Related papers: Ricci-flat graphs with girth four

Complex Ricci-flat (i.e., Calabi-Yau) hypersurfaces in spaces admitting a maximal (toric) $U(1)^n$ gauge symmetry of general type (encoded by certain non-convex and multi-layered multitopes) may degenerate, but can be smoothed by rational…

The incidence matrix of a graph is totally unimodular if and only if the graph is bipartite, i.e., it contains no odd cycles. We extend the characterization of total unimodularity to hypergraphs whose hyperedges of size at least four are…

In recent years extensions of manifold Ricci curvature to discrete combinatorial objects such as graphs and hypergraphs (popularly called as "network shapes"), have found a plethora of applications in a wide spectrum of research areas…

Curvature serves as a potent and descriptive invariant, with its efficacy validated both theoretically and practically within graph theory. We employ a definition of generalized Ricci curvature proposed by Ollivier, which Lin and Yau later…

We construct an infinite family of 6-regular graphs $\{G_n\}_{n\ge 3}$ by taking $n$ copies of the Petersen graph and wiring corresponding vertices according to an $n$-cycle permutation. Each $G_n$ has $10n$ vertices, $30n$ edges, and…

Finding Ricci-flat (Calabi-Yau) metrics is a long standing problem in geometry with deep implications for string theory and phenomenology. A new attack on this problem uses neural networks to engineer approximations to the Calabi-Yau metric…

In recent years, there has been significant interest in characterizing the induced subgraph obstructions to bounded treewidth and pathwidth. While this has recently been resolved for pathwidth, the case of treewidth remains open, and prior…

We study the long scale Ollivier-Ricci curvature of graphs as a function of the chosen idleness. As in the previous work on the short scale, we show that this idleness function is concave and piecewise linear with at most $3$ linear parts.…

In this work, we use the Ricci flow approach to study the gap phenomenon of Riemannian manifolds with non-negative curvature and sub-critical scaling invariant curvature decay. The first main result is a quantitative Ricci flow existence…

The spined cube $SQ_n$ is a variant of the hypercube $Q_n$, introduced by Zhou et al. in [Information Processing Letters 111 (2011) 561-567] as an interconnection network for parallel computing. A graph $\G$ is an $m$-Cayley graph if its…

For a given positive integer t we consider graphs having maximal independent sets of precisely t distinct cardinalities and restrict our attention to those that have no vertices of degree one. In the situation when t is four or larger and…

We prove that there is a unique graph with four edges which is the Gruenberg-Kegel graph of a solvable cut group. This contributes to the classification of the Gruenberg-Kegel graphs of solvable cut groups initiated in [A. B\"achle, A.…

Since the fundamental work of Chow-Luo \cite{CL03}, Ge \cite{Ge12,Ge17} et al., the combinatorial curvature flow methods became a basic technique in the study of circle pattern theory. In this paper, we investigate the combinatorial Ricci…

We investigate structural and combinatorial properties of Bi-Cayley graphs defined over cyclic groups of order $p^2q^2$, where $p$ and $q$ are distinct primes. We begin by describing their fundamental group-theoretic underpinnings. The main…

A graph $G$ is said to be crossing-critical if $cr(G-e)< cr(G)$ for every edge $e$ of $G$, where $cr(G)$ is the crossing number of $G$. Richter and Thomassen [Journal of Combinatorial Theory, Series B 58 (1993), 217-224] constructed an…

We characterise the quintic (i.e. 5-regular) multigraphs with the property that every edge lies in a triangle. Such a graph is either from a set of small graphs or is formed by adding a perfect matching to a line graph of a cubic graph as…

A graph $G$ is called an $L_1$-graph if $d(u)+d(v)\ge|N(u)\cup N(v)\cup N(w)|-1$ for every triple of vertices $u,v,w$ where $u$ and $v$ are at distance 2 and $w\in N(u)\cap N(v)$. Asratian et al. (1996) proved that all finite connected…

We say that a graph $G$ has an {\em odd $K_4$-subdivision} if some subgraph of $G$ is isomorphic to a $K_4$-subdivision and whose faces are all odd holes of $G$. For a number $\ell\geq 2$, let $\mathcal{G}_{\ell}$ denote the family of…

Whiteley \cite{wh} gives a complete characterization of the infinitesimal flexes of complete bipartite frameworks. Our work generalizes a specific infinitesimal flex to include joined graphs, a family of graphs that contain the complete…

We give topological conditions to ensure that a noncollapsed almost Ricci-flat 4-manifold admits a Ricci-flat metric. One sufficient condition is that the manifold is spin and has a nonzero A-hat genus. Another condition is that the…