Korean J. Math. Vol. 26 No. 4 (2018) pp.741-746
DOI: https://doi.org/10.11568/kjm.2018.26.4.741

Ihara zeta function of dumbbell graphs

Main Article Content

Sanghoon Kwon
Jung-Hyeon Park

Abstract

We study the Ihara zeta function of the dumbbell graph $D_{1,1,n}$ of type $(1,1,n)$ and $D_{1,2,n}$ of type $(1,2,n)$. Explicit formulas of the zeta functions of the graphs, their radius of convergence, and the connection with the number of closed cycles are given.


Article Details

Supporting Agencies

Catholic Kwandong University

References

[1] H. Bass, The Ihara-Selberg zeta function of a tree lattice, International. J. Math. 3 (1992), 717–797. Google Scholar

[2] M. D. Horton, H. M. Stark, and A. Terras, What are zeta functions of graphs and what are they good for ?, Contemporary Mathematics 415 (2006), Quantum Graphs and Their Applications; Edited by Gregory Berkolaiko, Robert Carlson, Stephen A. Fulling, and Peter Kuchment, 173–190. Google Scholar

[3] Y. Ihara, On discrete subgroups of the two by two projective linear group over p-adic fields, J. Math. Soc. Japan 18 (1966), 219–235. Google Scholar

[4] M. Kotani and T. Sunada, Zeta function of finite graphs, J. Math. Sci. Univ. Tokyo 7 (2000), 7–25. Google Scholar

[5] A. Terras, Zeta functions of graphs: a stroll through the garden, CambridgeX Studies in Advanced Mathematics, Vol. 128, Cambridge University Press, Cam- bridge, 2011, xii+239 pp Google Scholar