## Clique-inserted-graphs and spectral dynamics of clique-inserting.(English)Zbl 1148.05048

Summary: Motivated by studying the spectra of truncated polyhedra, we consider the clique-inserted-graphs. For a regular graph $$G$$ of degree $$r>0$$, the graph obtained by replacing every vertex of $$G$$ with a complete graph of order $$r$$ is called the clique-inserted-graph of $$G$$, denoted as $$C(G)$$. We obtain a formula for the characteristic polynomial of $$C(G)$$ in terms of the characteristic polynomial of $$G$$. Furthermore, we analyze the spectral dynamics of iterations of clique-inserting on a regular graph $$G$$. For any $$r$$-regular graph $$G$$ with $$r>2$$, let $$S(G)$$ denote the union of the eigenvalue sets of all iterated clique-inserted-graphs of $$G$$. We discover that the set of limit points of $$S(G)$$ is a fractal with the maximum $$r$$ and the minimum - 2, and that the fractal is independent of the structure of the concerned regular graph $$G$$ as long as the degree $$r$$ of $$G$$ is fixed. It follows that for any integer $$r>2$$ there exist infinitely many connected $$r$$-regular graphs (or, non-regular graphs with $$r$$ as the maximum degree) with arbitrarily many distinct eigenvalues in an arbitrarily small interval around any given point in the fractal. We also present a formula on the number of spanning trees of any $$k$$th iterated clique-inserted-graph and other related results.

### MSC:

 05C69 Vertex subsets with special properties (dominating sets, independent sets, cliques, etc.) 05C85 Graph algorithms (graph-theoretic aspects)
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