[6] **viXra:1411.0562 [pdf]**
*submitted on 2014-11-26 05:08:12*

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

**Comments:** 32 Pages.

Much research has involved the consideration of graphs which have sub-graphs of a particular kind, such as cliques. Known classes of graphs which are eigen-bi-balanced, i.e. they have a pair a,b of non-zero distinct eigenvalues, whose sum and product are integral, have been investigated. In this paper we will define ta new class of graphs, called q-cliqued graphs, on vertices, which contain q cliques each of order q connected to a central vertex, and then prove that these q-cliqued graphs are eigen-bi-balanced with respect to a conjugate pair whose sum is -1 and product 1-q. These graphs can be regarded as design graphs, and we use a specific example in an entomological experiment.
AMS Classification: 05C50
Key words: cliques, eigen-bi-balanced graphs, conjugate pair, designs.

**Category:** Combinatorics and Graph Theory

[5] **viXra:1411.0315 [pdf]**
*submitted on 2014-11-19 05:12:27*

**Authors:** Paul August Winter, Carol Lynne Jessop, Fadekemi Janet Adewusi

**Comments:** 20 Pages.

The complete graph is often used to verify certain graph theoretical definitions and applications. Regarding the adjacency matrix, associated with the complete graph, as a circulant matrix, we find its eigenvalues, and use this result to generate a trigonometrical unit-equations involving the sum of terms of the form , where a is odd. This gives rise to t-complete-eigen sequences and diagrams, similar to the famous Farey sequence and diagram. We show that the ratio, involving sum of the terms of the t-complete eigen sequence, converges to ½ , and use this ratio to find the t-complete eigen area. To find the eigenvalues, associated with the characteristic polynomial of complete graph, using induction, we create a general determinant equation involving the minor of the matrix associated with this characteristic polynomial.

**Category:** Combinatorics and Graph Theory

[4] **viXra:1411.0191 [pdf]**
*submitted on 2014-11-15 14:38:12*

**Authors:** Linfan Mao

**Comments:** 125 Pages. Edited By The Madis of Chinese Academy of Sciences and Beijing University of Civil Engineering and Architecture.

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 100-150 pages approx. per volume, which
publishes original research papers and survey articles in all aspects of Smarandachemulti-spaces, Smarandache geometries, mathematical combinatorics, non-euclidean geometry and topology and their applications to other sciences.

**Category:** Combinatorics and Graph Theory

[3] **viXra:1411.0190 [pdf]**
*submitted on 2014-11-15 14:40:18*

**Authors:** Linfan Mao

**Comments:** 129 Pages. Edited By The Madis of Chinese Academy of Sciences and Beijing University of Civil Engineering and Architecture.

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 100-150 pages approx. per volume, which publishes original research papers and survey articles in all aspects of Smarandachemulti-spaces, Smarandache geometries, mathematical combinatorics, non-euclidean geometry and topology and their applications to other sciences.

**Category:** Combinatorics and Graph Theory

[2] **viXra:1411.0189 [pdf]**
*submitted on 2014-11-15 14:41:29*

**Authors:** Linfan Mao

**Comments:** 111 Pages. Edited By The Madis of Chinese Academy of Sciences and Beijing University of Civil Engineering and Architecture.

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 100-150 pages approx. per volume, which publishes original research papers and survey articles in all aspects of Smarandachemulti-spaces, Smarandache geometries, mathematical combinatorics, non-euclidean geometry and topology and their applications to other sciences.

**Category:** Combinatorics and Graph Theory

[1] **viXra:1411.0050 [pdf]**
*replaced on 2015-05-31 03:30:25*

**Authors:** Ihsan Raja Muda Nasution

**Comments:** 2 Pages.

In this paper, we propose an idea of TSP-algorithm for any graph.

**Category:** Combinatorics and Graph Theory