Developments on Spectral Characterizations of Graphs

In [E.R. van Dam and W.H. Haemers, Which graphs are determined by their spectrum?, Linear Algebra Appl. 373 (2003), 241-272] we gave a survey of answers to the question of which graphs are determined by the spectrum of some matrix associated to the graph. In particular, the usual adjacency matrix and the Laplacian matrix were addressed. Furthermore, we formulated some research questions on the topic. In the meantime some of these questions have been (partially) answered. In the present paper we give a survey of these and other developments.2000 Mathematics Subject Classification: 05C50Spectra of graphs;Cospectral graphs;Generalized adjacency matrices;Distance-regular graphs

Graphs that are cospectral for the distance Laplacian

The distance matrix D(G) of a graph G is the matrix containing the pairwise distances between vertices, and the distance Laplacian matrix is DL(G)=T(G)−D(G), where T(G) is the diagonal matrix of row sums of D(G). We establish several general methods for producing DL-cospectral graphs that can be used to construct infinite families. We provide examples showing that various properties are not preserved by DL-cospectrality, including examples of DL-cospectral strongly regular and circulant graphs. We establish that the absolute values of coefficients of the distance Laplacian characteristic polynomial are decreasing, i.e., |δL1|≥⋯≥|δLn| where δLk is the coefficient of xk

Developments on Spectral Characterizations of Graphs

In [E.R. van Dam and W.H. Haemers, Which graphs are determined by their spectrum?, Linear Algebra Appl. 373 (2003), 241-272] we gave a survey of answers to the question of which graphs are determined by the spectrum of some matrix associated to the graph. In particular, the usual adjacency matrix and the Laplacian matrix were addressed. Furthermore, we formulated some research questions on the topic. In the meantime some of these questions have been (partially) answered. In the present paper we give a survey of these and other developments.2000 Mathematics Subject Classification: 05C50

Fractional isomorphism of graphs

AbstractLet the adjacency matrices of graphs G and H be A and B. These graphs are isomorphic provided there is a permutation matrix P with AP=PB, or equivalently, A=PBPT. If we relax the requirement that P be a permutation matrix, and, instead, require P only to be doubly stochastic, we arrive at two new equivalence relations on graphs: linear fractional isomorphism (when we relax AP=PB) and quadratic fractional isomorphism (when we relax A=PBPT). Further, if we allow the two instances of P in A=PBPT to be different doubly stochastic matrices, we arrive at the concept of semi-isomorphism.We present necessary and sufficient conditions for graphs to be linearly fractionally isomorphic, we prove that quadratic fractional isomorphism is the same as isomorphism and we relate semi-isomorphism to isomorphism of bipartite graphs

Developments on spectral characterizations of graphs

AbstractIn [E.R. van Dam, W.H. Haemers, Which graphs are determined by their spectrum? Linear Algebra Appl. 373 (2003), 241–272] we gave a survey of answers to the question of which graphs are determined by the spectrum of some matrix associated to the graph. In particular, the usual adjacency matrix and the Laplacian matrix were addressed. Furthermore, we formulated some research questions on the topic. In the meantime, some of these questions have been (partially) answered. In the present paper we give a survey of these and other developments

Diffusion Wavelet Embedding: a Multi-resolution Approach for Graph Embedding in Vector Space

In this article, we propose a multiscale method of embedding a graph into a vector space using diffusion wavelets. At each scale, we extract a detail subspace and a corresponding lower-scale approximation subspace to represent the graph. Representative features are then extracted at each scale to provide a scale-space description of the graph. The lower-scale is constructed using a super-node merging strategy based on nearest neighbor or maximum participation and the new adjacency matrix is generated using vertex identification. This approach allows the comparison of graphs where the important structural differences may be present at varying scales. Additionally, this method can improve the differentiating power of the embedded vectors and this property reduces the possibility of cospectrality typical in spectral methods, substantially. The experimental results show that augmenting the features of abstract levels to the graph features increases the graph classification accuracies in different datasets

An R-convolution Graph Kernel based on Fast Discrete-Time Quantum Walk

In this paper, a novel R-convolution kernel, named the fast quantum walk kernel (FQWK), is proposed for unattributed graphs. In FQWK, the similarity of the neighborhood-pair substructure between two nodes is measured via the superposition amplitude of quantum walks between those nodes. The quantum interference in this kind of local substructures provides more information on the substructures so that FQWK can capture finer-grained local structural features of graphs. In addition, to efficiently compute the transition amplitudes of multi-step discrete-time quantum walks, a fast recursive method is designed. Thus compared with all the existing kernels based on the quantum walk, FQWK has the highest computation speed. Extensive experiments demonstrate that FQWK outperforms state-of-the-art graph kernels in terms of classification accuracy for unattributed graphs. Meanwhile, it can be applied to distinguish a larger family of graphs including cospectral graphs, regular graphs, and even strong regular graphs which are not distinguishable by classical walkbased methods