Document Zbl 1390.05122

Bartholdi zeta functions of generalized join graphs. (English) Zbl 1390.05122

Graphs Comb. 34, No. 1, 207-222 (2018).

Summary: Let \(G=H[G_1,G_2,\dots,G_k]\) be the generalized join graph of \(G_1,G_2,\dots,G_k\) determined by \(H\). In this paper, we give a decomposition formula for the Bartholdi zeta function of \(G\). As applications, we obtain explicit formulae for Bartholdi zeta functions of some special kinds of graphs, such as the complete multipartite graph, the wheel, etc.

Cited in 1 Review

Cited in 1 Document

MSC:

05C50	Graphs and linear algebra (matrices, eigenvalues, etc.)
15A15	Determinants, permanents, traces, other special matrix functions

Keywords:

Bartholdi zeta function; generalized join graph

Cite Review PDF

Full Text: DOI

References:

[1]	Bartholdi, L.: Counting paths in graphs. Enseign. Math. 45, 83-131 (1999) · Zbl 0961.05032
[2]	Bass, H.: The IharaCSelberg zeta function of a tree lattice. Int. J. Math. 3, 717-797 (1992) · Zbl 0767.11025 · doi:10.1142/S0129167X92000357
[3]	Bayati, P., Somodi, M.: On the Ihara zeta function of cones over regular graphs. Graphs Combin. 29, 1633C1646 (2013) · Zbl 1282.05093
[4]	Cooper, Y.: Properties determined by the Ihara zeta function of a graph. Electron. J. Combin. 16(1), Research Paper 84, 14 (2009) · Zbl 1230.05199
[5]	Cui, S.-Y., Tian, G.-X.: The spectrum and the signless Laplacian spectrum of coronae. Linear Algebra Appl. 437, 1692-1703 (2012) · Zbl 1247.05135 · doi:10.1016/j.laa.2012.05.019
[6]	Cvetković, D.M., Rowlinson, P., Simić, H.: An Introduction to the Theory of Graph Spectra. Cambridge University Press, Cambridge (2010) · Zbl 1211.05002
[7]	Czarneski, D.: Zeta functions of finite graphs, ProQuest LLC, Ann Arbor, MI, 2005. PhD Thesis, Louisiana State University and Agricultural & Mechanical College (2005) · Zbl 1342.05086
[8]	Domingos, M.C., Maria, A.F., Enide, A.M., María, : Spectra of graphs obtained by a generalization of the join graph operation. Discret. Appl. Math. 313, 733-741 (2013) · Zbl 1259.05113
[9]	Foata, D., Zeilberger, D.: A combinatorial proof of Bass’s evaluations of the Ihara- Selberg zeta function for graphs. Trans. Am. Math. Soc. 351, 2257-2274 (1999) · Zbl 0921.05025 · doi:10.1090/S0002-9947-99-02234-5
[10]	Hashimoto, K.: Zeta Functions of Finite Graphs and Representations of p-Adic Groups, Advanced Studied in Pure Mathematics. Academic, New York (1989). 211C280 · Zbl 0709.22005
[11]	Hashimoto, K.: Artin-type L-functions and the density theorem for prime cycles on finite graphs. Int. J. Math. 3, 809-826 (1992) · Zbl 0767.11026 · doi:10.1142/S0129167X92000370
[12]	Ihara, Y.: On discrete subgroups of the two by two projective linear group over p-adic fields. J. Math. Soc. Japan 18, 219C235 (1966) · Zbl 0158.27702
[13]	Kim, H.K., Lee, J.: A generalized characteristic polynomial of a graph having a semifree action. Discret. Math. 308, 555C564 (2008) · Zbl 1130.05036
[14]	Kotani, M., Sunada, T.: Zeta functions of finite graphs. J. Math. Sci. Univ. Tokyo 7, 7-25 (2000) · Zbl 0978.05051
[15]	Mizuno, H., Sato, I.: Zeta functions of graph coverings. J. Comb. Theory Ser. B 80, 247-257 (2000) · Zbl 1023.05114 · doi:10.1006/jctb.2000.1983
[16]	Mizuno, H., Sato, I.: Bartholdi zeta functions of graph coverings. J. Comb. Theory Ser. B 89, 27-41 (2003) · Zbl 1023.05110 · doi:10.1016/S0095-8956(03)00043-1
[17]	MizunoI Sato, H.: Some weighted Bartholdi zeta function of a digraph. Linear Algebra Appl. 445, 1-17 (2014) · Zbl 1292.05177 · doi:10.1016/j.laa.2013.11.043
[18]	Sato, I.: A generalized Bartholdi zeta function for a hypergraph. Far East J. Math. Sci. 78, 93-130 (2013) · Zbl 1277.05110
[19]	Sato, I., Saito, S.: A generalized Bartholdi zeta function for a regular covering of a bipartite graph. Linear Algebra Appl. 438, 1025-1056 (2013) · Zbl 1257.05133 · doi:10.1016/j.laa.2012.09.007
[20]	SatoH MitsuhashiH Morita, I.: A generalized Bartholdi zeta function for a general graph. Linear Multilinear Algebra 64, 1-18 (2015)
[21]	Sato, I.: A new proof of a formula for the Bartholdi zeta function of a digraph. Graphs Comb. 32, 1571-1583 (2016) · Zbl 1342.05086 · doi:10.1007/s00373-015-1668-5
[22]	Schwenk, A.J.: Computing the characteristic polynomial of a graph. In: Bary R, Harary F (Eds.) Graphs Combinatorics, Lecture Notes in Mathematics, vol. 406, pp. 153-172. Springer, Berlin (1974) · Zbl 0308.05121
[23]	Scott, G., Storm, C.: The coefficients of the Ihara zeta function. Involve A J. Math. 1, 217C233 (2008) · Zbl 1228.05202
[24]	Setyadi, A., Storm, C.K.: Enumeration of graphs with the same Ihara zeta function. Linear Algebra Appl. 438, 564C572 (2013) · Zbl 1257.05064
[25]	Stark, H.M., Terras, A.A.: Zeta functions of finite graphs and coverings. Adv. Math. 121, 124-165 (1996) · Zbl 0874.11064 · doi:10.1006/aima.1996.0050
[26]	Stark, H.M., Terras, A.A.: Zeta functions of finite graphs and coverings, part II. Adv. Math. 154, 132-195 (2000) · Zbl 0972.11086 · doi:10.1006/aima.2000.1917
[27]	Sunada, T: L-functions in geometry and some applications. In: Lecture Notes in Math., vol. 1201, pp. 266-284 . Springer, New York (1986) · Zbl 0605.58046
[28]	Sunada, T.: Fundamental Groups and Laplacians (in Japanese). Kinokuniya, Tokyo (1988) · Zbl 0646.58027
[29]	Watanabe, Y., Fukumizu, K.: Graph zeta function in the Bethe free energy and loopy belief propagation. Adv. Neural Inf. Process. Syst. vol. 22, pp. 2017-2025. MIT Press (2010) · Zbl 0921.05025
[30]	Tahaei, M., Hashemi, S.: The coefficients of the reduced Bartholdi zeta function. Linear Algebra Appl. 509, 1-18 (2016) · Zbl 1346.05178 · doi:10.1016/j.laa.2016.07.004
[31]	Terras, A.: Zeta Functions of Graphs: A Stroll Through the Garden, Cambridge Studies in Advanced Mathematics, vol. 128. Cambridge University Press, Cambridge (2011) · Zbl 1206.05003
[32]	Zhang, F.Z.: The Schur Complement and Its Applications. Springer, New York (2005) · Zbl 1075.15002 · doi:10.1007/b105056

This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.