Related papers: Singularity Properties of Graph Varieties

In this paper, we investigate the Sobolev spaces W^{1,p}(V) and W_0^{1,p}(V) on a locally finite graph G=(V,E), which are fundamental tools when we apply the variational methods to partial differential equations on graphs. As a key…

Let $\Gamma$ be a simple connect graph on a finite vertex set $V$ and let $A$ be its adjacency matrix. Then $\Gamma$ is said to be \textit{singular} if and only if $0$ is an eigenvalue of $A.$ The \textit{nullity (singularity)} of $\Gamma,$…

Let $\Gamma$ be a simple undirected graph on a finite vertex set and let $A$ be its adjacency matrix. Then $\Gamma$ is {\it singular} if $A$ is singular. The problem of characterising singular graphs is easy to state but very difficult to…

Let $G$ be a graph. A set $S \subseteq V(G)$ is independent if its elements are pairwise non-adjacent. A vertex $v \in V(G)$ is shedding if for every independent set $S \subseteq V(G) \setminus N[v]$ there exists $u \in N(v)$ such that $S…

In this article we consider the connected component of the identity of $G$-character varieties of compact Riemann surfaces of genus $g > 0$, for connected complex reductive groups $G$ of type $A$ (e.g., $SL_n$ and $GL_n$). We show that…

A graph $G$ is called well-covered if all maximal independent sets of vertices have the same cardinality. A simplicial complex $\Delta$ is called pure if all of its facets have the same cardinality. Let $\mathcal G$ be the class of graphs…

The Wiener index $W(G)$ of a connected graph $G$ is a sum of distances between all pairs of vertices of $G$. In 1991, \v{S}olt\'{e}s formulated the problem of finding all graphs $G$ such that for every vertex $v$ the equation $W(G)=W(G-v)$…

In this note a new measure of irregularity of a simple undirected graph $G$ is introduced. It is named the total irregularity of a graph and is defined as $\irr_t(G) = 1/2\sum_{u,v \in V(G)} |d_G(u)-d_G(v)|$, where $d_G(u)$ denotes the…

Very sparse random graphs are known to typically be singular (i.e., have singular adjacency matrix), due to the presence of "low-degree dependencies'' such as isolated vertices and pairs of degree-1 vertices with the same neighbourhood. We…

Let $m:=|E(G)|$ sufficiently large and $s:=(m-1)/3$. We show that unless the maximum degree $\Delta > 2s$, there is a weighting $w:E\cup V\to \{0,1,...,s\}$ so that $w(uv)+w(u)+w(v)\ne w(u'v')+w(u')+w(v')$ whenever $uv\ne u'v'$ (such a…

We give characterizations for the failure of form uniqueness on weakly spherically symmetric graphs. The first characterization is in terms of the graph structure, the second involves the capacity of a Cauchy boundary. We also discuss the…

The total irregularity of a graph $G$ is defined as $\irr_t(G)=1/2 \sum_{u,v \in V(G)}$ $|d_G(u)-d_G(v)|$, where $d_G(u)$ denotes the degree of a vertex $u \in V(G)$. In this paper we give (sharp) upper bounds on the total irregularity of…

Let $q=p^e$, where $p$ is a prime and $e\geq 1$ is an integer. For $m\geq 1$, let $P$ and $L$ be two copies of the $(m+1)$-dimensional vector spaces over the finite field $\mathbb{F}_q$. Consider the bipartite graph $W_m(q)$ with partite…

For any graph $G$ on $n$ vertices and for any {\em symmetric} subgraph $J$ of $K_{n,n}$, we construct an infinite sequence of graphs based on the pair $(G,J)$. The First graph in the sequence is $G$, then at each stage replacing every…

We associate to a graphon $\gamma$ the sequence of $W$-random graphs $(G_n(\gamma))_{n \geq 1}$. We say that the graphon is singular if, for any finite graph $F$, the homomorphism density $t(F,G_n(\gamma))$ has a variance of order…

Let E be an elliptic curve defined over a number field K, V the complexification of the group of rational points of E over an algebraic closure L of K, and G the Galois group Gal(L/K). We show that for each root of unity w, the set of…

For a symmetric bounded measurable function W on [0,1]^2, "moments" of W can be defined as values t(F,W) indexed by simple graphs. We prove that every such function is determined by its moments up to a measure preserving transformation of…

In the branch of mathematics known as graph theory, graphs are considered as a set of points, called vertices, with connections between these points, called edges. The purpose of this paper is to study mappings between two graphs that have…

Let (G,w) be a weighted graph. The necessary and sufficient conditions under which a weight w : E(G)-->R^+ can be extended to a pseudoultrametric on V(G) are found. A criterion of the uniqueness of this extension is also obtained. It is…

The characteristic polynomial and the spectrum of the lexicographic product of graphs $H[G]$, a specific instance of the generalized composition (also called $H$-join), are explicitly determined for arbitrary graphs $H$ and $G$, in terms of…