Tomáš Vetrík

Cayley graphs of given degree and diameters 3, 4 and 5

Abstract Let C d , k be the largest number of vertices in a Cayley graph of degree d and diameter k . We show that C d , 3 ≥ 3 16 ( d − 3 ) 3 and C d , 5 ≥ 25 ( d − 7 4 ) 5 for any d ≥ 8 , and C d , 4 ≥ 32 ( d − 8 5 ) 4 for any d ≥ 10 . For sufficiently large d our graphs are the largest known Cayley graphs of degree d and diameters 3, 4 and 5.

WIENER INDEX OF TREES OF GIVEN ORDER AND DIAMETER AT MOST 6

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 \(6\). DOI: 10.1017/S0004972713000816

On the Gutman index and minimum degree

Abstract The Gutman index Gut ( G ) of a graph G is defined as ∑ { x , y } ⊆ V ( G ) deg ( x ) deg ( y ) d ( x , y ) , where V ( G ) is the vertex set of G , deg ( x ) , deg ( y ) are the degrees of vertices x and y in G , and d ( x , y ) is the distance between vertices x and y in G . We show that for finite connected graphs of order n and minimum degree δ , where δ is a constant, Gut ( G ) ≤ 2 4 ⋅ 3 5 5 ( δ + 1 ) n 5 + O ( n 4 ) . Our bound is asymptotically sharp for every δ ≥ 2 and it extends results of Dankelmann, Gutman, Mukwembi and Swart (2009) and Mukwembi (2012), whose bound is sharp only for graphs of minimum degree 2 .

List coloring of complete multipartite graphs

The choice number of a graph G is the smallest integer k such that for every assignment of a list L(v) of k colors to each vertex v of G, there is a proper coloring of G that assigns to each vertex v a color from L(v). We present upper and lower bounds on the choice number of complete multipartite graphs with partite classes of equal sizes and complete r-partite graphs with r − 1 partite classes of order two.

Large vertex-transitive and Cayley graphs with given degree and diameter

Abstract We present an upper bound on the number of vertices in graphs of given degree and diameter 3 that arise as lifts of dipoles with voltage assignments in Abelian groups. Further, we construct a family of Cayley graphs of degree d = 3 m − 1 and diameter k ⩾ 3 of order km k . By comparison with other available results in this area we show that, for sufficiently large d and k such that k ⩽ d − 2 , our family gives the current largest known Cayley graphs of degree d and diameter k.

Large Cayley digraphs and bipartite Cayley digraphs of odd diameters

Abstract Let C d , k be the largest number of vertices in a Cayley digraph of degree d and diameter k , and let B C d , k be the largest order of a bipartite Cayley digraph for given d and k . For every degree d ≥ 2 and for every odd k we construct Cayley digraphs of order 2 k ⌊ d 2 ⌋ k and diameter at most k , where k ≥ 3 , and bipartite Cayley digraphs of order 2 ( k − 1 ) ⌊ d 2 ⌋ k − 1 and diameter at most k , where k ≥ 5 . These constructions yield the bounds C d , k ≥ 2 k ⌊ d 2 ⌋ k for odd k ≥ 3 and d ≥ 3 k 2 k + 1 , and B C d , k ≥ 2 ( k − 1 ) ⌊ d 2 ⌋ k − 1 for odd k ≥ 5 and d ≥ 3 k − 1 k − 1 + 1 . Our constructions give the best currently known bounds on the orders of large Cayley digraphs and bipartite Cayley digraphs of given degree and odd diameter k ≥ 5 . In our proofs we use new techniques based on properties of group automorphisms of direct products of abelian groups.

An upper bound for graphs of diameter 3 and given degree obtained as abelian lifts of dipoles

We derive an upper bound on the number of vertices in graphs of diameter 3 and given degree arising from Abelian lifts of dipoles with loops and multiple edges.

Computing the metric dimension of the categorial product of some graphs

ABSTRACT A set of vertices W is a resolving set of a graph G if every two vertices of G have distinct representations of distances with respect to the set W. The number of vertices in a smallest resolving set is called the metric dimension. This invariant has extensive applications in robotics, since the metric dimension can represent the minimum number of landmarks, which uniquely determine the position of a robot moving in a graph space. Finding the metric dimension of a graph is a non-deterministic polynomial-time hard problem. We present exact values of the metric dimension of several networks, which can be obtained as categorial products of graphs.

The Degree-Diameter Problem for Claw-Free Graphs and Hypergraphs

We study the degree-diameter problem for claw-free graphs and 2-regular hypergraphs. Let be the largest order of a claw-free graph of maximum degree Δ and diameter D. We show that , where , for any D and any even . So for claw-free graphs, the well-known Moore bound can be strengthened considerably. We further show that for with (mod 4). We also give an upper bound on the order of -free graphs of given maximum degree and diameter for . We prove similar results for the hypergraph version of the degree-diameter problem. The hypergraph Moore bound states that the order of a hypergraph of maximum degree Δ, rank k, and diameter D is at most . For 2-regular hypergraph of rank and any diameter D, we improve this bound to , where . Our construction of claw-free graphs of diameter 2 yields a similar result for hypergraphs of diameter 2, degree 2, and any even rank .

On the metric dimension of directed and undirected circulant graphs

Abstract The undirected circulant graph Cn(±1, ±2, . . . , ±t) consists of vertices v0, v1, . . . , vn−1 and undirected edges vivi+j, where 0 ≤ i ≤ n − 1, 1 ≤ j ≤ t (2 ≤ t ≤ n2 {n \over 2} ), and the directed circulant graph Cn(1, t) consists of vertices v0, v1, . . . , vn−1 and directed edges vivi+1, vivi+t, where 0 ≤ i ≤ n − 1 (2 ≤ t ≤ n−1), the indices are taken modulo n. Results on the metric dimension of undirected circulant graphs Cn(±1, ±t) are available only for special values of t. We give a complete solution of this problem for directed graphs Cn(1, t) for every t ≥ 2 if n ≥ 2t2. Grigorious et al. [On the metric dimension of circulant and Harary graphs, Appl. Math. Comput. 248 (2014) 47–54] presented a conjecture saying that dim (Cn(±1, ±2, . . . , ±t)) = t + p − 1 for n = 2tk + t + p, where 3 ≤ p ≤ t + 1. We disprove it by showing that dim (Cn(±1, ±2, . . . , ±t)) ≤ t + p+12 {{p + 1} \over 2} for n = 2tk + t + p, where t ≥ 4 is even, p is odd, 1 ≤ p ≤ t + 1 and k ≥ 1.