An isomorphism between two graphs is a bijection between their vertices that preserves the edges. We consider the problem of determining whether two finite undirected weighted graphs are isomorphic, and finding an isomorphism relating them if the answer is positive. In this paper we introduce effective probabilistic linear programming (LP) heuristics to solve the graph isomorphism problem. We m...

The energy of a graph G, denoted by E(G), is defined as the sum of the absolute values of all eigenvalues of G. Let G be a graph of order n and rank(G) be the rank of the adjacency matrix of G. In this paper we characterize all graphs with E(G) = rank(G). Among other results we show that apart from a few families of graphs, E(G) ≥ 2max(χ(G), n − χ(G)), where n is the number of vertices of G, G ...

A graph is said to be $I$-eigenvalue free if it has no eigenvalues in the interval $I$ with respect adjacency matrix $A$. In this paper we present twoalgorithms for generating threshold graphs.

Abstract The Estrada index of a graph/network is defined as the trace adjacency matrix exponential. It has been extended to other graph-theoretic matrices, such Laplacian, distance, Seidel adjacency, Harary, etc. Here, we describe many these extensions, including new ones, Gaussian, Mittag–Leffler and Onsager ones. More importantly, contextualize all indices in physico-mathematical frameworks w...

Quantum algorithms for graph problems are considered, both in the adjacency matrix model and in an adjacency list-like array model. We give almost tight lower and upper bounds for the bounded error quantum query complexity of Connectivity, Strong Connectivity, Minimum Spanning Tree, and Single Source Shortest Paths. For example we show that the query complexity of Minimum Spanning Tree is in Θ(...

This paper describes a spectral method for graphmatching. We adopt a graphical models viewpoint in which the graph adjacency matrix is taken to represent the transition probability matrix of a Markov chain. The node-order of the steady state random walk associated with this Markov chain is determined by the co-efficent order of the leading eigenvector of the adjacency matrix. We match nodes in ...

Given a multimillion-node social network, how can we summarize connectivity pattern from the data, and how can we find unexpected user behavior? In this paper we study a complete graph from a large who-follows-whom network and spot lockstep behavior that large groups of followers connect to the same groups of followees. Our first contribution is that we study strange patterns on the adjacency m...

Generation of realistic topologies plays an important role in determining the accuracy and validity of simulation studies. This study presents a discussion to justify why, and how often randomly generated adjacency matrices may not not conform to wireless topologies in the physical world. Specifically, it shows through analysis and random trials that, more than 90% of times, a randomly generate...

Quantum algorithms for graph problems are considered, both in the adjacency matrix model and in an adjacency list-like array model. We give almost tight lower and upper bounds for the bounded error quantum query complexity of Connectivity, Strong Connectivity, Minimum Spanning Tree, and Single Source Shortest Paths. For example we show that the query complexity of Minimum Spanning Tree is in Θ(...