title: Common vertex matrix: a novel characterization of molecular graphs by counting.
abstract: We present a novel matrix representation of graphs based on the count of equal-distance common vertices to each pair of vertices in a graph. The element (i, j) of this matrix is defined as the number of vertices at the same distance from vertices (i, j). As illustrated on smaller alkanes, these novel matrices are very sensitive to molecular branching and the distribution of vertices in a graph. In particular, we show that ordered row sums of these novel matrices can facilitate solving graph isomorphism for acyclic graphs. This has been illustrated on all undecane isomers C11H24 having the same path counts (total of 25 molecules), on pair of graphs on 18 vertices having the same distance degree sequences (Slater's graphs), as well as two graphs on 21 vertices having identical several topological indices derived from information on distances between vertices.
authors.affiliation: Laboratory for Chemometrics, National Institute of Chemistry, Hajdrihova 19, Ljubljana, Slovenia. mrandic@msn.com
authors.forename: Milan
authors.initials: M
authors.lastname: Randić
authors.forename: Marjana
authors.initials: M
authors.lastname: Novič
authors.forename: Dejan
authors.initials: D
authors.lastname: Plavšić
mesh_terms: D000473:Alkanes / Q000737:chemistry*; D003196:Computer Graphics; D008956:Models, Chemical; D015394:Molecular Structure
chemical_list: D000473:Alkanes