2005/10/12 by Maarten Van den Nest, Bart De Moor, Nest, Maarten Van den +1 · 2 citations
Computer Science · Engineering · Mathematics · Physics and Astronomy · #Combinatorics (math.CO) #FOS: Mathematics #FOS: Physical sciences #Graph theory and applications #Matrix Theory and Algorithms #Quantum Physics (quant-ph) #graph theory and CDMA systems #math.CO #quant-ph
paper · pdf · doi:10.48550/arxiv.math/0510246
25 pages, 2 figures
arxiv created 2005/10/12 · openalex publication_date 2005/10/12 · arxiv updated 2009/12/01 · openalex created_date 2016/06/24 · openalex updated_date 2026/07/28
The local complement G*i of a simple graph G at one of its vertices i is obtained by complementing the subgraph induced by the neighborhood of i and leaving the rest of the graph unchanged. If e=i,j is an edge of G then G*e=((G*i)*j)*i is called the edge-local complement of G along the edge e. We call two graphs edge-locally equivalent if they are related by a sequence of edge-local complementations. The main result of this paper is an algebraic description of edge-local equivalence of graphs in terms of linear fractional transformations of adjacency matrices. Applications of this result include (i) a polynomial algorithm to recognize whether two graphs are edge-locally equivalent, (ii) a formula to count the number of graphs in a class of edge-local equivalence, and (iii) a result concerning the coefficients of the interlace polynomial, where we show that these coefficients are all even for a class of graphs; this class contains, as a subset, all strongly regular graphs with parameters (n, k, a, c), where k is odd and a and c are even.