Simple and efficient four-cycle counting on sparse graphs

We consider the problem of counting 4-cycles ($C_4$) in a general undirected graph $G$ of $n$ vertices and $m$ edges (in bipartite graphs, 4-cycles are also often referred to as $\textit{butterflies}$). There have been a number of previous algorithms for this problem; some of these are based on fast matrix multiplication, which is attractive theoretically but not practical, and some of these are based on randomized hash tables. We develop a new simpler algorithm for counting $C_4$ requiring $O(m\bar\delta(G))$ time and $O(n)$ space, where $\bar \delta(G) \leq O(\sqrt{m})$ is the $\textit{average degeneracy}$ parameter introduced by Burkhardt, Faber & Harris (2020). It has several practical improvements over previous algorithms; for example, it is fully deterministic, does not require any sorting of the adjacency list of the input graph, and avoids any expensive arithmetic in its inner loops. To the best of our knowledge, all previous efficient algorithms for $C_4$ counting have required $\Omega(m)$ space. The algorithm can also be adapted to count 4-cycles incident to each vertex and edge

A standard stellar library for evolutionary synthesis. III. Metallicity calibration

We extend the colour calibration of the widely used BaSeL standard stellar library (Lejeune, Cuisinier, & Buser 1997, 1998) to non-solar metallicities, down to [Fe/H] ~ -2.0 dex. Surprisingly, we find that at the present epoch it is virtually impossible to establish a unique calibration of UBVRIJHKL colours in terms of stellar metallicity [Fe/H] which is consistent simultaneously with both colour-temperature relations and colour-absolute magnitude diagrams (CMDs) based on observed globular cluster photometry data and on published, currently popular standard stellar evolutionary tracks and isochrones. The problem appears to be related to the long-standing incompleteness in our understanding of convection in late-type stellar evolution, but is also due to a serious lack of relevant observational calibration data that would help resolve, or at least further significant progress towards resolving this issue. In view of the most important applications of the BaSeL library, we here propose two different metallicity calibration versions: (1) the "WLBC 99" library, which consistently matches empirical colour-temperature relations and which, therefore, should make an ideal tool for the study of individual stars; and (2), the "PADOVA 2000" library, which provides isochrones from the Padova 2000 grid (Girardi et al., 2000) that successfully reproduce Galactic globular-cluster colour-absolute magnitude diagrams and which thus should prove particularly useful for studies of collective phenomena in stellar populations in clusters and galaxies.Comment: 15 pages, LaTeX, documentclass aa, accepted for A and