Saxagliptine : efficacité confirmée, même en cas d’insuffisance rénale chronique

Abstract

We say that a regular graph G of order n and degree r≥1 (which is not the complete graph) is strongly regular if there exist non-negative integers τ and θ such that |Si∩Sj|=τ for any two adjacent vertices i and j, and |Si∩Sj|=θ for any two distinct non-adjacent vertices i and j, where Sk denotes the neighborhood of the vertex k. Let i be a fixed vertex from the vertex set V(G)={1,2,…,n} and let Gi=G∖︀i be its vertex deleted subgraph. Let Hi be switching equivalent to Gi with respect to Si⊆V(Gi). We prove that Hi is a strongly regular graph of order n−1 and degree 2(r−θ) with τ(Hi)=r+τ−2θ and θ(Hi)=r−θ if and only if n=4r−2τ−2θ. Otherwise, we prove that Hi has exactly two main eigenvalues μ1 and μ2. In this case, the main eigenvalues of Hi are represented in the form We also prove that Gi and Hi are cospectral if and only if r=2θ. Finally, we show that if G is a strongly regular graph of order 2(2k2+2k+1) and degree k(2k+1) with τ=k2−1 and θ=k2 then Hi is a conference graph of order (2k+1)2 and degree 2k(k+1) with τ(Hi)=k2+k−1 and θ(Hi)=k(k+1). However, if G is a strongly regular graph of order 4(3k2+3k+1) and degree 3k(2k+1) with τ=3k2−k−2 and θ=k(3k+1) then Hi is a strongly regular graph (non-conference) of order 3(2k+1)2 and degree 2k(3k+2) with τ(Hi)=3k2−2 and θ(Hi)=k(3k+2).