Related papers: On the image of graph distance matrices

Let $G$ be a simple connected graph, and $D(G)$ be the distance matrix of $G$. Suppose that $D_{\max}(G)$ and $\lambda_1(G)$ are the maximum row sum and the spectral radius of $D(G)$, respectively. In this paper, we give a lower bound for…

The dissertation is related to combinatorial geometry with a strong probabilistic flavor. The main results can be split into three parts. The results of the first part guarantee that each "unit distance graph" in the plane has an induced…

A matchstick graph is a plane graph with edges drawn as unit distance line segments. This class of graphs was introduced by Harborth who conjectured that a matchstick graph on $n$ vertices can have at most $\lfloor 3n - \sqrt{12n -…

Let D be a simple digraph without loops or digons. For any v in V(D) let N_1(v) be the set of all nodes at out-distance 1 from v and let N_2(v) be the set of all nodes at out-distance 2. We provide sufficient conditions under which there…

Distance-hereditary graphs form an important class of graphs, from the theoretical point of view, due to the fact that they are the totally decomposable graphs for the split-decomposition. The previous best enumerative result for these…

Let G be a simple, connected graph on n vertices. Let d_G(u,v) denote the distance between vertices u and v in G. A subgraph H of G is isometric if d_H(u,v)=d_G(u,v) for every u,v in V(H). We say that G is a distance-preserving graph if G…

For a connected graph $G:=(V,E)$, the Steiner distance $d_G(X)$ among a set of vertices $X$ is the minimum size among all the connected subgraphs of $G$ whose vertex set contains $X$. The $k-$Steiner distance matrix $D_k(G)$ of $G$ is a…

In the random geometric graph model $\mathsf{Geo}_d(n,p)$, we identify each of our $n$ vertices with an independently and uniformly sampled vector from the $d$-dimensional unit sphere, and we connect pairs of vertices whose vectors are…

We propose the following model of a random graph on n vertices. Let F be a distribution in R_+^{n(n-1)/2} with a coordinate for every pair i$ with 1 \le i,j \le n. Then G_{F,p} is the distribution on graphs with n vertices obtained by…

A $k$-regular graph is called a divisible design graph (DDG for short) if its vertex set can be partitioned into $m$ classes of size $n$, such that two distinct vertices from the same class have exactly $\lambda_1$ common neighbors, and two…

We establish the following splitter theorem for graphs and its generalization for matroids: Let $G$ and $H$ be $3$-connected simple graphs such that $G$ has an $H$-minor and $k:=|V(G)|-|V(H)|\ge 2$. Let $n:=\left\lceil k/2\right\rceil+1$.…

Let $D(G)=(d_{ij})_{n\times n}$ denote the distance matrix of a connected graph $G$ with order $n$, where $d_{ij}$ is equal to the distance between vertices $v_{i}$ and $v_{j}$ in $G$. A graph is called distance integral if all eigenvalues…

A vertex with neighbours of degrees $d_1 \geq ... \geq d_r$ has {\em vertex type} $(d_1, ..., d_r)$. A graph is {\em vertex-oblique} if each vertex has a distinct vertex-type. While no graph can have distinct degrees, Schreyer, Walther and…

Consider d uniformly random permutation matrices on n labels. Consider the sum of these matrices along with their transposes. The total can be interpreted as the adjacency matrix of a random regular graph of degree 2d on n vertices. We…

Given a regular (connected) graph $\Gamma=(X,E)$ with adjacency matrix $A$, $d+1$ distinct eigenvalues, and diameter $D$, we give a characterization of when its distance matrix $A_D$ is a polynomial in $A$, in terms of the adjacency…

Let $P$ be a set of $n \geq 5$ points in convex position in the plane. The path graph $G(P)$ of $P$ is an abstract graph whose vertices are non-crossing spanning paths of $P$, such that two paths are adjacent if one can be obtained from the…

The unit distance graph $G_{\mathbb{R}^d}^1$ is the infinite graph whose nodes are points in $\mathbb{R}^d$, with an edge between two points if the Euclidean distance between these points is 1. The 2-dimensional version $G_{\mathbb{R}^2}^1$…

We study a generalization of the Erd\H{o}s-Falconer distance problem over finite fields. For a graph $G$, two embeddings $p, p': V(G) \to \mathbb{F}_q^d$ of a graph $G$ are congruent if for all edges $(v_i, v_j)$ of $G$ we have that…

We are interested in the general question: to what extent are the spectral properties of a graph connected to the distance function? Our motivation is a concrete example of this question that is due to Steinerberger. We provide some…

This article investigates the isomorphism problem for graphs derived from the four standard graph products: Cartesian, Kronecker (direct), strong, and lexicographic product. We provide a complete characterization of all simple connected…