Application of edge-magic total labelings

Import 04/07/2011Předkládaná práce se zabývá problematikou ukládání řídkých matic vysokých řádů pro možnost operací s nimi. Jelikož operace s řídkými maticemi vyšších řádů nejsou možné bez jejich úpravy, zabývá se práce možnostmi jejich efektivního ukládání s ohledem na minimalizaci časové náročnosti operací. Práce rovněž obsahuje popis několika známých způsobů ukládání řídkých matic a zaměřuje se především na možnosti efektivního využití magicky ohodnoceného grafu, který reprezentuje řídké matice vyšších řádů. Text navazuje na stávající práci zabývající se nalezením magického ohodnocení grafu počítačovým programem [3] a předkládá možnost dosažení lepších výsledků pomocí zobecněného magického ohodnocení za cenu omezení některých požadavků na ohodnocení, ale bez negativního dopadu na rychlost operace. Hlavním přínosem práce je konstruktivní důkaz existence předpisu zobecněného ohodnocení pro sítě Pm□Pn (pro libovolně velké m, n za podmínky, že m je liché) a možnost aplikovat nový systém ukládání řídkých matic pro realizaci operací, především násobení řídké matice vektorem zleva nebo zprava. Práce obsahuje rovněž výsledky autorem provedených experimentů, demonstrujících výhody zobecněného ohodnocení na praktických příkladech. Algoritmy byly realizovány v jazyce C++ v prostředí Visual Studio 2008.Presented work deals with sparse matrix storing, which is essential for a wide range of matrix operations. Operating with high-dimensional sparse matrices is not possible without their proper structure manipulation. This work tries to find effective sparse matrix storing with minimal time intensity. We compare known ways of sparse matrix storing with a new form based on edge-magic labeling form. Related with thesis [3], this text investigates the possibility of getting better results by using generalized edge-magic labeling form with lifted requirements, yet without time intensity increased. This work presents generalized edge-magic labeling of Pm□Pn for every integer m, n where m is odd. Also, we derived a generalized egde-magic storing and this work contains time intensity results of sparse matrix-vector operations and comparison time intensity of generalized edge-magic form with other known forms of storing.457 - Katedra aplikované matematikyvýborn

Vertex-Magic Graphs

In this paper, we will study magic labelings. Magic labelings were first introduced by Sedláček in 1963 [3]. At this time, the labels on the graph were only assigned to the edges. In 1970, Kotzig and Rosa defined what are now known as edge-magic total labelings, where both the vertices and the edges of the graph are labeled. Following this in 1999, MacDougall, Miller, Slamin, and Wallis introduced the idea of vertex-magic total labelings. There are many different types of magic labelings. In this paper will focus on vertex-magictotal labelings

Vertex-magic Labeling of Trees and Forests

A vertex-magic total labeling of a graph G(V,E) is a one-to-one map λ from E ∪ V onto the integers {1, 2, . . . , |E| + |V|} such that λ(x) + Σ λ(xy) where the sum is over all vertices y adjacent to x, is a constant, independent of the choice of vertex x. In this paper we examine the existence of vertex-magic total labelings of trees and forests. The situation is quite different from the conjectured behavior of edge-magic total labelings of these graphs. We pay special attention to the case of so-called galaxies, forests in which every component tree is a star

Expanding Super Edge-Magic Graphsâ

For a graph G, with the vertex set V(G) and the edge set E(G) an edge-magic total labeling is a bijection f from V(G)UE(G) to the set of integers {1,2,...., |V(G)|+|E(G)} with the property that f(u) + f(v) +f(uv) = k for each uv elemen E(G) and for a fixed integer k. An edge-magic total labeling f is called super edge-magic total labeling if f(E(G)) = {|V(G)+1, |V(G)+2,....., |V(G)+E(G)|}. In this paper we construct the expanded super edge-magic total graphs from cycles C, generalized Petersen graphs and generalized prisms

Super edge-magic total strength of some unicyclic graphs

Let $G$ be a finite simple undirected $(p,q)$-graph, with vertex set $V(G)$ and edge set $E(G)$ such that $p=|V(G)|$ and $q=|E(G)|$. A super edge-magic total labeling $f$ of $G$ is a bijection $f\colon V(G)\cup E(G)\longrightarrow \{1,2,\dots , p+q\}$ such that for all edges $u v\in E(G)$, $f(u)+f(v)+f(u v)=c(f)$, where $c(f)$ is called a magic constant, and $f(V(G))=\{1,\dots , p\}$. The minimum of all $c(f)$, where the minimum is taken over all the super edge-magic total labelings $f$ of $G$, is defined to be the super edge-magic total strength of the graph $G$. In this article, we work on certain classes of unicyclic graphs and provide shreds of evidence to conjecture that the super edge-magic total strength of a certain family of unicyclic $(p,q)$-graphs is equal to $2q+\frac{n+3}{2}$

Critical Set of Edge Magic Total Labeling of Expanding Cycle Graph *

This paper discusses about graph labeling. We will nd edge magic total labeling of cycle and expanding cycle graph. In the nal, we investigate the critical set of edge magic total labeling on cycleand expanding cycle graph

CRITICAL SET OF EDGE MAGIC TOTAL LABELING OF EXPANDING CYCLE GRAPH *

This paper discusses about graph labeling. We will nd edge magic total labeling of cycle and expanding cycle graph. In the nal, we investigate the critical set of edge magic total labeling on cycleand expanding cycle graph

New Methods for Magic Total Labelings of Graphs

University of Minnesota M.S. thesis. May 2015. Major: Mathematics. Advisors: Dalibor Froncek, Sylwia Cichacz-Przenioslo. 1 computer file (PDF); ix, 117 pages.A \textit{vertex magic total (VMT) labeling} of a graph $G=(V,E)$ is a bijection from the set of vertices and edges to the set of numbers defined by $\lambda:V\cup E\rightarrow\{1,2,\dots,|V|+|E|\}$ so that for every $x \in V$ and some integer $k$, $w(x)=\lambda(x)+\sum_{y:xy\in E}\lambda(xy)=k$. An \textit{edge magic total (EMT) labeling} is a bijection from the set of vertices and edges to the set of numbers defined by $\lambda:V\cup E\rightarrow\{1,2,\dots,|V|+|E|\}$ so that for every $xy \in E$ and some integer $k$, $w(xy)=\lambda(x)+\lambda(y)+\lambda(xy)=k$. Numerous results on labelings of many families of graphs have been published. In this thesis, we include methods that expand known VMT/EMT labelings into VMT/EMT labelings of some new families of graphs, such as unions of cycles, unions of paths, cycles with chords, tadpole graphs, braid graphs, triangular belts, wheels, fans, friendships, and more