For any s 2: 1 and t 2: (n, we prove that among all graphs with n vertices the graph that contains the maximal number of induced copies of K r , Hs for any fixed s 2: 1 and t 2: (n is K(n/2)+a. (n/2)-a for some function a = o(n). We show that this is not valid for t < (n.Analogous results for complete multipartite graphs are also obtained.

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Given a graph G whose edges are perfectly reliable and whose nodes each operate independently with probability p ∈ [0, 1], the node reliability of G is the probability that at least one node is operational and that the operational nodes can all communicate in the subgraph that they induce; it is the analogous node measure of robustness to the well studied all-terminal reliability, where the nodes are perfectly reliable but the edges fail randomly. In sharp contrast to what is known about the roots of the all-terminal reliability polynomial, we show that the node reliability polynomial of any connected graph on at least three nodes has a nonreal polynomial root, the collection of real roots of all node reliability polynomials is unbounded, and the collection of complex roots of all node reliability polynomials is dense in the entire complex plane.

The independence polynomial of a graph G is the function i(G, x) = k≥0 i k x k , where i k is the number of independent sets of vertices in G of cardinality k. We prove that real roots of independence polynomials are dense in (−∞, 0], while complex roots are dense in C, even when restricting to well covered or comparability graphs. Throughout, we exploit the fact that independence polynomials are essentially closed under graph composition.

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