Various results on the toughness of graphs

Let G be a graph, and let t 0 be a real number. Then G is t-tough if t!(G − S) jSj for all S V (G) with !(G − S) > 1, where !(G − S) denotes the number of components of G − S. The toughness of G, denoted by (G), is the maximum value of t for which G is t-tough (taking (Kn) = 1 for all n 1). G is minimally t-tough if (G) = t and (H) < t for every proper spanning subgraph H of G. We discuss how the toughness of (spanning) subgraphs of G and related graphs depends on (G), we give some sucient (degree) conditions implying (G) t, and we study which subdivisions of 2-connected graphs have minimally 2-tough squares

Toughness, Tenacity and Maximum Initial Strength of Rubber Modified Asphalt Binders

The toughness and tenacity test method, which was developed in the 1980s, is popular for evaluating a polymermodified binder. Several states like Nevada require performing this test to evaluate non-modified binder samples, as well as other types of modified binders. In this regard, a toughness and tenacity test was performed on rubber-modified samples produced from virgin binder PG58-28, PG64-16 and AC-20. In order to take the rubber size, type and content into account, two rubber sizes, mesh #20 and #40, two rubber types, ambient and cryogenic, and three rubber contents, 10%, 15%, and 20% were produced and tested. The results then were compared with polymer-modified and terminally blended rubber-modified samples. The results show improvement in the amount of initial maximum strength, and a decline in the magnitude of elongation, toughness and tenacity for the rubber-modified binder, compared to other types of binders

Edge-dominating cycles, k-walks and Hamilton prisms in 2K22K_2-free graphs

We show that an edge-dominating cycle in a $2K_2$-free graph can be found in polynomial time; this implies that every 1/(k-1)-tough $2K_2$-free graph admits a k-walk, and it can be found in polynomial time. For this class of graphs, this proves a long-standing conjecture due to Jackson and Wormald (1990). Furthermore, we prove that for any \epsilon>0 every (1+\epsilon)-tough $2K_2$-free graph is prism-Hamiltonian and give an effective construction of a Hamiltonian cycle in the corresponding prism, along with few other similar results.Comment: LaTeX, 8 page

The spectrum and toughness of regular graphs

In 1995, Brouwer proved that the toughness of a connected $k$-regular graph $G$ is at least $k/\lambda-2$, where $\lambda$ is the maximum absolute value of the non-trivial eigenvalues of $G$. Brouwer conjectured that one can improve this lower bound to $k/\lambda-1$ and that many graphs (especially graphs attaining equality in the Hoffman ratio bound for the independence number) have toughness equal to $k/\lambda$. In this paper, we improve Brouwer's spectral bound when the toughness is small and we determine the exact value of the toughness for many strongly regular graphs attaining equality in the Hoffman ratio bound such as Lattice graphs, Triangular graphs, complements of Triangular graphs and complements of point-graphs of generalized quadrangles. For all these graphs with the exception of the Petersen graph, we confirm Brouwer's intuition by showing that the toughness equals $k/(-\lambda_{min})$, where $\lambda_{min}$ is the smallest eigenvalue of the adjacency matrix of the graph.Comment: 15 pages, 1 figure, accepted to Discrete Applied Mathematics, special issue dedicated to the "Applications of Graph Spectra in Computer Science" Conference, Centre de Recerca Matematica (CRM), Bellaterra, Barcelona, June 16-20, 201

The effects of the minimum wage on wages and employment in Brazil: a menu of minimum wage variables

The international literature on minimum wage strongly lacks empirical evidence from developing countries. In Brazil, not only are increases in the minimum wage large and frequent - unlike the typically small increases focused upon in most of the existing literature - but also the minimum wage plays a central and complex role. In addition to its social role the minimum wage has been used as anti-inflationary policy, confirming its importance to the Brazilian Economy. This paper analyzes the effects of the minimum wage on both wages and employment using monthly household-level data (similar to the US CPS) over a reasonably long time period. A number of conceptual and identification questions is here discussed. Various strategies on how to best measure the effect of a constant (national) minimum wage are summarized in a “menu” of minimum wage variables. Also, an employment decomposition that separately estimates the hours worked and the number of jobs effects is used. Robust results indicate that an increase in the minimum wage strongly compresses the wages distribution with moderately small adverse effects on employment