In any connected, undirected graph G = (V, E), the distance d(x, y) between two vertices x and y of G is the minimum number of edges in a path linking x to y in G. A sphere in G is a set of the form Sr (x) = {y ∈ V : d(x, y) = r}, where x is a vertex and r is a nonnegative integer called the radius of the sphere. We first address in this paper the following question: What is the minimum number ...

A triangle can be specified by giving three independent geometric relationships defined between its elements. Usually, these relationships are distances between two vertices, angles between two sides, and heights. For each triangle specified by a set of three of such relationships, we present a procedure that constructs the triangle using ruler and compass alone. ∗While on leave in Computer Sci...

Let d, k be any two positive integers with k > d > 0. We consider a k-coloring of a graph G such that the distance between each pair of vertices in the same color-class is at least d. Such graphs are said to be (k,d)-colorable. The object of this paper is to determine the maximum size of (k, 3)-colorable, (k, 4)-colorable, and (k, k 1 )-colorable graphs. Sharp results are obtained for both (k, ...

Given a graph G = (V,E) with non-negative edge lengths whose vertices are assigned a label from L = {λ1, . . . , λl}, we construct a compact distance oracle that answers queries of the form: “What is δ(v, λ)?”, where v ∈ V is a vertex in the graph, λ ∈ L a vertex label, and δ(v, λ) is the distance (length of a shortest path) between v and the closest vertex labeled λ in G. We formalize this nat...

When a weighted graph is an instance of the Distance Geometry Problem (DGP), certain types of vertex orders (called discretization orders) allow the use of a very efficient, precise and robust discrete search algorithm (called Branch-and-Prune). Accordingly, finding such orders is critically important in order to solve DGPs in practice. We discuss three types of discretization orders, the compl...

In this paper, we study the parallel and the space complexity of the graph isomorphism problem (GI) for several parameterizations. Let H = {H1,H2, · · · ,Hl} be a finite set of graphs where |V (Hi)| ≤ d for all i and for some constant d. Let G be an H-free graph class i.e., none of the graphs G ∈ G contain any H ∈ H as an induced subgraph. We show that GI parameterized by vertex deletion distan...

We give a bound on the sizes of two sets of vertices at a given minimum distance in a graph in terms of polynomials and the Laplace spectrum of the graph. We obtain explicit bounds on the number of vertices at maximal distance and distance two from a given vertex, and on the size of two equally large sets at maximal distance. For graphs with four eigenvalues we find bounds on the number of vert...

let z2 = {0, 1} and g = (v ,e) be a graph. a labeling f : v → z2 induces an edge labeling f* : e →z2 defined by f*(uv) = f(u).f (v). for i ε z2 let vf (i) = v(i) = card{v ε v : f(v) = i} and ef (i) = e(i) = {e ε e : f*(e) = i}. a labeling f is said to be vertex-friendly if | v(0) − v(1) |≤ 1. the vertex balance index set is defined by {| ef (0) − ef (1) | : f is vertex-friendly}. in this paper ...

For any vertex v and any edge e in a non-trivial connected graph G, the distance sum d(v) of v is d(v) = ∑ u∈V d(v, u), the vertex-to-edge distance sum d1(v) of v is d1(v) = ∑ e∈E d(v, e), the edge-to-vertex distance sum d2(e) of e is d2(e) = ∑ v∈V d(e, v) and the edge-to-edge distance sum d3(e) of e is d3(e) = ∑ f∈E d(e, f). The set M(G) of all vertices v for which d(v) is minimum is the media...

The spectral excess theorem for distance-regular graphs states that a regular (connected) graph is distance-regular if and only if its spectral-excess equals its average excess. A bipartite graph Γ is distance-biregular when it is distance-regular around each vertex and the intersection array only depends on the stable set such a vertex belongs to. In this note we derive a new version of the sp...