"In theory there is no difference
beetween theory and practice.
In practice there is".
The critical ideals of a graph are determinantal ideals of the Laplacian matrix generalized of a graph. Critical Ideals generalize the characteristic polynomial of a graph and the concept of critical group. The critical group of a graph has been studied in different areas of mathematics. In algebraic geometry is known as the group of components, Picard group or as the Jacobian. In statistical mechanics, is known as the abelian model of sandpiles. And in Combinatorics is known as the group of sandpiles or the game of chip-firing.
On the other hand, allocation problem in bipartite graphs is one of the most studied problems in Combinatorial Optimization, and one of which has more potential applications. This part of my work was done in conjunction with the group of Cipriano Santos in the Hewlett-Packard Laboratories in Palo Alto, CA, and now we have 5 international patents.
Below you will find a list of my publications:
23. Critical ideals of signed graphs with twin vertices
Carlos A. Alfaro, Hugo Corrales and Carlos E. Valencia
ArXiv:1504.06257.
22. Critical ideals of trees
Hugo Corrales and Carlos E. Valencia.
ArXiv:1504.06239.
21. Optimum matchings in a weighted bipartite graph
Carlos E. Valencia and Marcos C. Vargas. Accepted for its publication in Boletín de la Sociedad Matemática Mexicana.
ArXiv:1403.5606.
20. Graphs with few trivial critical ideals
Carlos A. Alfaro and Carlos E. Valencia.
Accepted for its publication in Electronics Notes in Discrete Mathematics.
19. Third order nonlinear optical properties exhibited by a bilayer configuration of silver nanoparticles integrated to silicon nanocrystals embedded in ion-implanted silica
Alejandra López-Suárez, Carlos Torres-Torres, Bonifacio Can-Uc, Raúl Rangel-Rojo, Carlos E. Valencia, and Alicia Oliver. Journal of the Optical Society of America B 32-5 (2015), 805-811.
18. Improvement of titanium hydrogenation by low energy ion irradiation
Alejandra López-Suárez, Carlos E. Valencia, Juan López-Patiño, Marcos C. Vargas, and Beatriz Fuentes- Madariaga. International Journal of Hydrogen Energy 40-11 (2015), 4194-4199.
17. Graphs with two invariant factors equal to one
Carlos E. Valencia, Carlos A. Alfaro. Discrete Applied Mathematics 167 (2014) 33–44.
ArXiv:1304.4211
16. Dimension reduction in principal component analysis for trees
Carlos E. Valencia, C. A. Alfaro, B. Aydin, Elizabeth Bullitt, Alim Ladha. Computational Statistics and Data Analysis 74: 157-179, 2014.
ArXiv:1202.2371
15. On the critical ideals of graphs
Hugo Corrales, Carlos E. Valencia. Linear Algebra and its Applications 439 (12): 3870-3892, 2013.
ArXiv:1205.3105
14. 1-join composition for α -critical graphs
Carlos E. Valencia, Marcos I. Barrita. (Submitted for publication)
ArXiv:0707.4085
13. On the sandpile group of the cone of a graph
Carlos A. Alfaro, Carlos E. Valencia. Linear Algebra and Its Applications 436 (5): 1154-1176, 2012.
ArXiv:1004.3321
12. On the critical group of matrices
Hugo Corrales, Carlos E. Valencia. Bulletin Mathématique de la Socité des Sciences Mathématiques de Roumanie 54 (102): 213-236, 2011.
ArXiv:1008.5176
11. On bounds for some graph invariants
I. Gitler, Carlos E. Valencia. Boletín de la Sociedad Matemática Mexicana (3) 16:73-94, 2010.
ArXiv:0510387
10. A note on Rees algebras and the MFMC property
Carlos E. Valencia, I. Gitler, R. H. Villarreal. Beiträge zur Algebra und Geometrie Contributions to Algebra and Geometry 48 (1): 141-150, 2007.
ArXiv:0511307
9. On bounds for the stability number of graphs
Carlos E. Valencia, I. Gitler. Morfismos 10(2): 41-58, 2006.
8. Explicit representations by half spaces of the edge cone of a graph
Carlos E. Valencia, R. H. Villarreal. International Journal of Contemporary Mathematical Sciences 1(1-4): 53-66, 2006.
ArXiv:0506281
7. Multiplicities of edge subrings
Carlos E. Valencia, I. Gitler. Discrete Mathematics 302(1-3): 107-123, 2005.
6. A note on the Rees algebra of a bipartite graph
Carlos E. Valencia, I. Gitler, R. H. Villarreal. Journal of Pure and Applied Algebra 201: 17-24, 2005.
5. Bounds for invariants of edge-rings
Carlos E. Valencia, I. Gitler. Communications in Algebra 33: 1603-1616, 2005.
4. On packing and MFMC clutters, rings and polyhedra
Carlos E. Valencia, I. Gitler, E. Reyes, R. Villarreal. Memorias del Géométries Combinatories el Applications: Matroides Orientés, C.I.R.M.-LasVerg05, 2005.
3. Heuristic Approaches for Experimentation in Catalyst Optimization
Carlos E. Valencia, E. Possani, M. Viniegra, A. Montoya. INFORMS Annual meeting Denver, 2004.
2. Canonical modules of certain edge subrings
Carlos E. Valencia, R. H. Villarreal. European Journal of Combinatorics 24(5): 471-487, 2003.
1. Gráficas con una cubierta maximal independiente y cotas para algunos invariantes
Carlos E. Valencia, R. H. Villarreal. Morfismos 6(1): 57-71, 2002.