On zero-error codes produced by greedy algorithms

We present two greedy algorithms that determine zero-error codes and lower bounds on the zero-error capacity. These algorithms have many advantages, e.g., they do not store a whole product graph in a computer memory and they use the so-called distributions in all dimensions to get better approximations of the zero-error capacity. We also show an additional application of our algorithms.

CHARACTERISTICS OF SHACKLE GRAPH: Shack(Kn,v(j,i),t), Shack(Sn,v(j,i),t), & Shack(K(n,n),v(rj,i),t)

<p class="AfiliasiCxSpFirst" align="left"><strong>Abstrak:</strong></p><p class="AfiliasiCxSpMiddle">Operasi schackle adalah operasi antara dua atau lebih graf yang menghasilkan graf baru. Graf shackle dinotasikan adalah graf yang dihasilkan dari t salinan dari graf yang diberi simbol dengan dimana dan t bilangan asli. Operasi shackle ppada penelitian ini adalah shackle titik. Operasi shackle titik dinotasikan dengan artinya graf yang dibangun dari sembarang graf sebanyak salinan dan titik sebagai . Kelas graf yang akan di eksporasi karakterisinya dan bilangan kromatinya adalah <em>, S</em> <em>, &amp; S</em> . Hasil penelitiannya menunjukkan bahwa bilangan kromatik graf shackle sama dengan subgraf pembangunnya.</p><p class="AfiliasiCxSpMiddle" align="left"> </p><p class="AfiliasiCxSpLast" align="left"><strong>Kata Kunci</strong>:</p><p>Operasi Shackle, Shackle titik, graf shackle, bilangan kromatik.</p><p> </p><p class="AfiliasiCxSpFirst" align="left"><strong><em>Abstract:</em></strong></p><p class="AfiliasiCxSpMiddle"><em>A shackle operation is an operation between two or more graphs that results in a new graph. Shackle graph notated </em> <em> </em><em>is a product graph from </em> <em> copy of graph </em> <em> is denoted by </em> <em> where </em> <em> and </em> <em> are natural numbers. The shackle operation in this research is vertex shackle. Vertex shackle operation is denoted by </em> <em> which means that the graph is constructed from any graph </em> <em> as many as </em> <em> copies and vertex </em> <em> as linkage vertex. The class of graphs examined in this study are </em> <em>, S</em> <em>, &amp; S</em> <em>.</em> <em>The results show that the ch</em><em>r</em><em>omatic number of the shackle graph is the same as the subgraph that generates it</em><em>.</em></p><p class="AfiliasiCxSpMiddle" align="left"> </p><p class="AfiliasiCxSpLast" align="left"><strong><em>Keywords</em></strong><em>:</em></p><p><em>Shackle Operation, Vertex Shackle, Shackle Graph</em><em>, Chromatic Numbers.</em></p>

LIGHTS OUT! on graph products over the ring of integers modulo k

LIGHTS OUT! is a game played on a finite, simple graph. The vertices of the graph are the lights, which may be on or off, and the edges of the graph determine how neighboring vertices turn on or off when a vertex is pressed. Given an initial configuration of vertices that are on, the object of the game is to turn all the lights out. The traditional game is played over $\mathbb{Z}_2$, where the vertices are either lit or unlit, but the game can be generalized to $\mathbb{Z}_k$, where the lights have different colors. Previously, the game was investigated on Cartesian product graphs over $\mathbb{Z}_2$. We extend this work to $\mathbb{Z}_k$ and investigate two other fundamental graph products, the direct (or tensor) product and the strong product. We provide conditions for which the direct product graph and the strong product graph are solvable based on the factor graphs, and we do so using both open and closed neighborhood switching over $\mathbb{Z}_k$.

On the Wiener Index of the Dot Product Graph over Monogenic Semigroups

Algebraic study of graphs is a relatively recent subject which arose in two main streams: One is named as the spectral graph theory and the second one deals with graphs over several algebraic structures. Topological graph indices are widely-used tools in especially molecular graph theory and mathematical chemistry due to their time and money saving applications. The Wiener index is one of these indices which is equal to the sum of distances between all pairs of vertices in a connected graph. The graph over the nite dot product of monogenic semigroups has recently been dened and in this paper, some results on the Wiener index of the dot product graph over monogenic semigroups are given.

Total Domination in Rooted Product Graphs

During the last few decades, domination theory has been one of the most active areas of research within graph theory. Currently, there are more than 4400 published papers on domination and related parameters. In the case of total domination, there are over 580 published papers, and 50 of them concern the case of product graphs. However, none of these papers discusses the case of rooted product graphs. Precisely, the present paper covers this gap in the theory. Our goal is to provide closed formulas for the total domination number of rooted product graphs. In particular, we show that there are four possible expressions for the total domination number of a rooted product graph, and we characterize the graphs reaching these expressions.