相关论文: Polynomial and spectra factorization of graphs obt…
It is shown how a connected graph and a tree with partially prescribed spectrum can be constructed. These constructions are based on a recent result of Salez that every totally real algebraic integer is an eigenvalue of a tree. Our result…
This paper settles the computational complexity of model checking of several extensions of the monadic second order (MSO) logic on two classes of graphs: graphs of bounded treewidth and graphs of bounded neighborhood diversity. A classical…
Series solutions for a large family of single equation Dyson-Schwinger equations are given as expansions over decorated rooted connected chord diagrams. The analytic input to the new expansions are the expansions of the regularized…
We establish a connection between root multiplicities for Borcherds-Kac-Moody algebras and graph coloring. We show that the generalized chromatic polynomial of the graph associated to a given Borcherds algebra can be used to give a closed…
Graph learning from data represents a canonical problem that has received substantial attention in the literature. However, insufficient work has been done in incorporating prior structural knowledge onto the learning of underlying…
Graphs constructed to translate some graph problem into another graph problem are usually called auxiliary graphs. Specifically total graphs of simple graphs are used to translate the total colouring problem of the original graph into a…
Given a regular (connected) graph $\Gamma=(X,E)$ with adjacency matrix $A$, $d+1$ distinct eigenvalues, and diameter $D$, we give a characterization of when its distance matrix $A_D$ is a polynomial in $A$, in terms of the adjacency…
Compact and I/O-efficient data representations play an important role in efficient algorithm design, as memory bandwidth and latency can present a significant performance bottleneck, slowing the computation by orders of magnitude. While…
We introduce a new notion of graph sparsificaiton based on spectral similarity of graph Laplacians: spectral sparsification requires that the Laplacian quadratic form of the sparsifier approximate that of the original. This is equivalent to…
The notions of bounded expansion and nowhere denseness not only offer robust and general definitions of uniform sparseness of graphs, they also describe the tractability boundary for several important algorithmic questions. In this paper we…
Pairs of graded graphs, together with the Fomin property of graded graph duality, are rich combinatorial structures providing among other a framework for enumeration. The prototypical example is the one of the Young graded graph of integer…
A quick proof of Gallai's celebrated theorem on color-critical graphs is given from Gallai's simple, ingenious lemma on factor-critical graphs, in terms of partitioning the vertex-set into a minimum number of hyperedges of a hereditary…
We remark that Pearl's Graphoid intersection property, also called intersection property in Bayesian networks, is a particular case of a general intersection property, in the sense of intersection of coverings, for factorisation spaces,…
Given a graph and a representation of its fundamental group, there is a naturally associated twisted adjacency operator. The main result of this article is the fact that these operators behave in a controlled way under graph covering maps.…
Building upon [1], this study aims to introduce fractal geometry into graph theory, and to establish a potential theoretical foundation for complex networks. Specifically, we employ the method of substitution to create and explore…
Graph aggregation is the process of computing a single output graph that constitutes a good compromise between several input graphs, each provided by a different source. One needs to perform graph aggregation in a wide variety of…
Partitioning a graph into three pieces, with two of them large and connected, and the third a small ``separator'' set, is useful for improving the performance of a number of combinatorial algorithms. This is done using the second…
We construct infinite families of graphs that are determined by their generalized spectrum. This construction is based on new formulae for the determinant of the walk matrix of a graph. The graphs constructed here all satisfy a lower…
We prove a generalization of the Expander Mixing Lemma for arbitrary (finite) simplicial complexes. The original lemma states that concentration of the Laplace spectrum of a graph implies combinatorial expansion (which is also referred to…
We study a representation-theoretic refinement of the ordinary Laplacian spectrum of a graph. Given a graph $G$ on $n$ vertices, one may associate to it the element \[ X_G=\sum_{ij\in E(G)} (ij)\in \C[S_n]. \] The action of $X_G$ in…