We consider an edge-isoperimetric problem (EIP) on the cartesian powers of graphs. One of our objectives is to extend the list of graphs for whose cartesian powers the lexicographic order provides nested solutions for the EIP. We present several new classes of such graphs that include as special cases all presently known graphs with this property. Our new results are applied to derive best poss...

This paper concerns finite, edge-transitive direct and strong products, as well as infinite weak Cartesian products. We prove that the direct product of two connected, non-bipartite graphs is edge-transitive if and only if both factors are edgetransitive and at least one is arc-transitive, or one factor is edge-transitive and the other is a complete graph with loops at each vertex. Also, a stro...

A local coloring of a graph G is a function c : V (G) → N such that for each S ⊆ V (G), 2 ≤ |S| ≤ 3, there exist u, v ∈ S with |c(u) − c(v)| at least the size of the subgraph induced by S. The maximum color assigned by c is the value χl(c) of c, and the local chromatic number of G is χl(G) = min{χl(c) : c local coloring of G}. In this note the local chromatic number is determined for Cartesian ...

Abstract For a given graph G , the general position problem asks for largest set of vertices $$S \subseteq V(G)$$ S ⊆ V ( G ) such that no three distinct S belong to common shortest path . The Cartesian...

Many large graphs can be constructed from existing smaller graphs by using graph operations, for example, the Cartesian product and the lexicographic product. Many properties of such large graphs are closely related to those of the corresponding smaller ones. In this short note, we give some properties of the lexicographic products of vertextransitive and of edge-transitive graphs. In particula...

Let G be a simple connected graph. The first and second Zagreb indices have been introduced as  vV(G) (v)2 M1(G) degG and M2(G)  uvE(G)degG(u)degG(v) , respectively, where degG v(degG u) is the degree of vertex v (u) . In this paper, we define a new distance-based named HyperZagreb as e uv E(G) . (v))2 HM(G)     (degG(u)  degG In this paper, the HyperZagreb index of the Cartesian p...