A note on the generator subgraph of a graph
| Main Authors: | Mame, Neil Mores; College of Arts and Sciences, Batangas State University, Batangas City, Philippines, Gervacio, Severino Villanueva; Mathematics and Statistics Department, De La Salle University, Taft Avenue, Manila, Philippines |
|---|---|
| Other Authors: | PCIEERD-DOST, Batangas State University |
| Format: | Article info application/pdf eJournal |
| Bahasa: | eng |
| Terbitan: |
GTA Research Group, Univ. Newcastle, Indonesian Combinatorics Society and ITB
, 2020
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| Subjects: | |
| Online Access: |
https://www.ejgta.org/index.php/ejgta/article/view/659 https://www.ejgta.org/index.php/ejgta/article/view/659/pdf_123 https://www.ejgta.org/index.php/ejgta/article/downloadSuppFile/659/98 |
Daftar Isi:
- 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.