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 Formula: see text-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 Formula: see text-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 Formula: see text-connected graph with at least Formula: see text vertices other than a Wheel has a beta invariant of at least Formula: see text.