Small clique number graphs with three trivial critical ideals

The critical ideals of a graph are the determinantal ideals of the generalized Laplacian matrix associated to a graph. In this article we provide a set of minimal forbidden graphs for the set of graphs with at most three trivial critical ideals. Then we use these forbidden graphs to characterize the graphs with at most three trivial critical ideals and clique number equal to 2 and 3.Comment: 33 pages, 3 figure

On the critical group of matrices

Given a graph G with a distinguished vertex s, the critical group of (G,s) is the cokernel of their reduced Laplacian matrix L(G,s). In this article we generalize the concept of the critical group to the cokernel of any matrix with entries in a commutative ring with identity. In this article we find diagonal matrices that are equivalent to some matrices that generalize the reduced Laplacian matrix of the path, the cycle, and the complete graph over an arbitrary commutative ring with identity. We are mainly interested in those cases when the base ring is the ring of integers and some subrings of matrices. Using these equivalent diagonal matrices we calculate the critical group of the m-cones of the l-duplications of the path, the cycle, and the complete graph. Also, as byproduct, we calculate the critical group of another matrices, as the m-cones of the l-duplication of the bipartite complete graph with m vertices in each partition, the bipartite complete graph with 2m vertices minus a matching.Comment: 18 pages, 5 figure

Optimum matchings in weighted bipartite graphs

Given an integer weighted bipartite graph $\{G=(U\sqcup V, E), w:E\rightarrow \mathbb{Z}\}$ we consider the problems of finding all the edges that occur in some minimum weight matching of maximum cardinality and enumerating all the minimum weight perfect matchings. Moreover, we construct a subgraph $G_{cs}$ of $G$ which depends on an $\epsilon$-optimal solution of the dual linear program associated to the assignment problem on $\{G,w\}$ that allows us to reduced this problems to their unweighed variants on $G_{cs}$. For instance, when $G$ has a perfect matching and we have an $\epsilon$-optimal solution of the dual linear program associated to the assignment problem on $\{G,w\}$, we solve the problem of finding all the edges that occur in some minimum weight perfect matching in linear time on the number of edges. Therefore, starting from scratch we get an algorithm that solves this problem in time $O(\sqrt{n}m\log(nW))$, where $n=|U|\geq |V|$, $m=|E|$, and $W={\rm max}\{|w(e)|\, :\, e\in E\}$.Comment: 11 page

Uso y abuso de las redes sociales por parte de los adolescentes: un estudio en México

El uso abusivo y problemático de las diferentes tecnologías ha conducido al desarrollo de una línea de investigación centrada en el estudio de la adicción a estas y sus consecuencias. Por ello, algunos estudios se han enfocado en internet y su relación con problemas de tipo físico-biológico, social y familiar. Esta investigación se planteó examinar el nivel de adicción de los jóvenes mexicanos a las redes sociales online mediante la adaptación de la escala de Sahin (“Social Media Addiction Scale-Student Form”). Además, se analizó la existencia de respuestas diferenciadas en términos de género en la muestra utilizada que consistió en 605 estudiantes de preparatoria (296 hombres y 309 mujeres). Los resultados muestran que los estudiantes no se autoperciben como adictos a las redes sociales online. Por otro lado, no hay diferencias significativas en sus respuestas en términos de género. Estos resultados deben ser considerados dentro de sus limitaciones relacionadas con las características de la muestra y el hecho de que se examinaron las autopercepciones de los estudiantes.The abusive and problematic use of a variety of technologies has led to the development of research focusing on the study of addiction to such technologies and the consequences this brings. Therefore, some studies have focused on the Internet and its relationship with physicalbiological, social and family problems. This research sought to examine the extent of addiction of young Mexicans to online social media networks by adapting the Sahin scale (“Social Media Addiction Scale-Student Form”). In addition, we analyzed the existence of differentiated responses in terms of gender in the sample used, which consisted of 605 high school students (296 men and 309 women). The results indicate that students do not perceive themselves as addicted to online social media networks. In other aspects, there are no meaningful differences in registered responses due to the gender of the participant. These findings should be considered within their limitations related to the characteristics of the sample and the fact that the students’ selfperceptions were examined