Construction of \(r\)-regular singular graphs
type: Item
instance of: scholarly article
title: Construction of \(r\)-regular singular graphs
zbMATH Open document ID: 0958.05087
author: Severino V. Gervacio
author: Leonor Aquino-Ruivivar
published in: Indian Journal of Pure \& Applied Mathematics
publication date: 2001-03-30T00:00:00Z
review text: A graph \(G\) is singular if its adjacency matrix has 0 as an eigenvalue. Let \({\mathcal G}(n,r)\) denote the class of all connected \(r\)-regular graphs on \(n\) vertices. It is known that the only graph \(C_n\) in \({\mathcal G}(n,2)\) is singular if and only if \(n\equiv 0\text{ mod 4}\) and that the only graph \(K_n\) from \({\mathcal G}(n,n-1)\) is not singular. The main result of the paper under review is a construction of a singular graph in \({\mathcal G}(n,r)\) for every \(r>2\) and \(n>r+1\) such that \({\mathcal G}(n,r)\neq\varnothing\). Remarks of the reviewer: The result of the paper seems to be quite trivial. The statement at the end of page~499 that isomorphic graphs could have non-zero determinants of opposite signs is evidently wrong.
reviewed by: Valeriy B. Mnukhin
Mathematics Subject Classification ID: 05C50
zbMATH DE Number: 1484559
zbMATH Keywords: singular graph
zbMATH Keywords: adjacency matrix
zbMATH Keywords: eigenvalue
MaRDI profile type: Publication
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description: scientific article; zbMATH DE number 1484559