Group weighing matrices

Ph.DDOCTOR OF PHILOSOPH

Some Results On Spectrum And Energy Of Graphs With Loops

Let $G_S$ be a graph with loops obtained from a graph $G$ of order $n$ and loops at $S \subseteq V(G)$. In this paper, we establish a neccesary and sufficient condition on the bipartititeness of a connected graph $G$ and the spectrum Spec($G_S$) and Spec($G_{V(G)\backslash S}$). We also prove that for every $S \subseteq V(G)$, $E(G_S) \geq E(G)$ when $G$ is bipartite. Moreover, we provide an identification of the spectrum of complete graphs $K_n$ and complete bipartite graphs $K_{m,n}$ with loops. We characterize any graphs with loops of order n whose eigenvalues are all positive or non-negative, and also any graphs with a few distinct eigenvalues. Finally, we provide some bounds related to $G_S$.Comment: 16 pages, published versio

Parikh Matries of Words.

The notion of Parikh matrix of a word over and ordered alphabets was introduced by Mateescu Et al

Full Identification of Idempotens in Binary Abelian Group Rings

Every code in the latest study of group ring codes is a submodule thathas a generator. Study reveals that each of these binary group ring codes can havemultiple generators that have diverse algebraic properties. However, idempotentgenerators get the most attention as codes with an idempotent generator are easierto determine its minimal distance. We have fully identify all idempotents in everybinary cyclic group ring algebraically using basis idempotents. However, the conceptof basis idempotent constrained the exibilities of extending our work into the studyof identication of idempotents in non-cyclic groups. In this paper, we extend theconcept of basis idempotent into idempotent that has a generator, called a generatedidempotent. We show that every idempotent in an abelian group ring is either agenerated idempotent or a nite sum of generated idempotents. Lastly, we show away to identify all idempotents in every binary abelian group ring algebraically by fully obtain the support of each generated idempotent

Study of proper circulant weighing matrices with weight 9

10.1016/j.disc.2004.12.029Discrete Mathematics308132802-2809DSMH