The class of \(q\)-cliqued graphs: eigen-bi-balanced characteristic, designs, and an entomological experiment (Q499815)

Summary: Much research has involved the consideration of graphs which have subgraphs of a particular kind, such as cliques. Known classes of graphs which are eigen-bi-balanced, that is, they have a pair \(a\), \(b\) of nonzero distinct eigenvalues, whose sum and product are integral, have been investigated. In this paper we will define a new class of graphs, called \(q\)-cliqued graphs, on \(q^2+1\) 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.