Related papers: Sturm-Hurwitz Theorem for quantum graphs
The treewidth of a graph is an important invariant in structural and algorithmic graph theory. This paper studies the treewidth of line graphs. We show that determining the treewidth of the line graph of a graph $G$ is equivalent to…
For the classical Sturm-Liouville operators, we prove the sharp bounds for all nodes of eigenfunctions by regarding these nodes as nonlinear functionals of potential $q\in L^1[0,1]$. By studying the optimization problems to minimize or to…
Given a fixed graph H, we say that a graph G is H-free if G does not contain H as a subgraph. The Tur\'an number ex(n, H) of H is the maximum number of edges in an n-vertex H-free graph. The study of Tur\'an number of graphs is a central…
A large body of work has investigated the properties of graph neural networks and identified several limitations, particularly pertaining to their expressive power. Their inability to count certain patterns (e.g., cycles) in a graph lies at…
We give a sharp lower bound on the number of matchings of a given size in a bipartite graph. When specialized to regular bipartite graphs, our results imply Friedland's Lower Matching Conjecture and Schrijver's theorem proven by Gurvits and…
The Glazman-Povzner-Wienholtz theorem states that the completeness of a manifold, when combined with the semiboundedness of the Schr\"odinger operator $-\Delta + q$ and suitable local regularity assumptions on $q$, guarantees its essential…
We study the interplay between spectrum, geometry and boundary conditions for two distinguished self-adjoint realisations of the Laplacian on infinite metric graphs, the so-called riedrichs and Neumann extensions. We introduce a new…
We exploit a result by Nerman which shows that conditional limit theorems hold when a certain monotonicity condition is satisfied. Our main result is an application to vertex degrees in random graphs, where we obtain asymptotic normality…
This paper provides the generating series for the embedding of tree-like graphs of arbitrary number of vertices, accourding to their genus. It applies and extends the techniques of Chan, where it was used to give an alternate proof of the…
We prove that for every graph $G$ with $n$ vertices, the treewidth of $G$ plus the treewidth of the complement of $G$ is at least $n-2$. This bound is tight.
Lov\'asz (1967) showed that two graphs $G$ and $H$ are isomorphic if, and only if, they are homomorphism indistinguishable over all graphs, i.e., $G$ and $H$ admit the same number of number of homomorphisms from every graph $F$.…
In this paper, we discuss the concept of quantum graphs with transparent vertices by considering the case where the graph interacts with an external time-independent field. In particular, we address the problem of transparent boundary…
We define a range of new coarse geometric invariants based on various graph-theoretic measures of complexity for finite graphs, including: treewidth, pathwidth, cutwidth and bandwidth. We prove that, for bounded degree graphs, these…
We study upper bounds for the first non-zero eigenvalue of the Steklov problem defined on finite graphs with boundary. For finite graphs with boundary included in a Cayley graph associated to a group of polynomial growth, we give an upper…
One powerful method for upper-bounding the largest independent set in a graph is the Hoffman bound, which gives an upper bound on the largest independent set of a graph in terms of its eigenvalues. It is easily seen that the Hoffman bound…
We establish new bounds on the minimum number of distinct eigenvalues among real symmetric matrices with nonzero off-diagonal pattern described by the edges of a graph and apply these to determine the minimum number of distinct eigenvalues…
Various generating functions of simple Hurwitz numbers of the projective line are known to satisfy many properties. They include a heat equation, the Eynard-Orantin topological recursion, an infinite-order differential equation called a…
The scramble number of a graph provides a lower bound for gonality and an upper bound for treewidth, making it a graph invariant of interest. In this paper we study graphs of scramble number at most two, and give a classification of all…
The spectral theory of graphs provides a bridge between classical signal processing and the nascent field of graph signal processing. In this paper, a spectral graph analogy to Heisenberg's celebrated uncertainty principle is developed.…
We establish a Harnack inequality for finite connected graphs with non-negative Ricci curvature. As a consequence, we derive an eigenvalue lower bound, extending previous results for Ricci flat graphs.