A note on the generator subgraph of a graph

Abstract

Graphs considered in this paper are  finite simple undirected graphs. Let G = (V(G), E(G)) be a graph with E(G) = {e₁, e₂,..., e_m}, 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 E_H(G), is the set of all elements of ℰ(G) that induces a subgraph isomorphic to H. The subspace of ℰ(G) generated by  ℰ_H(G) shall be denoted by ℰ_H(G). If E_H(G) is a generating set, that is ℰ_H(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.