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- JOHN MACHACEK
- 2011

The critical group K(G) of a graph G is a finite abelian group whose order is the number of spanning forests of the graph. Here we investigate the relationship between the critical group of a regular bipartite graph G and its line graph lineG. The relationship between the two is known completely for regular nonbipartite graphs. We compute the critical group… (More)

- Carolina Benedetti, Joshua Hallam, John Machacek
- SIAM J. Discrete Math.
- 2016

We consider a Hopf algebra of simplicial complexes and provide a cancellation-free formula for its antipode. We then obtain a family of combinatorial Hopf algebras by defining a family of characters on this Hopf algebra. The characters of these combinatorial Hopf algebras give rise to symmetric functions that encode information about colorings of simplicial… (More)

- John Machacek
- 2016

We define a new type of vertex coloring which generalizes vertex coloring in graphs, hypergraphs, and simplicial complexes. To this coloring there is an associated symmetric function in noncommuting variables for which we give a deletion-contraction formula. In the case of graphs our symmetric function in noncommuting variables agrees with the chromatic… (More)

- John Machacek
- Journal of Automata, Languages and Combinatorics
- 2017

We consider the language consisting of all words such that it is possible to obtain the empty word by iteratively deleting powers. It turns out that in the case of deleting squares in binary words this language is regular, and in the case of deleting squares in words over a larger alphabet the language is not regular. However, for deleting squares over any… (More)

- John Machacek, Shafiu Jibrin
- J. Applied Mathematics
- 2012

We investigate solving semidefinite programs SDPs with an interior point method called SDPCUT, which utilizes weighted analytic centers and cutting plane constraints. SDP-CUT iteratively refines the feasible region to achieve the optimal solution. The algorithm uses Newton’s method to compute the weighted analytic center. We investigate different stepsize… (More)

- John Machacek
- Electr. J. Comb.
- 2017

We define a new type of vertex coloring which generalizes vertex coloring in graphs, hypergraphs, and simplicial complexes. This coloring also generalizes oriented coloring, acyclic coloring, and star coloring. There is an associated symmetric function in noncommuting variables for which we give a deletion-contraction formula. In the case of graphs this… (More)

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