Related papers: The Curvature of Graph Products

We establish for the first time the explicit curvature formulas for the horizontal and vertical edges of the strong product of two regular graphs. We complement this result with showing that there does not exist an analogous formula for the…

We study the Bakry-\'Emery curvature function $\mathcal{K}_{G,x}:(0,\infty]\to \mathbb{R}$ of a vertex $x$ in a locally finite graph $G$ systematically. Here $\mathcal{K}_{G,x}(\mathcal{N})$ is defined as the optimal curvature lower bound…

In this paper, we reformulate the Bakry-\'Emery curvature on a weighted graph in terms of the smallest eigenvalue of a rank one perturbation of the so-called curvature matrix using Schur complement. This new viewpoint allows us to show…

The direct product of graphs $G=(V(G),E(G))$ and $H=(V(H),E(H))$ is the graph, denoted as $G\times H$, with vertex set $V(G\times H)=V(G)\times V(H)$, where vertices $(x_1,y_1)$ and $(x_2,y_2)$ are adjacent in $G\times H$ if $x_1x_2\in…

In this paper, we compare Ollivier Ricci curvature and Bakry-\'Emery curvature notions on combinatorial graphs and discuss connections to various types of Ricci flatness. We show that non-negativity of Ollivier Ricci curvature implies…

Given a graph $G=(V,E)$, an integer $k$, and a function $f_G:V^k \times V^k \to {0,1}$, the $k^{th}$ graph product of $G$ w.r.t $f_G$ is the graph with vertex set $V^k$, and an edge between two vertices $x=(x_1,...,x_k)$ and…

We introduce a notion of curvature on finite, combinatorial graphs. It can be easily computed by solving a linear system of equations. We show that graphs with curvature bounded below by $K>0$ have diameter bounded by $\mbox{diam}(G) \leq…

In this paper we introduce a product-like operation that generalizes the construction of generalized Sierpi\'nski graphs. Let $G,H$ be graphs and let $f: V(G) \to V(H)$ be a function. Then the Sierpi\'nski product of $G$ and $H$ with…

The power graph $P(G)$ of a finite group $G$ is the graph with vertex set $G$ and two distinct vertices are adjacent if either of them is a power of the other. Here we show that the power graph $P(G_1 \times G_2)$ of the direct product of…

We characterise the form of all simple, finite graphs for which the girth of the graph is equal to the circumference of the graph. We apply this to prove a bound on the number of edges in such a graph.

The k-fold Cartesian product of a graph G is defined as a graph on k-tuples of vertices, where two tuples are connected if they form an edge in one of the positions and are equal in the rest. Starting with G as a single edge gives G^k as a…

In this paper we study a new product of graphs called {\em tight product}. A graph $H$ is said to be a tight product of two (undirected multi) graphs $G_1$ and $G_2$, if $V(H)=V(G_1)\times V(G_2)$ and both projection maps $V(H)\to V(G_1)$…

We construct a Cartesian product G x H for finite simple graphs. It satisfies the Kuenneth formula: H^k(G x H) is a direct sum of tensor products H^i(G) x H^j(G) with i+j=k and so p(G x H,x) = p(G,x) p(H,y) for the Poincare polynomial…

The thickness of a graph $G$ is the minimum number of planar subgraphs whose union is $G$. In this paper, we present sharp lower and upper bounds for the thickness of the Kronecker product $G\times H$ of two graphs $G$ and $H$. We also give…

The lexicographic product $G[H]$ of two graphs $G$ and $H$ is obtained from $G$ by replacing each vertex with a copy of $H$ and adding all edges between any pair of copies corresponding to adjacent vertices of $G$. We consider also the…

Associated to a finite graph $X$ is its quantum automorphism group $G(X)$. We prove a formula of type $G(X*Y)=G(X)*_wG(Y)$, where $*_w$ is a free wreath product. Then we discuss representation theory of free wreath products, with the…

Given a finite simple graph G=(V,E) with chromatic number c and chromatic polynomial C(x). Every vertex graph coloring f of G defines an index i_f(x) satisfying the Poincare-Hopf theorem sum_x i_f(x)=chi(G). As a variant to the index…

We investigate Cayley graphs of graph products by showing that graph products with vertex groups that have isomorphic Cayley graphs yield isomorphic Cayley graphs.

In this paper we prove that: Any graph product of finitely many groups, all of them satisfying weak Haagerup property with $\Lambda_{WH}=1$, also satisfies weak Haagerup property and as a corollary of this result we obtain that the free…

For warped products with harmonic curvature, nonconstant warping functions $\phi$, and compact two-dimensional bases $(M,h)$, we establish a dichotomy: either the Gaussian curvature $K$ of the metric $g=\phi^{-2}h$ is constant and negative,…