## The Class of Q-Cliqued Graphs: Eigen-bi-Balanced Characteristic, Designs and an Entomological Experiment

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

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.

**Comments:** 32 Pages.

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### Submission history

[v1] 2014-11-26 05:08:12

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