On the dimension to represent a graph by a unit distance graph

It is proved that if a graphG has maximum degreed, then its vertices can be represented by distinct unit vectors inR ^2d so that two vectors are orthogonal if and only if the corresponding vertices are adjacent. As a corollary it follows that if a graph has maximum degreed, then it is isomorphic to a unit distance graph inR ^2d.     

On the degree distance of a graph

If G is a connected graph with vertex set V, then the degree distance of G, D^′(G), is defined as , where degw is the degree of vertex w, and d(u,v) denotes the distance between u and v. We prove the asymptotically sharp upper bound for graphs of order n and diameter d. As a corollary we obtain the bound for graphs of order n. This essentially proves a conjecture by Tomescu [I. Tomescu, Some extremal properties of the degree distance of a graph, Discrete Appl. Math. (98) (1999) 159-163].     

The average Steiner distance of a graph

The average distance μ(G) of a graph G is the average among the distances between all pairs of vertices in G. For n ≥ 2, the average Steiner n-distance μ_n(G) of a connected graph G is the average Steiner distance over all sets of n vertices in G. It is shown that for a connected weighted graph G, μ_n(G) ≤ μ_k(G) + μ_n+1−k(G) where 2 ≤ k ≤ n − 1. The range for the average Steiner n-distance of a connected graph G in terms of n and |V(G)| is established. Moreover, for a tree T and integer k, 2 ≤ k ≤ n − 1, it is shown that μ_n(T) ≤ (n/k)μ_k(T) and the range for μ_n(T) in terms of n and |V(T)| is established. Two efficient algorithms for finding the average Steiner n-distance of a tree are outlined. © 1996 John Wiley & Sons, Inc.     

Convex graph invariant relaxations for graph edit distance

Mathematical Programming - The edit distance between two graphs is a widely used measure of similarity that evaluates the smallest number of vertex and edge deletions/insertions required to...     

On the distance matrix of a directed graph

In this note, we show how the determinant of the distance matrix D(G) of a weighted, directed graph G can be explicitly expressed in terms of the corresponding determinants for the (strong) blocks G_i of G. In particular, when cof D(G), the sum of the cofactors of D(G), does not vanish, we have the very attractive formula .     

On the distance function of a connected graph

An axiomatic characterization of the distance function of a connected graph is given in this note. The triangle inequality is not contained in this characterization.     

On the least distance eigenvalue of a graph

Denote by D(G)=_(di,j)n×n$D(G)={(d_{i,j})}_{n\times n}$ the distance matrix of a connected graph G with n   vertices, where _dij$d_{ij}$ is equal to the distance between vertices _vi$v_i$ and _vj$v_j$ in G  . The least eigenvalue of D(G)$D(G)$ is called the least distance eigenvalue of G  , denoted by _λn${\lambda }_n$. In this paper, we determine all the graphs with _λn∈[−2.383,0]${\lambda }_n\in [-2.383,0]$.     

Bounding the distance among longest paths in a connected graph

It is easy to see that in a connected graph any $2$ longest paths have a vertex in common. For $k\ge 7$, Skupień in 1966 obtained a connected graph in which some $k$ longest paths have no common vertex, but every $k?1$ longest paths have a common vertex. It is not known whether every $3$ longest paths in a connected graph have a common vertex and similarly for 4, 5, and $6$ longest path. Fujita et al. in 2015 give an upper bound on distance among $3$ longest paths in a connected graph. In this paper we give a similar upper bound on distance between $4$ longest paths and also for $k$ longest paths, in general.     

Mean distance and minimum degree

We prove that in a graph of order n and minimum degree d, the mean distance μ must satisfy . This asymptotically confirms, and improves, a conjecture of the computer program GRAFFITI. The result is close to optimal; examples show that for any d, μ may be larger than n/(d + 1). © 1997 John Wiley & Sons, Inc. J Graph Theory 25: 95–99, 1997     

On the probability of the occurrence of a copy of a fixed graph in a random distance graph

The threshold probability of the occurrence of a copy of a balanced graph in a random distance graph is obtained. The technique used by P. Erd?s and A. Rényi for determining the threshold probability for the classical random graph could not be applied in the model under consideration. In this connection, a new method for deriving estimates of the number of copies of a balanced graph in a complete distance graph is developed.     

Inverse of the distance matrix of a block graph

A connected graph G, whose 2-connected blocks are all cliques (of possibly varying sizes) is called a block graph. Let D be its distance matrix. By a theorem of Graham, Hoffman and Hosoya, we have det(D)?≠?0. We give a formula for both the determinant and the inverse, D ^?1 of D.     

Independence numbers of random subgraphs of a distance graph

We consider the so-called distance graph G(n, 3, 1), whose vertices can be identified with three-element subsets of the set {1, 2,..., n}, two vertices being joined by an edge if and only if the corresponding subsets have exactly one common element. We study some properties of random subgraphs of G(n, 3, 1) in the Erd?s–Rényi model, in which each edge is included in the subgraph with some given probability p independently of the other edges. We find the asymptotics of the independence number of a random subgraph of G(n, 3, 1).     

Mean distance of Brownian motion on a Riemannian manifold

Consider the mean distance of Brownian motion on Riemannian manifolds. We obtain the first three terms of the asymptotic expansion of the mean distance by means of stochastic differential equation for Brownian motion on Riemannian manifold. This method proves to be much simpler for further expansion than the methods developed by Liao and Zheng (Ann. Probab. 23(1) (1995) 173). Our expansion gives the same characterizations as the mean exit time from a small geodesic ball with regard to Euclidean space and the rank 1 symmetric spaces.     

The center and the distance center of a ptolemaic graph

The points of a distance center in a Ptolemaic graph G induce a complete subgraph of G, and the points of a center in a chordal graph constitute a convex.     

Approximately locating an invisible agent in a graph with relative distance queries

In a pursuit evasion game on a finite, simple, undirected, and connected graph $G$, a first player visits vertices $m_1,m_2,\dots$ of $G$, where $m_{i+1}$ is in the closed neighborhood of $m_i$ for every $i$, and a second player probes arbitrary vertices $c_1,c_2,\dots$ of $G$, and learns whether or not the distance between $c_{i+1}$ and $m_{i+1}$ is at most the distance between $c_i$ and $m_i$. Up to what distance $d$ can the second player determine the position of the first? For trees of bounded maximum degree and grids, we show that $d$ is bounded by a constant. We conjecture that $d=O(logn)$ for every graph $G$ of order $n$, and show that $d=0$ if $m_{i+1}$ may differ from $m_i$ only if $i$ is a multiple of some sufficiently large integer.