Fred Buckley

On the euclidean dimension of a wheel

AbstractFollowing Erdös, Harary, and Tutte, the euclidean dimension of a graphG is the minimumn such thatG can be embedded in euclideann-spaceRn so that each edge ofG has length 1. We present constructive proofs which give the euclidean dimension of a wheel and of a complete tripartite graph. We also define the generalized wheelWm,n as the join

A note on graphs with diameter-preserving spanning trees

\bar K_m + C_n

Radius-edge-invariant and diameter-edge-invariant graphs

and determine the euclidean dimension of all generalized wheels.

Unsolved Problems on Distance in Graphs

A graph G is called invertible if its adjacency matrix A has an inverse which is the adjacency matrix of some graph H. All such graphs were shown by Harary and Minc to have the form nK2. We now introduce signed invertible (or briefly s-invertible) graphs G as those whose inverse H is a signed graph. We identify two infinite classes of s-invertible graphs: the paths P2n of even order, and the corona of any graph with K1. We then characterize s-invertible trees.

Diameter-essential edges in a graph

The distance between a pair of vertices u, v in a graph G is the length of a shortest path joining u and v. The diameter diam(G) of G is the maximum distance between all pairs of vertices in G. A spanning tree T of G is diameter preserving if diam(T) = diam(G). In this note, we characterize graphs that have diameter-preserving spanning trees.

Highly irregular m chromatic graphs

ABSTRACT Let S be a subset of the vertex set V(G) of a nontrivial connected graph G. The geodetic closure (S) of S is the set of all vertices on geodesics between two vertices in S. The first player A chooses a vertex v1 of G. The second player B then picks v2 ≠ v1 and forms the geodetic closure (S2) = ({v1, v2}). Now A selects v3 e V—S2 and forms (S3) = ({v1, v2, v3}), etc. The player who first selects a vertex vn such that (Sn) = V wins the achievement game, but loses the avoidance game. These geodetic achievement and avoidance games are solved for several families of graphs by determining which player is the winner.

Regularizing irregular graphs

The eccentricity e(v) of v is the distance to a farthest vertex from v. The radius r(G) is the minimum eccentricity among the vertices of G and the diameter d(G) is the maximum eccentricity. For graph G-e obtained by deleting edge e in G, we have r(G-e)>=r(G) and d(G-e)>=d(G). If for all e in G, r(G-e)=r(G), then G is radius-edge-invariant. Similarly, if for all e in G, d(G-e)=d(G), then G is diameter-edge-invariant. In this paper, we study radius-edge-invariant and diameter-edge-invariant graphs and obtain characterizations of radius-edge-invariant graphs and diameter-edge-invariant graphs of diameter two.

Distance in graphs

Abstract The distance between a pair of nodes of a graph G is the length of a shortest path connecting them. The eccentricity of a node v is the greatest distance between v and another node. The radius and diameter of a graph are, respectively, the smallest and largest eccentricities among its nodes. The status of v is the sum of the distances from v to all other nodes. We shall discuss various conjectures and unsolved problems concerning distance concepts in graphs. These problems involve radius, diameter, and status, as well as other distance concepts such as distance sequences, domination, distance in digraphs, and convexity.