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### Distance graphs

Chen Avin
2008 Proceedings of the fifth international workshop on Foundations of mobile computing - DIAL M-POMC '08
To explain the similarities in the connectivity thresholds, we introduce an extension of the random geometric graphs: the random distance graph, D(n, g): A graph resulting from placing n points uniformly  ...  A random geometric graph G(n, r) is a graph resulting from placing n points uniformly at random on the unit area disk, and connecting two points iff their Euclidean distance is at most the radius r(n).  ...  that is based on distance graphs.  ...

### Distanced graphs

Lawrence S. Moss
1992 Discrete Mathematics
These are graphs with additional relation symbols for finite distances. Every embedding of distanced graphs is an isometric embedding.  ...  ., Distanced graphs, Discrete Mathematics 102 (1992) 287-305. This paper studies model theoretic conditions that arise in the study of distanced graphs.  ...  For distanced graphs, we verify the Amalgamation Property (iii) in Section 3. Condition (v), specialized to the case of distanced graphs, requires that every distanced graph be distance finite.  ...

### Distances between directed graphs

1987 Časopis pro pěstování matematiky
Instead of speaking about the distance between isomorphism classes of graphs we shall sometimes speak about the distance between graphs.  ...  The distance between the graphs G l9 G 2 is the distance between isomorphism classes © l9 © 2 such that G x e © 1? G 2 e © 2 . Now we define the degree vectors of a digraph.  ...  Let © A , © 2 be two isomorphism classes of directed graphs with the same number n of vertices and the same number m of edges. Then 4(®i. ©2) -S m -1 . References  ...

### Steiner distance in graphs

Gary Chartrand, Ortrud R. Oellermann, Song Lin Tian, Hung Bin Zou
1989 Časopis pro pěstování matematiky
INTRODUCTION One of the most basic concepts associated with a graph is distance.  ...  THE SIZE OF (n; p) GRAPHS Given a nonempty subset S of the vertex set of a connected graph G, the distance d(S) is the minimum size of a connected graph whose vertex set contains S.  ...

### Signed distance graphs

Miroslav Fiedler
1969 Journal of Combinatorial Theory
Signed graphs are assigned to systems of points in a metric space. Special cases are investigated.  ...  It is easy to see that this definition is in conformity with the definition of the distance graphs for the case of a spherical (r-1)-space and the distance d = 89 and dasses which can easily be derived  ...  by a positive edge if and only if the distance p(p~, p~) < d, and by a negative edge if and only if p(Pi, Pk) ~ d (hence they are not joined if and only if p(Pi , P~) = d).  ...

### Distance Preserving Graphs [article]

2015 arXiv   pre-print
We say a graph G is distance preserving (dp) if it has an isometric subgraph of every possible order up to the order of G.  ...  We consider how to add a vertex to a dp graph so that the result is a dp graph. This condition implies that chordal graphs are dp.  ...  The definition of a distance-preserving graph is similar to the one for distance-hereditary graphs, but is less restrictive.  ...

### Integral distance graphs

Jer‐Jeong Chen, Gerard J. Chang, Kuo‐Ching Huang
1997 Journal of Graph Theory
The distance graph G(Z, D) with distance set D is the graph with vertex set Z, and two vertices x and y are adjacent if and only if |x − y| ∈ D.  ...  For each subset D of dist(S), the distance graph G(S, D) with distance set D is the graph with vertex set S and edge set {{x, y}: d(x, y) ∈ D}.  ...  Motivated by the graph coloring problem, Eggleton  considered the following general problem. We refer  for general notation and terminology in graphs.  ...

### Distance in Graphs [chapter]

Wayne Goddard, Ortrud R. Oellermann
2010 Structural Analysis of Complex Networks
The distance between two vertices is the basis of the definition of several graph parameters including diameter, radius, average distance and metric dimension.  ...  We also discuss characterizations of graph classes described in terms of distance or shortest paths. Finally, generalizations are considered.  ...  Acknowledgments We would like to thank Peter Dankelmann for sharing his thoughts on average distance with us.  ...

### Distance-residual graphs [article]

Primoz Luksic, Tomaz Pisanski
2006 arXiv   pre-print
If we are given a connected finite graph G and a subset of its vertices V_0, we define a distance-residual graph as a graph induced on the set of vertices that have the maximal distance from V_0.  ...  The relations between the distance-residual graphs of product graphs and their factors are shown.  ...  Since every graph can be a distance-residual graph, it is a challenge to find wellknown graphs as distance residuals of some other well-known graphs.  ...

### Distance-regular graphs [article]

Edwin R. van Dam, Jack H. Koolen, Hajime Tanaka
2016 arXiv   pre-print
This is a survey of distance-regular graphs.  ...  We present an introduction to distance-regular graphs for the reader who is unfamiliar with the subject, and then give an overview of some developments in the area of distance-regular graphs since the  ...  A graph is called distance degree regular if each distance-i graph is regular.  ...

### ?-Unit Distance Graphs

Geoffrey Exoo
2004 Discrete & Computational Geometry
We consider a variation on the problem of determining the chromatic number of the Euclidean plane and define the ε-unit distance graph to be the graph whose vertices are the points of E 2 , in which two  ...  points are adjacent whenever their distance is within ε of 1.  ...  4 . 4 An ε-unit distance graph with chromatic number 5.  ...

### Distance-Transitive Graphs [chapter]

Andries E. Brouwer, Arjeh M. Cohen, Arnold Neumaier
1989 Distance-Regular Graphs
Distance-Regular Graphs Definition 2.4.1 We can always find a distance partition for any graph (e.g. example 2.2.2).  ...  Antipodal Graphs Antipodal graphs are a class of imprimitive distance-transitive graphs.  ...  They showed that there is only one cubic graph that is distance-regular but not distance-transitive, this being Tutte's (3, 12) -cage (see  ).  ...

### Distance-hereditary graphs

Hans-Jürgen Bandelt, Henry Martyn Mulder
1986 Journal of combinatorial theory. Series B (Print)
Distance-hereditary graphs (sensu Howorka) are connected graphs in which all induced paths are isometric. Examples of such graphs are provided by complete multipartite graphs and ptolemaic graphs.  ...  Moreover, distance-hereditary graphs are characterized in terms of the distance function d, or via forbidden isometric subgraphs. t-1  ...  That is, the distance of any two vertices in an induced path equals their distance in the graph.  ...

### Distance-Balanced Graphs

Janja Jerebic, Sandi Klavžar, Douglas F. Rall
2008 Annals of Combinatorics
Distance-balanced graphs are introduced as graphs in which every edge uv has the following property: the number of vertices closer to u than to v is equal to the number of vertices closer to v than to  ...  Distance-balanced Cartesian and lexicographic products of graphs are also characterized. Several open problems are posed along the way.  ...  Distance-balanced product graphs In this section we study the conditions under which the standard graph products produce a distance-balanced graph.  ...

### Integral distance graphs

Jer-Jeong Chen, Gerard J. Chang, Kuo-Ching Huang
1997 Journal of Graph Theory
The distance graph G(Z, D) with distance set D is the graph with vertex set Z, and two vertices x and y are adjacent if and only if |x − y| ∈ D.  ...  For each subset D of dist(S), the distance graph G(S, D) with distance set D is the graph with vertex set S and edge set {{x, y}: d(x, y) ∈ D}.  ...  Motivated by the graph coloring problem, Eggleton  considered the following general problem. We refer  for general notation and terminology in graphs.  ...
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