## Faculty Research Work

#### Title

A note on the generator subgraph of a graph

#### College

College of Science

#### Department/Unit

Mathematics and Statistics Department

Article

#### Source Title

Electronic Journal of Graph Theory and Applications

8

1

17

27

1-1-2020

#### Abstract

© 2020 Indonesian Combinatorics Society. 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 Δ Y, the symmetric difference of sets X and Y, for X, Y j 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.

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#### Digitial Object Identifier (DOI)

10.5614/EJGTA.2020.8.1.3