For any d ≥ 5 and k ≥ 3 we construct a family of Cayley graphs of degree d, diameter k, and order at least k((d−3)/3) k . By comparison with other available results in this area we show that our family gives the

Let CC d,k be the largest possible number of vertices in a cyclic Cayley graph of degree d and diameter k, and let AC d,k be the largest order in an Abelian Cayley graph for given d and k. We show that CC d,2 ≥ 13 36 (d + 2)(d − 4) for any d = 6p − 2 where p is a prime such that p = 13, p ≡ 1 (mod 13), and− 3 where q is a prime power.

The long-standing open problem of finding an upper bound for the Wiener index of a graph in terms of its order and diameter is addressed. Sharp upper bounds are presented for the Wiener index, and the related degree distance and Gutman index, for trees of order n and diameter at most 6.2010 Mathematics subject classification: primary 05C12; secondary 05C05, 05C07. Keywords and phrases: Wiener index, degree distance, Gutman index, tree, diameter. IntroductionLet G be a graph with vertex set V(G) and order n. We denote the distance between two vertices u, v in G by d G (u, v) (or simply d(u, v)); the diameter of G will be denoted by d(G) (or d), the eccentricity of a vertex v will be denoted by ec(v) and the degree of v will be denoted by deg (v). Let N G i (v) (or simply N i (v)) be the set of vertices at distance i from v in G. Let u, v be two adjacent (nonadjacent) vertices of a graph G. Then G = G − uv (G = G + uv) is obtained by removing the edge uv from G (by adding the edge uv to G).The Wiener index is the oldest topological index. It has been investigated in the mathematical, chemical and computer science literature since the 1940s. The Wiener index W(G) of a connected graph G is defined as the sum of the distances between all unordered pairs of vertices. The minimum value of the Wiener index of a graph (of a tree) of given order is attained by the complete graph (by the star), and the maximum value is attained by the path.The degree distance, a variant of the Wiener index, is defined as and the Gutman index is defined asThe smallest value of the degree distance and Gutman index of graphs of order n is attained by stars (see [1,11]). Turning to upper bounds on the degree distance, in 1999 Tomescu [11] conjectured the asymptotic upper bound D (G) ≤ (1/27)n 4 + O(n 3 ). Nine years later, Bucicovschi and Cioabǎ [2] commented that Tomescu's conjecture 'seems difficult at present time'. In the following year Dankelmann et al. [3] considered this problem and though they came close to proving the conjecture, their proof was inadequate to meet the O(n 3 ) error term. Recently, Morgan et al., in a submitted paper ('On a conjecture by Tomescu'), salvaged enough from the proof given in [3] and solved Tomescu's conjecture completely. There one can also find upper bounds on the degree distance of graphs of given order and diameter. Upper bounds on the Gutman index of a graph of given order and diameter were studied in [4,9]. In [9] it was proved, that Gut(. In this paper we study the indices mentioned above for trees of given order and diameter. Since Klein et al. [7] showed that for every tree T of order n,and in [6] Gutman proved thatany result on W(T ) yields a similar result on D (T ) and Gut(T ). It is not difficult to show that the extremal tree, which has the minimum Wiener index among trees of order n and diameter d, is the path of length d (containing d + 1 vertices) with the central vertex joined to the other n − d − 1 vertices; see [12]. The problem of finding an upper bound on the Wiener index of a ...

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