We obtain sharp asymptotic estimates on the number of n × $n \times n$ contingency tables with two linear margins C $Cn$ and B $BCn$ . The results imply a second-order phase transition such tables, critical value at c : = 1 + / $B_{c}:=1 \sqrt {1+1/C}$ As consequence, for > $B>B_{c}$ , we prove that classical independence heuristic leads to large undercounting.

Let mGI denote the number of Laplacian eigenvalues a graph G in an interval I and let α(G) independence G. In this paper, we determine classes graphs that satisfy condition mG[0,n−α(G)]=α(G) when α(G)=2 α(G)=n−2, where n is order When α(G)=2, G≅K1∇Kn−m∇Km−1 for some m≥2. there are two types B(p,q,r) B′(p,q,r) = p + q r 2, which call binary star graphs. Also, show with determined by their spectra.

The Birkhoff graph B n is the Cayley of symmetric group S , where two permutations are adjacent if they differ by a single cycle. Our main result tighter upper bound on independence number ? ( ) namely, we show that ? O ! / 1 . 9 7 improving previous known 2 Kane et al. (2017). approach combines higher-order version their representation theoretic techniques with linear programming. With an expl...