Recently the invariant (copoint) pre-hull number ph(G) of a graph G that measures the nonconvexity of a convex space was introduced by Polat and Sabidussi in [18]. We introduce a similar invariant called convex pre-hull number which is a natural upper bound for the copoint pre-hull number and consider in this work both on the lexicographic product of graphs. We present exact values with respect...

Let G be a connected graph and H be an arbitrary graph. In this paper, we study the identifying codes of the lexicographic product G[H] of G and H. We first introduce two parameters of H, which are closely related to identifying codes of H. Then we provide the sufficient and necessary condition for G[H] to be identifiable. Finally, if G[H] is identifiable, we determine the minimum cardinality o...

For any two linear orderings L and M , the representability number reprM (L) of L in M is the least ordinal α such that L can be orderembedded into the lexicographic power Mα lex. We study reprM (L) in the particular case that L is a lexicographic product and M is a Dedekindcomplete chain. Mathematics Subject Classification: Primary: 06A05; Secondary: 03E05, 54F05, 91B16

Consider a simple connected graph $$G = (V,E)$$ of order p and size q. For bijection $$f : E \to \{1,2,\ldots,q\}$$ , let $$f^+(u) \sum_{e\in E(u)} f(e)$$ where $$E(u)$$ is the set edges incident to u. We say f local antimagic labeling G if for any two adjacent vertices u v, we have \ne f^+(v)$$ . The minimum number distinct values $$f^+$$ taken over all denoted by $$\chi_{la}(G)$$ Let $$G[H]$$...

In this paper, we introduce the notion of Pythagorean Anti fuzzy graph. We then define Cartesian product and Lexicographic on It is proved that two graphs graph general, lexicographic regular need not be Necessary sufficient conditions for to are determined. Further, concept isomorphism with suitable example.

In this paper, we prove that the connectivity and the edge connectivity of the lexicographic product of two graphs G1 and G2 are equal to κ1v2 and min{λ1v 2 2 , δ2 +δ1v2}, respectively, where δi, κi, λi and vi denote the minimum degree, the connectivity, the edge-connectivity and the number of vertices of Gi, respectively. We also obtain that the edge-connectivity of the direct product of K2 an...

where wd(J ) = J i 1J i 2 . . ., P is the insertion tableaux map, and we assume that all the evaluations are in lexicographic order and all the row words of an evaluation are in lexicographic order. The tableau P ( ∏ J wd(J )1wd(J)2 . . .wd(JJ )λJ ) has k|I| boxes and is equal to the (plactic) product of k tableaux of shape I, and these k tableaux are in lexicographic order. If H is the shape o...

There are four standard products of graphs: the direct product, the Cartesian product, the strong product and the lexicographic product. The chromatic number turned out to be an interesting parameter on all these products, except on the Cartesian one. A survey is given on the results concerning the chromatic number of the three relevant products. Some applications of product colorings are also ...

A coloring c of the vertices of a graph G is nonrepetitive if there exists no path v1v2 . . . v2l for which c(vi) = c(vl+i) for all 1 ≤ i ≤ l. Given graphs G and H with |V (H)| = k, the lexicographic product G[H ] is the graph obtained by substituting every vertex of G by a copy of H , and every edge of G by a copy of Kk,k. We prove that for a sufficiently long path P , a nonrepetitive coloring...