Novel Graph Distance Matrix

doi:10.1002/jcc.21461

Novel Graph Distance Matrix

2010-03-17

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DOI：10.1002/jcc.21461

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We have introduced novel distance matrix for graphs, which is based on interpretation of columns of the adjacency matrix of a graph as a set of points in n-dimensional space, n being the number of vertices in the graph. Numerical values for the distances are based on the Euclidean distance between n points in n-dimensional space. In this way, we have combined the traditional representation of graphs (drawn as 2D object of no fixed geometry) with their representation in n-dimensional space, defined by a set of n-points that lead to a representation of definite geometry. The novel distance matrix, referred to as natural distance matrix, shows some structural properties and offers novel graph invariants as molecular descriptors for structure-property-activity studies. One of the novel graph descriptors is the modified connectivity index in which the bond contribution for (in, n) bond-type is given by 1/root(m + n), where in and n are the valence of the end vertices of the bond. The novel distance matrix (ND) can be reduced to sparse distance-adjacency matrix (DA), which can be viewed as specially weighted adjacency matrix of a graph. The quotient of the leading eigenvalues of novel distance-adjacency matrix and novel distance matrix, as illustrated on a collection of graphs of chemical interest, show parallelism with a simple measure of graph density, based on the quotient of the number of edges in a graph and the maximal possible number of edges for graphs of the same size. (c) 2010 Wiley Periodicals, Inc. J Comput Chem 31: 1832-1841, 2010

Keywords:

Graph Theory

Chemical Graph Theory

distance matrix

distance-adjacency matrix

the connectivity index

natural distance matrix

Novel Graph Distance Matrix

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