18 research outputs found
Betti diagrams from graphs
The emergence of Boij-S\"oderberg theory has given rise to new connections
between combinatorics and commutative algebra. Herzog, Sharifan, and Varbaro
recently showed that every Betti diagram of an ideal with a k-linear minimal
resolution arises from that of the Stanley-Reisner ideal of a simplicial
complex. In this paper, we extend their result for the special case of 2-linear
resolutions using purely combinatorial methods. Specifically, we show bijective
correspondences between Betti diagrams of ideals with 2-linear resolutions,
threshold graphs, and anti-lecture hall compositions. Moreover, we prove that
any Betti diagram of a module with a 2-linear resolution is realized by a
direct sum of Stanley-Reisner rings associated to threshold graphs. Our key
observation is that these objects are the lattice points in a normal reflexive
lattice polytope.Comment: To appear in Algebra and Number Theory, 15 pages, 7 figure
Linear algebraic techniques for spanning tree enumeration
Kirchhoff's Matrix-Tree Theorem asserts that the number of spanning trees in
a finite graph can be computed from the determinant of any of its reduced
Laplacian matrices. In many cases, even for well-studied families of graphs,
this can be computationally or algebraically taxing. We show how two well-known
results from linear algebra, the Matrix Determinant Lemma and the Schur
complement, can be used to elegantly count the spanning trees in several
significant families of graphs.Comment: This paper presents unweighted versions of the results in
arXiv:1903.03575 with more concrete and concise proofs. It is intended for a
broad audience and has extra emphasis on exposition. It will appear in the
American Mathematical Monthl
Graded Expectations: Betti numbers and anti-lecture hall compositions of random threshold graphs
This paper examines the one-to-one-to-one correspondence between threshold
graphs, Betti numbers of quotients of polynomial rings by -linear ideals,
and anti-lecture hall compositions. In particular, we establish new explicit
combinatorial mappings between each of these classes of objects and calculate
the expected values of the Betti numbers and anti-lecture hall composition
corresponding to a random threshold graph.Comment: 24 pages, 3 figures, and 4 table
An Evolutionary Method for the Minimum Toll Booth Problem: the Methodology
This paper considers the minimum toll booth problem (MINTB) for determining a tolling strategy in a transportation network that requires the least number of toll locations, and simultaneously causes the most efficient use of the network. The paper develops a methodology for using the genetic algorithm to solve MINTB and presents the algorithm GAMINTB. The proposed method is tested and validated through a computational study with six example networks. Additional numerical test discovers some interesting properties for the proposed method, and provides guidelines for further application of the GAMINTB