A k-ended tree is a tree with at most k endvertices. Broersma and Tuinstra [3] have proved that for k ≥ 2 and for a pair of nonadjacent vertices u, v in a graph G of order n with degG u + degG v ≥ n − 1, G has a spanning k-ended tree if and only if G + uv has a spanning k-ended tree. The distant area for u and v is the subgraph induced by the set of vertices that are not adjacent with u or v. W...

A graph is walk-regular if the number of cycles of length ` rooted at a given vertex is a constant through all the vertices. For a walk-regular graph G with d + 1 different eigenvalues and spectrally maximum diameter D = d, we study the geometry of its d-cliques, that is, the sets of vertices which are mutually at distance d. When these vertices are projected onto an eigenspace of its adjacency...

In a graph G, the distance d(u, v) between a pair of vertices u and v is the length of a shortest path joining them. The eccentricity e(u) of a vertex u is the distance to a vertex farthest from u. The minimum eccentricity is called the radius of the graph and the maximum eccentricity is called the diameter of the graph. The radial graph R(G) based on G has the vertex set as in G, two vertices ...

Let G be a simple connected graph. The terminal distance matrix of G is the distance matrix between all pendant vertices of G. In this paper, we introduce some general transformations that increase the terminal distance spectral radius of a connected graph and characterize the extremal trees with respect to the terminal distance spectral radius among all trees with a fixed number of pendant ver...

The packing chromatic number χρ(G) of a graph G is the smallest integer k needed to proper color the vertices of G in such a way the distance between any two vertices having color i be at least i + 1. We obtain χρ(Hq,m) for m = 3, where Hq,m is the Hamming graph of words of length m and alphabet with q symbols, and tabulate bounds of them for m ≥ 4 up to 10000 vertices. We also give a polynomia...

We show here that the Kirchhoff index of a network is the average of the Wiener capacities of its vertices. Moreover, we obtain a closed-form formula for the effective resistance between any pair of vertices when the considered network has some symmetries which allows us to give the corresponding formulas for the Kirchhoff index. In addition, we find the expression for the Foster’s n-th Formula.

We present a deterministic local routing scheme that is guaranteed to find a path between any pair of vertices in a halfθ6-graph whose length is at most 5/ √ 3 = 2.886... times the Euclidean distance between the pair of vertices. The half-θ6graph is identical to the Delaunay triangulation where the empty region is an equilateral triangle. Moreover, we show that no local routing scheme can achie...

A geometric t-spanner on a set of points in Euclidean space is a graph containing for every pair of points a path of length at most t times the Euclidean distance between the points. Informally, a spanner is O(k)-robust if deleting k vertices only harms O(k) other vertices. We show that on any one-dimensional set of n points, for any ε > 0, there exists an O(k)-robust 1-spanner with O(n1+ε) edg...

Let T be a tree on n vertices and let the n− 1 edges e1, e2, . . . , en−1 have weights that are s× s matrices W1,W2, . . . ,Wn−1, respectively. For two vertices i, j, let the unique ordered path between i and j be pi,j = er1er2 . . . erk . Define the distance between i and j as the s × s matrix Ei,j = ∏k p=1Wep . Consider the ns × ns matrix D whose i, j-th block is the matrix Ei,j . We give a f...

Given a set of points in the plane, we show that the θ-graph with 5 cones is a geometric spanner with spanning ratio at most √ 50 + 22 √ 5 ≈ 9.960. This is the first constant upper bound on the spanning ratio of this graph. The upper bound uses a constructive argument that gives a (possibly self-intersecting) path between any two vertices, of length at most √ 50 + 22 √ 5 times the Euclidean dis...