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Mirka Miller
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Control & Systems Engineering Mathematics

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On The Existence of Non-Diregular Digraphs of Order Two Less than the Moore Bound S Slamin; Mirka Miller
Jurnal ILMU DASAR Vol 12 No 1 (2011)
Publisher : Fakultas Matematika dan Ilmu Pengetahuan Alam Universitas Jember

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Abstract

A communication network can be modelled as a graph or a directed graph, where each processing element is represented by a vertex and the connection between two processing elements is represented by an edge (or, in case of directed connections, by an arc). When designing a communication network, there are several criteria to be considered. For example, we can require an overall balance of the system. Given that all the processing elements have the same status, the flow of information and exchange of data between processing elements will be on average faster if there is a similar number of interconnections coming in and going out of each processing element, that is, if there is a balance (or regularity) in the network. This means that the in-degree and out-degree of each vertex in a directed graph (digraph) must be regular. In this paper, we present the existence of digraphs which are not diregular (regular out-degree, but not regular in-degree) with the number of vertices two less than the unobtainable upper bound for most values of out-degree and diameter, the so-called Moore bound.
On Total Vertex Irregularity Strength of Cocktail Party Graph Kristiana Wijaya; S Slamin; Mirka Miller
Jurnal ILMU DASAR Vol 12 No 2 (2011)
Publisher : Fakultas Matematika dan Ilmu Pengetahuan Alam Universitas Jember

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Abstract

A vertex irregular total k-labeling of a graph G is a function λ from both the vertex and the edge sets to {1,2,3,,k} such that for every pair of distinct vertices u and x, λ(u)+∑λ(uv)≠λ(x)+∑λ(xy). uv∈E xy∈E. The integer k is called the total vertex irregularity strength, denoted by tvs (G ) , is the minimum value of the largest label over all such irregular assignments. In this paper, we prove that the total vertex irregularity strength of the Cocktail Party graph H2,n ,that is tvs(H2,n )= 3 for n ≥ 3.
Co-Authors Kristiana Wijaya S Slamin