We generalize the Tutte polynomial of a matroid to morphism matroids via K-theory flag varieties. introduce two different generalizations, and demonstrate that each has its own merits, where trade-off is between ease combinatorics geometry. One generalization recovers Las Vergnas matroids, which admits corank-nullity formula deletion-contraction recursion. The other does not, but better reflect...

In this thesis, we make some contributions at the interface between ‘algebra’ and ‘graph theory’. In Chapter 1, we give an overview of the topics and also the definitions and preliminaries. In Chapter 2, we estimate the number of possible types degree patterns of k-lacunary polynomials of degree t < p which split completely modulo p. The result is based on a rather unusual combination of two te...

A graph describes the zero-nonzero pattern of a family of matrices, with the type of graph (undirected or directed, simple or allowing loops) determining what type of matrices (symmetric or not necessarily symmetric, diagonal entries free or constrained) are described by the graph. The minimum rank problem of the graph is to determine the minimum among the ranks of the matrices in this family; ...

Barrett et al. asked in [W. Barrett et al. Minimum rank of edge subdivisions of graphs. Electronic Journal of Linear Algebra, 18:530–563, 2009.], whether the maximum nullity is equal to the zero forcing number for all complete subdivision graphs. We prove that this equality holds. Furthermore, we compute the value of M(F, G̊) = Z(G̊) by introducing the bridge tree of a connected graph. Since this...

For any matroid M realizable over Q, we give a combinatorial interpretation of the Tutte polynomial T M (x; y) which generalizes many of its known interpretations and specializations, including Tutte's coloring and ow interpretations of T M (1 ? t; 0); T M (0; 1 ? t), Crapo and Rota's nite eld interpretation of T M (1 ? q k ; 0), the interpretation in terms of the Whitney corank-nullity polynom...