TY - JOUR

T1 - Clique-inserted-graphs and spectral dynamics of clique-inserting

AU - Zhang, Fuji

AU - Chen, Yi Chiuan

AU - Chen, Zhibo

N1 - Funding Information:
E-mail addresses: fjzhang@jingxian.xmu.edu.cn (F. Zhang), YCChen@math.sinica.edu.tw (Y.-C. Chen), zxc4@psu.edu (Z. Chen). 1 Supported by NSFC 10671162. 2 Partially supported by NSC 95-2115-M-001-023. 3 The work was done when Z. Chen was on sabbatical in China.

PY - 2009/1/1

Y1 - 2009/1/1

N2 - 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 kth iterated clique-inserted-graph and other related results.

AB - 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 kth iterated clique-inserted-graph and other related results.

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U2 - 10.1016/j.jmaa.2008.08.036

DO - 10.1016/j.jmaa.2008.08.036

M3 - Article

AN - SCOPUS:52749095294

SN - 0022-247X

VL - 349

SP - 211

EP - 225

JO - Journal of Mathematical Analysis and Applications

JF - Journal of Mathematical Analysis and Applications

IS - 1

ER -