@article{be87f9f3c36e4d36ad6862e597d7a42b,

title = "Tutte Polynomial, Complete Invariant, and Theta Series",

abstract = "In this study, we present two results that relate Tutte polynomials. First, we provide new and complete polynomial invariants for graphs. We note that the number of variables of our polynomials is one. Second, let L1 and L2 be two non-isomorphic lattices. We state that L1 and L2 are theta series equivalent if those theta series are the same. The problem of identifying theta series equivalent lattices is discussed in Prof. Conway{\textquoteright}s book The Sensual (Quadratic) Form with the title “Can You Hear the Shape of a Lattice?” In this study, we present a method to find theta series equivalent lattices using matroids and their Tutte polynomials.",

keywords = "Code, Lattice, Matroid, Theta series, Tutte polynomial, Weight enumerator",

author = "Misaki Kume and Tsuyoshi Miezaki and Tadashi Sakuma and Hidehiro Shinohara",

note = "Funding Information: It is a great honor to have the opportunity to publish in the issue celebrating the 75th birthday of Prof. Eiichi Bannai and Prof. Hikoe Enomoto. The second named author was Prof. Bannai{\textquoteright}s last PhD student at Kyushu University. When he was a graduate school student, he had the impression that Prof. Bannai believed that interesting and significant mathematics arise at the intersection of several different mathematical areas. He would like to keep this in mind as he continues his studies in mathematics. The authors thank Prof. Akihiro Munemasa and Prof. Manabu Oura for their helpful discussions on this research. The authors would also like to thank the anonymous reviewers for their beneficial comments on an earlier version of the manuscript. This work was supported by JSPS KAKENHI (18K03217, 18K03388). Publisher Copyright: {\textcopyright} 2020, Springer Japan KK, part of Springer Nature.",

year = "2020",

doi = "10.1007/s00373-020-02238-w",

language = "English",

journal = "Graphs and Combinatorics",

issn = "0911-0119",

publisher = "Springer Japan",

}