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We give a short proof that every finite graph (or matroid) has a tree-decomposition that displays all maximal tangles. This theorem for graphs is a central result of the graph minors project of Robertson and Seymour and the extension to…
A natural representation of random graphs is the random measure. The collection of product random measures, their transformations, and non-negative test functions forms a general representation of the collection of non-negative weighted…
We use two variational techniques to prove upper bounds for sums of the lowest several eigenvalues of matrices associated with finite, simple, combinatorial graphs. These include estimates for the adjacency matrix of a graph and for both…
In here, I present a series of combinatorial equalities derived using a graph based approach. Different nodes in the graphs are visited following probabilistic dynamics of a moving dot. The results are presented in such a way that the…
In this article we consider resistance matrix of a connected graph. For unweighted graph we study some necessary and sufficient conditions for resistance regular graphs. Also we find some relationship between Laplacian matrix and resistance…
A typical result in graph theory says that a graph $G$, satisfying certain conditions, has some property $\cal P$. Once such a theorem is established, it is natural to ask how strongly $G$ satisfies $\cal P$. Can one strengthen the result…
We study the inertia of distance matrices of weighted graphs. Our novel congruence-based proof of the inertia of weighted trees extends to a proof for the inertia of weighted unicyclic graphs whose cycle is a triangle. Partial results are…
Recently Baranyai described a necessary combinatorial characterisation of graph rigidity for dimension 3. In this short note we provide a counter-example to the converse of the condition. Additionally, we provide an alternative proof to the…
Bond percolation on Cayley graphs provides examples of random graphs. Other examples arise from the dynamical study of proper repetitive subgraphs of Cayley graphs. In this paper we demonstrate that these two families have mutually singular…
This paper investigates some combinatorial and algebraic properties of a Witt type formula for graphs.
We study topological properties of the graph topology.
In an earlier paper the authors proved that limits of convergent graph sequences can be described by various structures, including certain 2-variable real functions called graphons, random graph models satisfying certain consistency…
We determine the probability thresholds for the existence of monotone paths, of finite and infinite length, in random oriented graphs with vertex set $\mathbb N^{[k]}$, the set of all increasing $k$-tuples in $\mathbb N$. These graphs…
The analysis of large simple graphs with extreme values of the densities of edges and triangles has been extended to the statistical structure of typical graphs of fixed intermediate densities, by the use of large deviations of Erdoes-Renyi…
Two finitely generated monoids are constructed, one finitely presented the other not, whose (directed, unlabelled) Cayley graphs are isomorphic.
The monotonicity properties of remainder of Stirling's formula for the gamma function are simply obtained by using the integral transforms with series.
We prove a general duality theorem for tangle-like dense objects in combinatorial structures such as graphs and matroids. This paper continues, and assumes familiarity with, the theory developed in [6]
We prove that if a unimodular random graph is almost surely planar and has finite expected degree, then it has a combinatorial embedding into the plane which is also unimodular. This implies the claim in the title immediately by a theorem…
We prove that a graph G is asymptotically isomorphic to the ray if and only if G is uniformly spherically bounded and is of bounded local degrees. This problem arouse in combinatorics and was posed in [3] (Problem 10.1).
Although false for general graphs, this note gives an elementary proof of the bunkbed conjecture for any acyclic graph. The argument is short and self-contained, and may be of educational interest.