Mirka Miller

On irregular total labellings

Two new graph characteristics, the total vertex irregularity strength and the total edge irregularity strength, are introduced. Estimations on these parameters are obtained. For some families of graphs the precise values of these parameters are proved.

A Note on Large Graphs of Diameter Two and Given Maximum Degree

Letvt(d,2) be the largest order of a vertex-transitive graph of degreedand diameter 2. It is known thatvt(d,2)=d2+1 ford=1,2,3, and 7; for the remaining values ofdwe havevt(d,2)?d2?1. The only knowngenerallower bound onvt(d,2), valid forall d, seems to bevt(d,2)??(d+2)/2? ?(d+2)/2?. Using voltage graphs, we construct a family of vertex-transitive non-Cayley graphs which shows thatvt(d,2)?(8/9)(d+12)2for alldof the formd=(3q?1)/2, whereqis a prime power congruent with 1 (mod 4). The construction generalizes to all prime powers and yields large highly symmetric graphs for other degrees as well. In particular, ford=7 we obtain as a special case the Hoffman?Singleton graph, and ford=11 andd=13 we have new largest graphs of diameter 2, and degreedon 98 and 162 vertices, respectively.

The train marshalling problem

Abstract The problem considered in this paper arose in connection with the rearrangement of railroad cars in China. In terms of sequences the problem reads as follows: Train Marshalling Problem: Given a partition S of {1,…,n} into disjoint sets S1,…,St, find the smallest number k=K(S) so that there exists a permutation p(1),…,p(t) of {1,…,t} with the property: The sequence of numbers 1,2,…,n,1,2,…,n,…,1,2,…,n where the interval 1,2,…,n is repeated k times contains all the elements of Sp(1), then all elements of S p(2) ,… , etc., and finally all elements of Sp(t). The aim of this paper is to show that the decision problem: “Given numbers n,k and a partition S of {1,2,…,n}, is K(S)⩽k?” is NP-complete. In light of this, we give a general upper bound for K(S) in terms of n.

Randoms variance reduction in 3D PET

In positron emission tomography (PET), random coincidence events must be removed from the measured signal in order to obtain quantitatively accurate data. The most widely implemented technique for estimating the number of random coincidences on a particular line of response is the delayed coincidence channel method. Estimates obtained in this way are subject to Poisson noise, which then propagates into the final image when the estimates are subtracted from the prompt signal. However, this noise may be reduced if variance reduction techniques similar to those used in normalization of PET detectors are applied to the randoms estimates prior to use. We have investigated the effects of randoms variance reduction on noise-equivalent count (NEC) rates on a whole-body PET camera operating in 3D mode. NEC rates were calculated using a range of phantoms representative of situations that might be encountered clinically. We have also investigated the properties of three randoms variance reduction methods (based on algorithms previously used for normalization) in terms of their systematic accuracy and their variance reduction efficacy, both in phantom studies and in vivo. Those algorithms investigated that do not make assumptions about the spatial distribution of random coincidences give the best estimates of the randoms distribution. With the camera used, which has a limited axial extent (10.8 cm) and a large ring diameter (102 cm), the gains in image signal-to-noise ratio obtained with this technique ranged from approximately 5% to approximately 15%, depending on object size, activity distribution and the amount of activity in the field of view. Larger gains would be expected if this technique were to be employed on cameras of greater axial extent and smaller ring diameter.

A characterization of strongly chordal graphs

Abstract In this paper, we present a simple charactrization of strongly chordal graphs. A chordal graph is strongly chordal if and only if every cycle on six or more vertices has an induced triangle with exactly two edges of the triangle as the chords of the cycle.

Maximum h -colourable subgraph problem in balanced graphs

The k-fold clique transversal problem is to locate a minimum set Ω of vertices of a graph such that every maximal clique has at least k elements of Ω. The maximum h-colourable subgraph problem is to find a maximum subgraph of a graph which is h-colourable. We show that the k-fold clique transversal problem and the maximum h-colourable subgraph problem are polynomially solvable on balanced graphs. We also provide a polynomial algorithm to recognize balanced graphs.

Edge-antimagic graphs

For a graph G=(V,E), a bijection g from V(G)@?E(G) into {1,2,...,|V(G)|+|E(G)|} is called (a,d)-edge-antimagic total labeling of G if the edge-weights w(xy)=g(x)+g(y)+g(xy), xy@?E(G), form an arithmetic progression starting from a and having common difference d. An (a,d)-edge-antimagic total labeling is called super (a,d)-edge-antimagic total if g(V(G))={1,2,...,|V(G)|}. We study super (a,d)-edge-antimagic properties of certain classes of graphs, including friendship graphs, wheels, fans, complete graphs and complete bipartite graphs.

( a,d )-edge-antimagic total labelings of caterpillars

For a graph G = (V,E), a bijection g from V(G) ∪ E(G) into { 1,2, ..., ∣ V(G) ∣ + ∣ E(G) ∣ } is called (a,d)-edge-antimagic total labeling of G if the edge-weights w(xy) = g(x) + g(y) + g(xy), xy ∈ E(G), form an arithmetic progression with initial term a and common difference d. An (a,d)-edge-antimagic total labeling g is called super (a,d)-edge-antimagic total if g(V(G)) = { 1,2,..., ∣ V(G) ∣ } . We study super (a,d)-edge-antimagic total properties of stars Sn and caterpillar Sn1,n2,...,nr.

On the Structure of Digraphs with Order Close to the Moore Bound

Abstract. The Moore bound for a diregular digraph of degree E5>, k and diameter k is . It is known that digraphs of order do not exist for d>1 and k>1 ([24] or [6]). In this paper we study digraphs of degree E5>, k, diameter k and order , denoted by (d, k)-digraphs. Miller and Fris showed that (2, k)-digraphs do not exist for k≥3 [22]. Subsequently, we gave a necessary condition of the existence of (3, k)-digraphs, namely, (3, k)-digraphs do not exist if k is odd or if k+1 does not divide [3]. The (E5>, k, 2)-digraphs were considered in [4]. In this paper, we present further necessary conditions for the existence of (d, k)-digraphs. In particular, for , we show that a (d, k)-digraph contains either no cycle of length k or exactly one cycle of length k.

Vertex-antimagic total labelings of graphs

In this paper we introduce a new type of graph labeling, the