Homology of the line graph and of related graph-valued functions
type: Item
instance of: scholarly article
title: Homology of the line graph and of related graph-valued functions
zbMATH Open document ID: 0696.05046
DOI: 10.1007/BF01198228
author: Erich Prisner
published in: Archiv der Mathematik
publication date: 1991-01-01T00:00:00Z
review text: Let \(G^{\uparrow}\) denote the simplicial complex having the vertex sets of all complete subgraphs of the finite graph G as simplices. Then the first Betti numbers of \(G^{\uparrow}\) and of the complex \(L(G)^{\uparrow}\) of its line graph coincide. The second Betti number of \(G^{\uparrow}\) is not greater than that of \(L(G)^{\uparrow}\). Furthermore, the n-th Betti number of \(L(G)^{\uparrow}\) vanishes for every \(n\geq 3\). Similar or related results are obtained for the total graph, the middle graph, the n-simplex graph, the clique-line graph, and the line-clique graph.
reviewed by: Erich Prisner
Mathematics Subject Classification ID: 05C99
zbMATH DE Number: 4139794
zbMATH Keywords: graph-valued function
zbMATH Keywords: simplicial complex
zbMATH Keywords: line graph
zbMATH Keywords: total graph
zbMATH Keywords: middle graph
zbMATH Keywords: n-simplex graph
zbMATH Keywords: clique-line graph
zbMATH Keywords: line-clique graph
MaRDI profile type: Publication
cites work: Q3695292
cites work: Q5672654
cites work: Homology groups of relations
cites work: Connected cutsets of a graph and triangle bases of the cycle space
cites work: Über iterierte Clique-Graphen
cites work: Convergence of iterated clique graphs
description: scientific article; zbMATH DE number 4139794