Let Γ be a finite simple graph with vertex set V , so the edges may be taken to be unordered pairs of distinct elements of V . Assume that, for every v ∈ V , there is assigned a group Gv. The graph product GΓ is the quotient of the free product ∗v∈V Gv by the normal subgroup generated by all [Gu,Gv] for which {u,v} is an edge. (As usual, [Gu,Gv] means the subgroup generated by { x−1y−1xy | x ∈ ...

The first and second Zagreb indices of a graph are equal, respectively, to the sum of squares of the vertex degrees, and the sum of the products of the degrees of pairs of adjacent vertices. We now consider analogous graph invariants, based on the second degrees of vertices (number of their second neighbors), called leap Zagreb indices. A number of their basic properties is established.

A k-matching M of a graph G = (V,E) is subset ⊆ E such that each connected component in the subgraph F (V,M) either single-vertex or k-regular, i.e., vertex has degree k. In this contribution, we are interested k-matchings four standard products: Cartesian, strong, direct and lexicographic product. As shall see, problem finding non-empty (k ≥ 3) products NP-complete. Due to general intractabili...