Related papers: A note on Steinerberger's curvature for graphs

We develop the notions of connections and curvature for general Lie-Rinehart algebras without using smoothness assumptions on the base space. We present situations when a connection exists. E.g., this is the case when the underlying module…

In this paper, we consider the problem of approximately aligning/matching two graphs. Given two graphs $G_{1}=(V_{1},E_{1})$ and $G_{2}=(V_{2},E_{2})$, the objective is to map nodes $u, v \in G_1$ to nodes $u',v'\in G_2$ such that when $u,…

We study the volume growth of metric balls as a function of the radius in discrete spaces, and focus on the relationship between volume growth and discrete curvature. We improve volume growth bounds under a lower bound on the so-called…

The Wiener index of a graph is the sum of all the distances between any pair of vertices. We aim to describe graphs which minimize the Wiener index among all unicyclic graphs with fixed girth and given degree sequence. Depending on where…

We consider the motion by mean curvature of an $n$-dimensional graph over a time-dependent domain in $\mathbb{R}^n$, intersecting $\mathbb{R}^n$ at a constant angle. In the general case, we prove local existence for the corresponding…

We prove that any graph of multicurves satisfying certain natural properties is either hyperbolic, relatively hyperbolic, or thick. Further, this geometric characterization is determined by the set of subsurfaces that intersect every vertex…

We introduce the weighted orthogonal Ricci curvature -- a two-parameter version of Ni--Zheng's orthogonal Ricci curvature. This curvature serves as a very natural object in the study of the relationship between the Ricci curvature(s) and…

In this note, we give answers to three questions from the paper [A. Das, Triameter of graphs, Discuss. Math. Graph Theory, 41 (2021), 601--616]. Namely, we obtain a tight lower bound for the triameter of trees in terms of order and number…

We obtain a Bonnet-Myers theorem under a spectral condition: a closed Riemannian manifold $(M^n,g)$ for which the lowest eigenvalue of the Ricci tensor $\rho$ is such that the Schr\"odinger operator $(n-2)\Delta + \rho$ is positive has…

In this paper, we investigate various algebraic and graph theoretic properties of the distance matrix of a graph. Two graphs are $D$-cospectral if their distance matrices have the same spectrum. We construct infinite pairs of $D$-cospectral…

The Theorem of Bonnet--Myers implies that manifolds with topology $M^{n-1} \times \mathbb{S}^1$ do not admit a metric of positive Ricci curvature, while the resolution of Geroch's conjecture implies that the torus $\mathbb{T}^n$ does not…

In this paper, we provide a simple proof for the fact that two simplicial complexes are isomorphic if and only if their associated Stanley-Reisner rings, or their associated facet rings are isomorphic as $K$-algebras. As a consequence, we…

We study the properties of Ricci curvature of ${\mathfrak{g}}$-manifolds with particular attention paid to higher dimensional abelian Lie algebra case. The relations between Ricci curvature of the manifold and the Ricci curvature of the…

A mixed graph is obtained by orienting some edges of a simple graph. The positive inertia index of a mixed graph is defined as the number of positive eigenvalues of its Hermitian adjacency matrix, including multiplicities. This matrix was…

In this second part of a sequence of two papers, we discuss the implementation of a curvature flow on weighted graphs based on the Bakry-\'Emery calculus. This flow can be adapted to preserve the Markovian property and its limits as time…

For Einstein four-manifolds with positive scalar curvature, we derive relations among various positivity conditions on the curvature tensor, some of which are of great importance in the study of the Ricci flow. These relations suggest…

An almost self-centered graph is a connected graph of order $n$ with exactly $n-2$ central vertices, and an almost peripheral graph is a connected graph of order $n$ with exactly $n-1$ peripheral vertices. We determine (1) the maximum girth…

We study autocorrelation inequalities, in the spirit of Barnard and Steinerberger's work. In particular, we obtain improvements on the sharp constants in some of the inequalities previously considered by these authors, and also prove…

Motivated and inspired by the recent work of Colding [5] and Colding-Minicozzi [6] we derive several families of monotonicity formulas for manifolds with nonnegative Bakry-Emery Ricci curvature, extending the formulas in [5, 6].

In `A survey of two-graphs' \cite{Sei}, J.J. Seidel lays out the connections between simple graphs, two-graphs, equiangular lines and strongly regular graph. It is well known that there is a one-to-one correspondence between regular…