In this article we study the modular graph product, ♦, that is known to preserve the property of being 3-existentially closed (i.e., 3-e.c.). We produce new families of 3-e.c. graphs G♦H such that neither G nor H is required to be 3-e.c. Assuming that G is weakly 3-existentially closed with certain adjacency properties, we find the sufficient conditions on the adjacency properties of H such tha...

In this paper, we generalise Magnus’ Freiheitssatz and solution to the word problem for one-relator groups by considering one relator quotients of certain classes of right-angled Artin groups and graph products of locally indicable polycyclic groups.

A transitive decomposition is a pair (Γ,P) where Γ is a graph and P is a partition of the arc set of Γ such that there is a subgroup of automorphisms of Γ which leaves P invariant and transitively permutes the parts in P. In an earlier paper we gave a characterisation of G-transitive decompositions where Γ is the graph product Km×Km and G is a rank 3 group of product action type. This character...

The first step in the analysis of a structure is to generate its configuration. Different means are available for this purpose. The use of graph products is an example of such tools. In this paper, the use of product graphs is extended for the formation of different types of structural models. Here weighted graphs are used as the generators and the connectivity properties of different models ar...

We introduce a modification of the Harary index where the contributions of vertex pairs are weighted by the sum of their degrees. After establishing basic mathematical properties of the new invariant, we proceed by finding the extremal graphs and investigating its behavior under several standard graph products.

We give examples of vertex-transitive graphs with non-monotonic chromatic difference sequences, disproving a conjecture of Albertson and Collins on the monotonicity of the chromatic difference sequence of vertex-transitive graphs, and answering a question of Zhou on the achievability of circulants.

Equivalence relations on the edge set of a graph G that satisfy restrictive conditions on chordless squares play a crucial role in the theory of Cartesian graph products and graph bundles. We show here that such relations in a natural way induce equitable partitions on the vertex set of G, which in turn give rise to quotient graphs that can have a rich product structure even if G itself is prime.