No abstract

In this paper, we show that a connected graph with smallest eigenvalue at least −3 and large enough minimal degree is 2-integrable. This result generalizes a 1977 result of Hoffman for connected graphs with smallest eigenvalue at least −2.

Abstract:In this paper, we first discuss the development of the field of tribology, and highlight some of the main problems encountered in this area, such as lack of systematicness, loose correlation, and inadequate focus on the microscopic perspective. Then, we provide basic formulas of frictional mechanics while considering the friction effect on classical mechanics formulae. In order to carry out the frictional mechanics analysis, we first classify the interface. According to the size analysis of surface films, the manufacturing roughness of the surface, the contact width, and the roller radius of the rolling contact bearing, frictional mechanics has the features of interface mechanics, while interfaces are classified based on the presence or absence of a medium. Based on the classification, we further analyze the pressure and frictional stress of sliding and rolling friction problems without a medium, such as a slider, wedge key, and V belt. We also analyze problems with a medium, such as journal and rolling contact bearings. By comparing these results with those of classical mechanics without considering friction, we see that (1) friction causes deviations in the result for classical mechanics which does not consider friction, and (2) if the frictional stress and normal pressure affect each other, their interaction should be considered simultaneously. Finally, we summarize the friction problems, namely, sliding and rolling, with and without a medium, and deformed and non-deformed. From our analysis, we propose two conclusions. First, the frictional mechanics problem is a deviation of the classical mechanics problem, and secondly, frictional stress and normal pressure influence each other.

No abstract

In 1977, Hoffman gave a characterization of graphs with smallest eigenvalue at least −2. In this paper we generalize this result to graphs with smaller smallest eigenvalue. For the proof, we use a combinatorial object named Hoffman graph, introduced by Woo and Neumaier in 1995. Our result says that for every λ ≤ −2, if a graph with smallest eigenvalue at least λ satisfies some local conditions, then it is highly structured. We apply our result to graphs which are cospectral with the Hamming graph H(3, q), the Johnson graph J(v, 3) and the 2clique extension of grids, respectively.