Isometric Hamming embeddings of weighted graphs.

A mapping α:V(G)→V(H) from the vertex set of one graph G to another graph H is an isometric embedding if the shortest path distance between any two vertices in G equals the distance between their images in H. Here, we consider isometric embeddings of a weighted graph G into unweighted Hamming graphs, called Hamming embeddings, when G satisfies the property that every edge is a shortest path between its endpoints. Using a Cartesian product decomposition of G called its canonical isometric representation, we show that every Hamming embedding of G may be partitioned into a canonical partition, whose parts provide Hamming embeddings for each factor of the canonical isometric representation of G. This implies that G permits a Hamming embedding if and only if each factor of its canonical isometric representation is Hamming embeddable. This result extends prior work on unweighted graphs that showed that an unweighted graph permits a Hamming embedding if and only if each factor is a complete graph. When a graph G has nontrivial isometric representation, determining whether G has a Hamming embedding can be simplified to checking embeddability of two or more smaller graphs.

Factorization and pseudofactorization of weighted graphs.

For unweighted graphs, finding isometric embeddings of a graph G is closely related to decompositions of G into Cartesian products of smaller graphs. When G is isomorphic to a Cartesian graph product, we call the factors of this product a factorization of G. When G is isomorphic to an isometric subgraph of a Cartesian graph product, we call those factors a pseudofactorization of G. Prior work has shown that an unweighted graph's pseudofactorization can be used to generate a canonical isometric embedding into a product of the smallest possible pseudofactors. However, for arbitrary weighted graphs, which represent a richer variety of metric spaces, methods for finding isometric embeddings or determining their existence remain elusive, and indeed pseudofactorization and factorization have not previously been extended to this context. In this work, we address the problem of finding the factorization and pseudofactorization of a weighted graph G, where G satisfies the property that every edge constitutes a shortest path between its endpoints. We term such graphs minimal graphs, noting that every graph can be made minimal by removing edges not affecting its path metric. We generalize pseudofactorization and factorization to minimal graphs and develop new proof techniques that extend the previously proposed algorithms due to Graham and Winkler [Graham and Winkler, '85] and Feder [Feder, '92] for pseudofactorization and factorization of unweighted graphs. We show that any n-vertex, m-edge graph with positive integer edge weights can be factored in O(m2) time, plus the time to find all pairs shortest paths (APSP) distances in a weighted graph, resulting in an overall running time of O(m2+n2 log log n) time. We also show that a pseudofactorization for such a graph can be computed in O(mn) time, plus the time to solve APSP, resulting in an O(mn+n2 log log n) running time.

Effects of serum phosphorus on vascular calcification in a healthy, adult population: A systematic review.

Cardiovascular disease has been associated with elevated serum phosphorus levels, which have been associated with cardiovascular mortality. This is commonly seen in the chronic kidney disease (CKD) population where studies have shown that high phosphorus levels cause coronary artery calcification. Although studies have independently associated vascular stiffness and serum phosphorus in those with and without CKD, there are fewer data in individuals without CKD. Therefore, the aim of this systematic review was to analyze whether serum phosphorus levels are associated with cardiovascular calcification in healthy individuals. A systematic review of the literature that was conducted revealed 10 articles, all cross-sectional studies, that met eligibility criteria. These criteria were peer-reviewed studies on a healthy, adult population written in the English language. Studies lacking data on serum phosphorus and measured to assess its association with vascular calcification were excluded. Studies on subjects with CKD, other chronic diseases, or on children were also excluded. Of the 10 studies located, 8 indicated an association between serum phosphorus and vascular calcification. One study did not indicate an association. One study indicated a statistically significant association between serum phosphorus and vascular calcification prevalence, but not incidence. Studies were limited since no randomized controlled trials were available. This systematic review generates gaps in research. Due to considerable amounts of phosphorus additives in the food supply, there may be a connection to dietary phosphorus and vascular calcification. Additionally, phosphorus binders may assist in the prevention of vascular calcification but have not been studied in a healthy population. Further study on both dietary phosphorus restriction and phosphorus binders is needed. While 8 out of 10 cross-sectional studies found an association in this systematic review, the topic of vascular calcification and serum phosphorus needs further study if a cause and effect relationship is to be detected.