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Title: | The adjacency matrices of complete and nutful graphs |
Authors: | Sciriha, Irene Farrugia, Alexander Gauci, John Baptist |
Keywords: | Graph theory Matrices |
Issue Date: | 2014 |
Publisher: | University of Kragujevac. Faculty of Science |
Citation: | Sciriha, I., Farrugia, A., & Gauci, J. B. (2014). The adjacency matrices of complete and nutful graphs. Match, 72(1), 165-178. |
Abstract: | A real symmetric matrix G with zero entries on its diagonal is an adjacency matrix associated with a graph G (with weighted edges and no loops) if and only if the non-zero entries correspond to edges of G. An adjacency matrix G belongs to a generalized-nut graph G if every entry in a vector in the nullspace of G is non-zero. A graph G is termed NSSD if it corresponds to a non-singular adjacency matrix G with a singular deck {G- v}, where G- v is the submatrix obtained from G by deleting the vth row and column. An NSSD G whose deck consists of generalized- nut graphs with respect to G is referred to as a G-nutful graph. We prove that a G-nutful NSSD is equivalent to having a NSSD with G-1 as the adjacency matrix of the complete graph. If the entries of G for a G-nutful graph are restricted to 0 or 1, then the graph is known as nuciferous, a concept that has arisen in the context of the quantum mechanical theory of the conductivity of non-singular Carbon molecules according to the SSP model. We characterize nuciferous graphs by their inverse and the nullities of their one- and two-vertex deleted subgraphs. We show that a G-nutful graph is a NSSD which is either K2 or has no pendant edges. Moreover, we reconstruct a labelled NSSD either from the nullspace generators of the ordered one-vertex deleted subgraphs or from the determinants of the ordered two-vertex deleted subgraphs. |
URI: | https://www.um.edu.mt/library/oar//handle/123456789/28288 |
ISSN: | 03406253 |
Appears in Collections: | Scholarly Works - FacSciMat |
Files in This Item:
File | Description | Size | Format | |
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Adj Mat Comp Nutful Gphs (match72n1_165-178) PUBLISHED.PDF | 243.79 kB | Adobe PDF | View/Open |
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