[4] **viXra:1503.0228 [pdf]**
*submitted on 2015-03-28 15:32:12*

**Authors:** J Gregory Moxness

**Comments:** 321 Pages.

For each of the 480 unique octonion Fano plane mnemonic multiplication tables, there are 7 split octonions (one for each of 7 triads in the parent octonion). This PDF is a comprehensive list of all 3840=480+3360 (octonions + split octonions), their Fano planes, and multiplication tables. They are organized in pairs of 240 parent octonions=(8-bit sign mask)*(30 canonical sets of 7 triads). The pairs of parent octonions are created by flipping (reversing) the first triad (center circular line) creating a unique Fano plane mnemonic.

**Category:** Combinatorics and Graph Theory

[3] **viXra:1503.0148 [pdf]**
*submitted on 2015-03-19 03:21:55*

**Authors:** Paul August Winter, Carol Lynne Jessop

**Comments:** 21 Pages.

The separate study of the two concepts of energy and vertex coverings of graphs has opened many avenues of research. In this paper we combine these two concepts in a ratio, called the eigen-cover ratio, to investigate the domination effect of the subgraph induced by a vertex covering of a graph (called the cover graph of ), on the original energy of , where large number of vertices are involved. This is referred to as the eigen-cover domination and has relevance, in terms of conservation of energy, when a molecule’s atoms and bonds are mapped onto a graph with vertices and edges, respectively. If this energy-cover ratio is a function of , the order of graphs belonging to a class of graph, then we discuss its horizontal asymptotic behavior and attach the graphs average degree to the Riemann integral of this ratio, thus associating eigen-cover area with classes of graphs. We found that the eigen-cover domination had a strongest effect on the complete graph, while this chromatic-cover domination had zero effect on star graphs. We show that the eigen-cover asymptote of discussed classes of graphs belong to the interval [0,1], and conjecture that the class of complete graphs has the largest eigen-cover area of all classes of graphs.

**Category:** Combinatorics and Graph Theory

[2] **viXra:1503.0124 [pdf]**
*submitted on 2015-03-16 04:59:31*

**Authors:** Paul August Winter

**Comments:** 18 Pages.

The graph theoretical ratio, the tree-cover ratio, involving spanning trees of a graph G, and a 2-vertex covering (a minimum set S of vertices such that every edge (or path on 2 vertices) of G has at least vertex end in S) of G has been researched. In this paper we introduce a ratio, called the tree-3-covering ratio with respect to S, involving spanning trees and a 3-vertex covering (a minimum set S of vertices of G such that every path on 3 vertices has at least one vertex in S) of graphs. We discuss the asymptotic convergence of this tree-3-cover ratio for classes of graphs, which may have application in ideal communication situations involving spanning trees and 3-vertex coverings of extreme networks. We show that this asymptote lies on the interval with the dumbbell graph (a complete graph on n-1 vertices appended to an end vertex) has tree-3-cover asymptotic convergence of 1/e, identical to the convergence in the secretary problem, and the tree-cover asymptotic convergence of complete graphs. We also introduce the idea of a tree-3-cover area by integrating this tree-3-cover ratio.
AMS classification: 05C99
Key words: spanning trees of graphs, vertex cover, 3-vertex cover, ratios, social interaction, network communication, convergence, asymptotes.

**Category:** Combinatorics and Graph Theory

[1] **viXra:1503.0046 [pdf]**
*submitted on 2015-03-07 06:42:57*

**Authors:** Paul August Winter

**Comments:** 23 Pages.

Much research has been done involving the chromatic number of a graph involving the least number of colors, that the vertices of a graph can be colored, so that no two adjacent vertices have the same color. The idea of how the chromatic number of a vertex cover of a graph dominates the vertex cover of the original graph, where a large number of vertices are involved, has been investigated. The difference between the energy of the complete graph,, and the energy of any other graph G. has been studied, in terms of a ratio. The complete graph, on n vertices, has chromatic number n, and is significant in terms of its easily accessible graph theoretical properties, such as its high level of connectivity and robustness. In this paper, we introduce a ratio, the chromatic-complete difference ratio, involving the difference between the chromatic number of the complete graph, and the chromatic number of any other connected graph G, on the same number n of vertices. This allowed for the investigation of the effect of the chromatic number of G, with respect to the complete graph, when a large number of vertices are involved - referred to as the chromatic-complete difference domination effect. The value of this domination effect lies on the interval [0,1], with most classes of graphs taking on the right hand end-point, while graphs with a large clique takes on the left hand end-point. When this ratio is a function f(n), of the order of a graph, we attach the average degree of G to the Riemann integral to investigate the chromatic-complete difference area aspect of classes of graphs. We applied these chromatic-complete difference aspects to complements of classes of graphs.
AMS Classification: 05C50
1Corresponding author: Paul August Winter: Department of Mathematics, Howard College, University of KwaZulu-Natal, Glenwood, Durban, 4041, South Africa;
ORCID ID: 0000-0003-3539;
email: winterp@ukzn.ac.za
Key words: Chromatic number, domination, ratios, domination, asymptotes, areas

**Category:** Combinatorics and Graph Theory