TRANSACTIONS ON COMBINATORICS
Year:2013 | Volume:2 | Issue:1|Papers Count:6

In this paper, we define the common minimal common neighborhood dominating signed graph (or common minimal CN-dominating signed graph) of a given signed graph and offer a structural characterization of common minimal CN-dominating signed graphs. In the sequel, we also obtained switching equivalence characterization:  `S~CMCN (S), where `S and CMCN (S) are complementary signed graph and common minimal CN-signed graph of Sigma respectively.

A set S of vertices in a graph G is a dominating set if every vertex of V-S is adjacent to some vertex in S. The domination number g (G) is the minimum cardinality of a dominating set in G. The annihilation number a (G) is the largest integer k such that the sum of the first k terms of the non-decreasing degree sequence of G is at most the number of edges in G. In this paper, we show that for any tree T of order n³2, g (T)£3a (T)+2/4, and we characterize the trees achieving this bound.

We construct two classes of Gray maps, called type-I Gray map and type-II Gray map, for a finite p-group G. Type-I Gray maps are constructed based on the existence of a Gray map for a maximal subgroup H of G. When G is a semidirect product of two finite p-groups H and K, both H and K admit Gray maps and the corresponding homomorphism  y: H®Aut (K) is compatible with the Gray map of K in a sense which we will explain, we construct type-II Gray maps for G. Finally, we consider group codes over the dihedral group D8 of order 8 given by the set of their generators, and derive a representation and an encoding procedure for such codes.

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Numerous problems encountered in real life cannot be actually formulated as a single objective problem; hence the requirement of Multi-Objective Optimization (MOO) had arisen several years ago. Due to the complexities in such type of problems powerful heuristic techniques were needed, which has been strongly satisfied by Swarm Intelligence (SI) techniques. Particle Swarm Optimization (PSO) has been established in 1995 and became a very mature and most popular domain in SI. Multi-Objective PSO (MOPSO) established in 1999, has become an emerging field for solving MOOs with a large number of extensive literature, software, variants, codes and applications. This paper reviews all the applications of MOPSO in miscellaneous areas followed by the study on MOPSO variants in our next publication. An introduction to the key concepts in MOO is followed by the main body of review containing survey of existing work, organized by application area along with their multiple objectives, variants and further categorized variants.

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