A note on the generator subgraph of a graph - Neil Mores Mame; Severino Villanueva Gervacio (Electronic Journal of Graph Theory and Applications (EJGTA), 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.