Some Topics of Special Interest in Graph Theory

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Some Investigations in the Theory of Graphs

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Discussion On Some Important Topics in Graph Theory

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On the beta-number of forests with isomorphic components

The beta-number, β (G), of a graph G is defined to be either the smallest positive integer n for which there exists an injective function f : V (G) → {0, 1, . . . , n} such that each uv ∈ E (G) is labeled |f (u) − f (v)| and the resulting set of edge labels is {c, c+ 1, . . . , c+|E (G)| −1} for some positive integer c or +∞ if there exists no such integer n. If c = 1, then the resulting beta-number is called the strong beta-number of G and is denoted by βs (G). In this paper, we show that if G is a bipartite graph and m is odd, then β (mG) ≤ mβ (G) + m − 1. This leads us to conclude that β (mG) = m |V (G)| − 1 if G has the additional property that G is a graceful nontrivial tree. In addition to these, we examine the (strong) beta-number of forests whose components are isomorphic to either paths or stars

International Journal of Mathematical Combinatorics, Vol.2

The International J.Mathematical Combinatorics (ISSN 1937-1055) is a fully refereed international journal, sponsored by the MADIS of Chinese Academy of Sciences and published in USA quarterly comprising 460 pages approx. per volume, which publishes original research papers and survey articles in all aspects of Smarandache multi-spaces, Smarandache geometries, mathematical combinatorics, non-euclidean geometry and topology and their applications to other sciences

Method Meditation: An Experimental Demonstration of Systemization in Architecture

Method Meditation is an architectural design method developed during my exploration of systemization in the design process. Systems have been used throughout architectural history in an attempt to create space that can affect an occupant exactly the way the architect intended. However, these attempts have had inconsistent outcomes. This inconsistency has been attributed to several factors, from the variety to individual experiences that skew an observer’s viewpoint, to the lack of provable, causal relationships between environment and behavior. Due to these obstacles, other designers have used systems not to create perfect results, but to push their designs to new extents, and to produce unprecedented outcomes. To explore the relationship between systemization and design, I create my own method that would further incorporate the occupant in the design process, testing whether more thorough collaboration between the architect and occupant could produce a more desirable result. To do so, I examined the design preferences of thirty participants, twenty of whom are outside of the architectural field. The participants answered questions on two separate surveys that asked for preferences based on sixty two-dimensional illustrations. These illustrations are based in part on the work of Christopher Alexander, a proponent of systemization as a means of better design. The results of these surveys showed discrepancies between the choices of the participants within, and outside, of the architectural profession. Though the scope of this method is limited, its findings suggest the possible benefits of a closer relationship between the architect and occupants during the design process