A note on the generator subgraph of a graph - Neil Mores Mame; Severino Villanueva Gervacio (GTA Research Group, Univ. Newcastle, Indonesian Combinatorics Society and ITB, 2020)
Graphs considered in this paper are finite simple undirected graphs. Let G = (V(G), E(G)) be a graph with E(G) = e1, e2,..., em, for some positive integer m. The edge space of G, denoted by ℰ(G), is a vector space over the field ℤ2. The elements of ℰ(G) are all the subsets of E(G). Vector addition is defined as X+Y = X ∆ Y, the symmetric difference of sets X and Y, for X,Y ∈ ℰ(G). Scalar multiplication is defined as 1.X =X and 0.X = ∅ for X ∈ ℰ(G). Let H be a subgraph of G. The uniform set of H with respect to G, denoted by EH(G), is the set of all elements of ℰ(G) that induces a subgraph isomorphic to H. The subspace of ℰ(G) generated by ℰH(G) shall be denoted by ℰH(G). If EH(G) is a generating set, that is ℰH(G)= ℰ(G), then H is called a generator subgraph of G. This study determines the dimension of subspace generated by the set of all subsets of E(G) with even cardinality and the subspace generated by the set of all k-subsets of E(G), for some positive integer k, 1 ≤ k ≤ m. Moreover, this paper determines all the generator subgraphs of star graphs. Furthermore, it gives a characterization for a graph G so that star is a generator subgraph of G.