Zeta functions of quaternion weighted graphs (Research on algebraic combinatorics and representation theory of finite groups and vertex operator algebras)グラフのゼータ関数は, 伊原[6]により定義された伊原ゼータ関数が起源である. 伊原ゼータ関数は, PGL(2, mathbb{Q}_{p})の捻れのない余コンパクトな離散部分群rから定まるセルバーグゼータ関数の類似であり, 母関数型表示と行列式表示を持つことが[6]において示された. その後, Serre[15]により, 伊原ゼータ関数は, SL(2, mathbb{Q}_{p})に付随したBruhat-Tits tree(無限正則木)のrによる商グラフ(有限正則グラフ)のゼータ関数であることが示唆され, 砂田[17, 18]によってグラフのゼータ関数が確立された. その後多くの研究者の貢献により, グラフのゼータ関数は大きく発展した.
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