Given a finite, simple, connected graph with , we consider the associated graph Laplacian matrix with eigenvalues . One can also consider the same graph equipped with positive edge weights
Given a finite, simple, connected graph with , we consider the associated graph Laplacian matrix with eigenvalues . One can also consider the same graph equipped with positive edge weights
In this article, we investigate the problem of finding in a given weighted hypergraph a subhypergraph with the maximum possible density. Using the notion of a support matrix we prove that the density of an optimal subhypergraph is equal to for an optimal support matrix . Alternatively, the maximum density of a subhypergraph is equal to the solution of a minimax problem for column sums of support matrices. We study the density decomposition of a hypergraph and show that it is a significant refinement of the Dulmage–Mendelsohn decomposition. Our theoretical results yield an efficient algorithm for finding the maximum density subhypergraph and more generally, the density decomposition for a given weighted hypergraph.
For a graph and a list assignment with for all , an -packing consists of -colorings such that
Let be a graph, and let be functions. Let be the union of edges shared by two cycles of order at most four, and let
We define the average -TSP distance of a graph as the average length of a shortest closed walk visiting vertices, that is, the expected length of the solution for a random TSP instance with uniformly random chosen vertices. We prove relations with the average -Steiner distance and characterize the cases where equality occurs. We also give sharp bounds for given the order of the graph.
Carmesin and Gollin proved that every finite graph has a canonical tree-decomposition of adhesion less than that efficiently distinguishes every two distinct -profiles, and which has the further property that every separable -block is equal to the unique part of in which it is contained. We give a shorter proof of this result by showing that such a tree-decomposition can in fact be obtained from any canonical tight tree-decomposition of adhesion less than . For this, we decompose the parts of such a tree-decomposition by further tree-decompositions. As an application, we also obtain a generalization of Carmesin and Gollin's result to locally finite graphs.
An independent set of size in a finite undirected graph is a set of vertices of the graph, no two of which are connected by an edge. Let be the number of independent sets of size in the graph and let
One of the most fundamental results in graph theory is Mantel's theorem which determines the maximum number of edges in a triangle-free graph of order . Recently, a colorful variant of this problem has been solved. In this variant we consider graphs on a common vertex set, think of each graph as edges in a distinct color, and want to determine the smallest number of edges in each color which guarantees the existence of a rainbow triangle. Here, we solve the analogous problem for directed graphs without rainbow triangles, either directed or transitive, for any number of colors. The constructions and proofs essentially differ for and and the type of the forbidden triangle. Additionally, we also solve the analogous problem in the setting of oriented graphs.
Fiol, Garriga, and Yebra introduced the notion of pseudo-distance-regular vertices, which they used to come up with a new characterization of distance-regular graphs. Building on that work, Fiol and Garriga developed the spectral excess theorem for distance-regular graphs. We extend both these characterizations to distance-biregular graphs and show how these characterizations can be used to study bipartite graphs with distance-regular halved graphs and graphs with the spectrum of a distance-biregular graph.
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