“School of Mathematics”
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| Paper IPM / M / 112 |
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| Abstract: | |||||
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Let G be a simple graph. We assign a polynomial C(G;x) to G, called the clique polynomial, where the coefficient of xi,i > 0, is the number of cliques of G with i vertices and the constant term is 1. Fisher and Solow (1990), proved that this polynomial always has a real root.
We prove this result by a simple and elementary method, which also implies
the following results. If ζG is the greatest real root of C(G;x) then for an induced subgraph H of G,ζH ≤ ζG, and for a spanning subgraph H of G, ζH ≥ ζG. As a consequence of the first inequality we have α(G) ≤ −1/ζG, where α(G) denotes the independence number of G.
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