In this paper we prove that any distance-balanced graph G with ∆(G) ≥ |V (G)| − 3 is regular. Also we define notion of distancebalanced closure of a graph and we find distance-balanced closures of trees T with ∆(T ) ≥ |V (T )| − 3.

We show that regular homogeneous two-weight Z pk -codes where p is odd and k > 2 with dual Hamming distance at least four do not exist. The proof relies on existence conditions for the strongly regular graph built on the cosets of the dual code.

A colouring of a strongly regular graph is an allocation of colours (or treatments) to the vertices of the graph. Such a colouring is balanced if every pair of distinct colours occurs equally often on the ends of an edge. When the graph is the complete regular multipartite graph a balanced colouring is just a balanced incomplete-block design, or 2-design. Some constructions are given. For examp...

The concept of strongly balanced graph is introduced. It is shown that there exists a strongly balanced graph with u vertices and e edges if and only if I s u 1 s e s (2"). This result is applied to a classic question of Erdos and Renyi: What is the probability that a random graph on n vertices contains a given graph? A rooted version of this problem is also solved.

A graph G is said to be distance-balanced if for any edge uv of G, the number of vertices closer to u than to v is equal to the number of vertices closer to v than to u. Let GP (n, k) be a generalized Petersen graph. Jerebic, Klavžar, and Rall [Distance-balanced graphs, Ann. Comb. 12 (2008) 71–79] conjectured that: For any integer k > 2, there exists a positive integer n0 such that the GP (n, k...

For any 2-distance set X in the n-dimensional binary Hamming spaceHn, let ΓX be the graph with X as the vertex set and with two vertices adjacent if and only if the distance between them is the smaller of the two nonzero distances in X. The binary spherical representation number of a graph Γ, or bsr Γ , is the least n such that Γ is isomorphic to ΓX , where X is a 2-distance set lying on a sphe...

A graph Γ with diameter d is strongly distance-regular if Γ is distanceregular and its distance-d graph Γd is strongly regular. The known examples are all the connected strongly regular graphs (i.e. d = 2), all the antipodal distanceregular graphs, and some distance-regular graphs with diameter d = 3. The main result in this paper is a characterization of these graphs (among regular graphs with...

Let G be a strongly connected, weighted directed graph. We define a product distance η(i, j) for pairs i, j of vertices and form the corresponding product distance matrix. We obtain a formula for the determinant and the inverse of the product distance matrix. The edge orientation matrix of a directed tree is defined and a formula for its determinant and its inverse, when it exists, is obtained....

Let Γ be a distance-regular graph with diameter d and Kneser graph K = Γd, the distance-d graph of Γ. We say that Γ is partially antipodal when K has fewer distinct eigenvalues than Γ. In particular, this is the case of antipodal distance-regular graphs (K with only two distinct eigenvalues), and the so-called half-antipodal distance-regular graphs (K with only one negative eigenvalue). We prov...

Inspired by some intriguing examples, we study uniform association schemes and uniform coherent configurations. Perhaps the most important subclass of these objects is the class of cometric Q-antipodal association schemes. The concept of a cometric association scheme (the dual version of a distance-regular graph) is well-known; however, until recently it has not been studied well outside the ar...