Abstract : Crapo's beta invariant was defined by Henry Crapo in the 1960s. For a matroid $M$, the invariant $\beta(M)$ is the non-negative integer that is the coefficient of the $x$ term of the Tutte polynomial. Crapo proved that $\beta(M)>0$ if and only if $M$ is connected and $M$ is not a loop, and Brylawski proved that $M$ is the matroid of a series-parallel network if and only if $M$ is a co-loop or $\beta(M)=1.$ In this talk, we present several generalizations of the beta invariant to combinatorial structures that are not matroids. We concentrate on posets, chordal graphs, and finite subsets of Euclidean space. In each case, our definition of $\beta$ measures the number of "interior'' elements.

Keywords : Crapo's beta invariant; matroid; Tutte polynomial; co-loop; chordal graphs; posets; Euclidean space