index of Cartesian product of graphs. This extends a result by Klavzar [On the PI index: PI-partitions and Cartesian product...

In many real world problems, data sometimes comes from n agents (n≥2), multipolar information exists. For issues that are associated with uncertainty, this information can not be showed with the values of crisp, fuzzy, intuitionistic or bipolar. Graph is one of the mathematical models widely used in different sciences. Ambiguity in a graph where data depends on the n parameter can not be showed...

In a recent paper we showed that every connected graph can be written as a weak cartesian product of a family of indecomposable rooted graphs and that this decomposition is unique to within isomorphisms. Using this unique prime factorization theorem we prove that if a graph X can be written as a product of connected rooted graphs, which are pairwise relatively prime, then the automorphism group...

Equivalence relations on the edge set of a hypergraph that satisfy the “grid-property” (a certain restrictive condition on diagonal-free grids that can be seen as a generalization of the more familiar “square property” on graphs) play a crucial role in the theory of Cartesian hypergraph products. In particular, every convex relation with the grid property induces a factorization w.r.t. the Cart...

A nowhere-zero k-flow on a graph G is an assignment of a direction and a non-zero integer in absolute value smaller than k to each edge of G in such a way that, for each vertex, the sum of incoming values equals the sum of outgoing values. Nowhere-zero flow problems evolved from flowcolouring duality to a theory which has a central role in graph theory. There are many important flow problems wh...

In this work we investigate the behavior of various geodesic convexity parameters with respect to the Cartesian product operation for graphs. First, we show that the convex sets arising from geodesic convexity in a Cartesian product of graphs are exactly the same as the convex sets arising from the usual binary operation ⊕ for making a convexity space out of the Cartesian product of any two con...

The Wiener index is a graph invariant that has found extensive application in chemistry. In addition to that a generating function, which was called the Wiener polynomial, who’s derivate is a q-analog of the Wiener index was defined. In an article, Sagan, Yeh and Zhang in [The Wiener Polynomial of a graph, Int. J. Quantun Chem., 60 (1996), 959969] attained what graph operations do to the Wiene...

By the sorting method of vertices, the equitable chromatic number and the equitable chromatic threshold of the Cartesian products of wheels with bipartite graphs are obtained. Key–Words: Cartesian product, Equitable coloring, Equitable chromatic number, Equitable chromatic threshold

We study linkedness of the Cartesian product of graphs and prove that the product of an alinked and a b-linked graphs is (a+b−1)-linked if the graphs are su ciently large. Further bounds in terms of connectivity are shown. We determine linkedness of products of paths and products of cycles.