Related papers: Analytic connectivity of $k$-uniform hypergraphs

Let $G$ be a connected graph with minimum degree $\delta(G)$ and vertex-connectivity $\kappa(G)$. The graph $G$ is $k$-connected if $\kappa(G)\geq k$, maximally connected if $\kappa(G) = \delta(G)$, and super-connected (or super-$\kappa$)…

The algebraic connectivity of a graph $G$ in a finite dimensional real normed linear space $X$ is a geometric counterpart to the Fiedler number of the graph and can be regarded as a measure of the rigidity of the graph in $X$. We analyse…

A $k$-connectivity oracle for a graph $G=(V,E)$ is a data structure that given $s,t \in V$ determines whether there are at least $k+1$ internally disjoint $st$-paths in $G$. For undirected graphs, Pettie, Saranurak & Yin [STOC 2022, pp.…

The (k,d)-hypersimplex is a (d-1)-dimensional polytope whose vertices are the (0,1)-vectors that sum to k. When k=1, we get a simplex whose graph is the complete graph with d vertices. Here we show how many of the well known graph…

A $k$-connected set in an infinite graph, where $k > 0$ is an integer, is a set of vertices such that any two of its subsets of the same size $\ell \leq k$ can be connected by $\ell$ disjoint paths in the whole graph. We characterise the…

k-connectivity of random graphs is a fundamental property indicating reliability of multi-hop wireless sensor networks (WSN). WSNs comprising of sensor nodes with limited power resources are modeled by random graphs with unreliable nodes,…

An arc of a graph is an oriented edge and a 3-arc is a 4-tuple $(v,u,x,y)$ of vertices such that both $(v,u,x)$ and $(u,x,y)$ are paths of length two. The 3-arc graph of a graph $G$ is defined to have vertices the arcs of $G$ such that two…

This paper generalizes and unifies the existing spectral bounds on the $k$-independence number of a graph, which is the maximum size of a set of vertices at pairwise distance greater than $k$. The previous bounds known in the literature…

Fix an integer $k \ge 3$. A $k$-uniform hypergraph is simple if every two edges share at most one vertex. We prove that there is a constant $c$ depending only on $k$ such that every simple $k$-uniform hypergraph $H$ with maximum degree $\D$…

Substituting each edge of a simple connected graph $G$ by a path of length 1 and $k$ paths of length 5 generates the $k$-hexagonal graph $H^k(G)$. Iterative graph $H^k_n(G)$ is produced when the preceding constructions are repeated $n$…

We generalize a result of Balister, Gy{\H{o}}ri, Lehel and Schelp for hypergraphs. We determine the unique extremal structure of an $n$-vertex, $r$-uniform, connected, hypergraph with the maximum number of hyperedges, without a…

Given two graphs $H_1$ and $H_2$, a graph is $(H_1,H_2)$-free if it contains no induced subgraph isomorphic to $H_1$ nor $H_2$. A graph $G$ is $k$-vertex-critical if every proper induced subgraph of $G$ has chromatic number less than $k$,…

For a $k$-uniform hypergraph $H$, we obtain some trace formulas for the Laplacian tensor of $H$, which imply that $\sum_{i=1}^nd_i^s$ ($s=1,\ldots,k$) is determined by the Laplacian spectrum of $H$, where $d_1,\ldots,d_n$ is the degree…

The Lagrangian density of an $r$-uniform hypergraph $H$ is $r!$ multiplying the supremum of the Lagrangians of all $H$-free $r$-uniform hypergraphs. For an $r$-uniform graph $H$ with $t$ vertices, it is clear that $\pi_{\lambda}(H)\ge…

The first three authors [European J. Combin. 33 (2012), 62--71] established a relationship between the transversal number and the domination number of uniform hypergraphs. In this paper, we establish a relationship between the total…

Directed graphs are widely used in modelling of nonsymmetric relations in various sciences and engineering disciplines. We discuss invariants of strongly connected directed graphs - minimal number of vertices or edges necessary to remove to…

We introduce and study elementary properties of graph homology of algebras. This new homology theory shares many features of cyclic and Hochschild homology. We also define a graph K-theory together with an analog of Chern character.

In this paper we consider two natural notions of connectivity for hypergraphs: weak and strong. We prove that the strong vertex connectivity of a connected hypergraph is bounded by its weak edge connectivity, thereby extending a theorem of…

We establish a sharp edge-connectivity estimate for graphs with non-negative Bakry-\'Emery curvature. This leads to a geometric criterion for the existence of a perfect matching. Precisely, we show that any regular graph with non-negative…

Let $G = (V, E)$ be a graph with non-empty set of vertices $V$ and set of edges $E$. The \emph{eccentric connectivity index} of the graph $G$ is defined as $$\displaystyle{\xi^C(G) = \sum_{u \in V} d_u \;ecc(u)}$$ where $d_u$ is the degree…