Let D = (V,E) be a digraph and u, v ∈ V. The metric maximum distance is defined by md(u,v) max {d⃗(u,v), d⃗(v,u)}, where d⃗(u,v) denote the length of shortest directed u − path in D. m-eccentricity vertex mecc(v) {md(v,u) : V(D)}. m-center mC(D) strongly connected consists all vertices with minimum m-eccentricity, m-periphery mPer(D) relationship between two digraphs their lexicographic product st...

A set of vertices W resolves a graph G if every vertex is uniquely determined by its coordinate of distances to the vertices in W . The minimum cardinality of a resolving set of G is called themetric dimension of G. In this paper, we consider a graphwhich is obtained by the lexicographic product between two graphs. The lexicographic product of graphs G and H , which is denoted by G ◦ H , is the...

A graph is said to be well-dominated if all its minimal dominating sets are of the same size. In this work, we introduce the notion of an irreducible dominating set, a variant of dominating set generalizing both minimal dominating sets and minimal total dominating sets. Based on this notion, we characterize the family of minimal dominating sets in a lexicographic product of two graphs and deriv...

Let ‘G’ be a graph. If u,v ∈V, then u-v geodetic of G is the shortest path between u and v. The closed interval I[u, v] consists all vertices lying in some G. For S⊆V(G) set I[S] union sets I [u, for u,v∈S. A S if I[S]=V(G). cardinality minimum number G, denoted by g(G). graph split 〈V-S〉 disconnected, g_s (G) In this paper study restrained strong product lexicographic graphs. subgraph disconne...

Consider two graphs G and H. Let H[G] be the lexicographic product of H and G, where H is the lexicographic product of the graph H by itself k times. In this paper, we determine the spectrum of H[G] and H whenG andH are regular and the Laplacian spectrum ofH[G] andH for G and H arbitrary. Particular emphasis is given to the least eigenvalue of the adjacency matrix in the case of lexicographic p...