Bounding the beta invariant of 3-connected graphs

The beta invariant is closely related to the Chromatic and Tutte Polynomials and has been extensively studied, see Brylawski [A combinatorial model for series-parallel networks, Trans. Amer. Math. Soc. 154 (1971) 1–22], Crapo [A higher invariant for matroids, J. Combin. Theory 2 (1967) 406–417], Lee and Wu [Bounding the beta invariant of 3-connected matroids, Discrete Math. 354 (2022) 1–11], Oxley [On Crapo’s beta invariant for matroids, Stud. Appl. Math. 66(3) (1982) 267–277], and others. Lee and Wu [Bounding the beta invariant of 3-connected matroids, Discrete Math. 354 (2022) 1–11] established that a 3-connected graph with [Formula: see text] vertices possesses a beta invariant of at least [Formula: see text], reaching equality only when the graph is a wheel or the Prism. Additionally, Oxley [On Crapo’s beta invariant for matroids, Stud. Appl. Math. 66(3) (1982) 267–277] provided characterizations for matroids with beta invariant values of two, three, and four. This paper extends the findings of Lee and Wu by offering a comprehensive characterization of all 3-connected graphs with [Formula: see text] vertices that have a beta invariant of either [Formula: see text] or [Formula: see text]. As a consequence, we prove that any 3-connected graph with at least [Formula: see text] vertices other than a Wheel has a beta invariant of at least [Formula: see text].