Related papers: The dual Cheeger-Buser inequality for graphons

Previously, Biggs has conjectured that the resistance between any two points on a distance-regular graph of valency greater than 2 is bounded by twice the resistance between adjacent points. We prove this conjecture, give the sharp constant…

We show that there exists only one duality pair for ordered graphs. We will also define a corresponding definition of $\chi^<$-boundedness for ordered graphs and show that all ordered graphs are $\chi^<$-bounded and prove an analogy of…

We define the Euler number of a bipartite graph on $n$ vertices to be the number of labelings of the vertices with $1,2,...,n$ such that the vertices alternate in being local maxima and local minima. We reformulate the problem of computing…

Gross, Mansour and Tucker introduced the partial-dual orientable genus polynomial and the partial-dual Euler genus polynomial. They showed that the partial-dual genus polynomial for an orientable ribbon graph is interpolating and gave an…

In this paper we consider higher isoperimetric numbers of a (finite directed) graph. In this regard we focus on the $n$th mean isoperimetric constant of a directed graph as the minimum of the mean outgoing normalized flows from a given set…

Let $G$ be a bipartite graph on $n$ vertices with the Laplacian matrix $L_G$. When $G$ is a tree, inequalities involving coefficients of immanantal polynomials of $L_G$ are known as we go up $GTS_n$ poset of unlabelled trees with $n$…

In this paper, we give tight bounds for the normalized Laplacian eigenvalues of hypergraphs that are not necessarily uniform, and provide an edge version interlacing theorem, a Cheeger inequality, and a discrepancy inequality that are…

In this paper we study the maximum value of the largest eigenvalue for simple bipartite graphs, where the number of edges is given and the number of vertices on each side of the bipartition is given. We state a conjectured solution, which…

We prove a conjecture of Shteiner and Shteyner stating that for a bipartite graph $G=(V,E)$, the number of forests in $G$ equals the number of degree sequences arising from its spanning subgraphs. In the process, we provide several…

We address a question posed by Fessler-Jensen-Kelsey-Owen regarding graphs whose second gonality is greater than the first by exactly 1. We answer the question affirmatively under a stronger condition, thereby characterising the entire…

A basic fact in algebraic graph theory is that the number of connected components in an undirected graph is equal to the multiplicity of the eigenvalue 1 in the normalized adjacency matrix of the graph. In particular, the graph is…

A bipartite graph $G=(V,E)$ with $V=V_1\cup V_2$ is biregular if all the vertices of each stable set, $V_1$ and $V_2$, have the same degree, $r$ and $s$, respectively. This paper studies difference sets derived from both Abelian and…

We generalize the Cheeger inequality, a lower bound on the first nontrivial eigenvalue of a Laplacian, to the case of geometric sub-Laplacians on rank-varying Carnot-Carath\'eodory spaces and we describe a concrete method to lower bound the…

A dominating set in a graph $G$ is a set $S$ of vertices such that every vertex of $G$ is either in $S$ or is adjacent to a vertex in $S$. Nordhaus-Gaddum inequailties relate a graph $G$ to its complement $\bar{G}$. In this spirit Wagner…

The structural parameter twin-width was introduced by Bonnet et al. in [FOCS 2020], and already this first paper included an asymptotic argument bounding the twin-width of planar graphs by a non-explicit constant. Quite recently, we have…

In this paper, we use machine learning to show that the Cheeger constant of a connected regular graph has a predominant linear dependence on the largest two eigenvalues of the graph spectrum. We also show that a trained deep neural network…

We propose a Laplacian based on general inner product spaces, which we call the inner product Laplacian. We show the combinatorial and normalized graph Laplacians, as well as other Laplacians for hypergraphs and directed graphs, are special…

K\"orner introduced the notion of graph entropy in 1973 as the minimal code rate of a natural coding problem where not all pairs of letters can be distinguished in the alphabet. Later it turned out that it can be expressed as the solution…

We introduce a new optimization problem regarding embeddings of a graph into a Euclidean space and discuss its relation to the two, mutually dual, optimizations problems introduced by Goering-Helmberg-Wappler. We prove that the Laplace…

While the adjacency tensor of a bipartite 2-graph is a nonnegative biquadratic tensor, it is inherently reducible. To address this limitation, we introduce the concept of quasi-irreducibility in this paper. The adjacency tensor of a…