In this paper we study the behaviour of conjugacy languages in virtual graph products, extending results by Ciobanu et. al. [13]. We focus primarily on products form a semi-direct product. First, twisted representatives right-angled Artin and Coxeter groups. prove regularity geodesic language for certain cases, highlight properties spherical language, depending automorphism ordering generating ...

The second Zagreb coindex is a well-known graph invariant defined as the total degree product of all non-adjacent vertex pairs in a graph. The second Zagreb eccentricity coindex is defined analogously to the second Zagreb coindex by replacing the vertex degrees with the vertex eccentricities. In this paper, we present exact expressions or sharp lower bounds for the second Zagreb eccentricity co...

A homogeneous factorisation of a digraph Γ consists of a partition P = {P1, . . . , Pk} of the arc set AΓ and two vertex-transitive subgroups M 6 G 6 Aut(Γ) such that M fixes each Pi setwise while G leaves P invariant and permutes its parts transitively. Given two graphs Γ1 and Γ2 we consider several ways of taking a product of Γ1 and Γ2 to form a larger graph, namely the direct product, cartes...

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 colouring of the vertices of a graph (or hypergraph) G is adapted to a given colouring of the edges of G if no edge has the same colour as both (or all) its vertices. The adaptable chromatic number of G is the smallest integer k such that each edge-colouring of G by colours 1, 2, . . . , k admits an adapted vertex-colouring of G by the same colours 1, 2, . . . , k. (The adaptable chromatic nu...

In this paper we study a new product of graphs called tight product. A graph H is said to be a tight product of two (undirected multi) graphs G1 and G2, if V (H) = V (G1) × V (G2) and both projection maps V (H) → V (G1) and V (H) → V (G2) are covering maps. It is not a priori clear when two given graphs have a tight product (in fact, it is NP -hard to decide). We investigate the conditions unde...