Compression and sparsification algorithms are frequently applied in a preprocessing step before analyzing or optimizing large networks/graphs. In this paper we propose and study a new framework for contracting edges of a graph (merging vertices into super-vertices) with the goal of preserving pairwise distances as accurately as possible. Formally, given an edge-weighted graph, the contraction s...

Let G be a connected graph with vertex set V.G/. The degree Kirchhoff index of G is defined as S .G/D P fu;vg V.G/ d.u/d.v/R.u;v/, where d.u/ is the degree of vertex u, and R.u;v/ denotes the resistance distance between vertices u and v. In this paper we obtain some upper and lower bounds for the degree Kirchhoff index of graphs. We also obtain some bounds for the Nordhaus-Gaddum-type result fo...

Abstract In this paper, the exact formulae for the Harary indices of join, disjunction, symmetric difference, strong product of graphs are obtained. Also, the Schultz and modified Schultz indices of join and strong product of graphs are computed. We apply some of our results to compute the Harary, Schultz and modified Schultz indices of fan graph, wheel graph, open fence and closed fence graphs.

The tau constant is an important invariant of a metrized graph. It has connections to other graph invariants such as Kirchhoff index, and it has applications to arithmetic properties of algebraic curves. We show how the tau constant of a metrized graph changes under successive edge contractions and deletions. We prove identities which we call “contraction”, “deletion”, and “contraction-deletion...

It is necessary to generate the automorphism group of a chemical graph in computer-aided structure elucidation. An Euclidean graph associated with a molecule is defined by a weighted graph with adjacency matrix M = [dij], where for i≠j, dij is the Euclidean distance between the nuclei i and j. In this matrix dii can be taken as zero if all the nuclei are equivalent. Otherwise, one may introduce...

We develop a theory to measure the variance and covariance of probability distributions defined on nodes graph, which takes into account distance between nodes. Our approach generalizes usual (co)variance setting weighted graphs retains many its intuitive desired properties. Interestingly, we find that number famous concepts in graph network science can be reinterpreted this as variances covari...