Let Fn,tr(n) denote the family of all graphs on n vertices and tr(n) edges, where tr(n) is the number of edges in the Turán’s graph Tr(n) – the complete r-partite graph on n vertices with partition sizes as equal as possible. For a graph G and a positive integer λ, let PG(λ) denote the number of proper vertex colorings of G with at most λ colors, and let f(n, tr(n), λ) = max{PG(λ) : G ∈ Fn,tr(n...

Extremal optimization is a new general-purpose method for approximating solutions to hard optimization problems. We study the method in detail by way of the computationally hard (NP-hard) graph partitioning problem. We discuss the scaling behavior of extremal optimization, focusing on the convergence of the average run as a function of run time and system size. The method has a single free para...

We prove that the number of multigraphs with vertex set {1, . . . , n} such that every four vertices span at most nine edges is a +o(n) where a is transcendental (assuming Schanuel’s conjecture from number theory). This is an easy consequence of the solution to a related problem about maximizing the product of the edge multiplicities in certain multigraphs, and appears to be the first explicit ...

Let F (n, r, k) denote the maximum number of edge r-colorings without a monochromatic copy of Kk that a graph with n vertices can have. Addressing two questions left open by Alon, Balogh, Keevash, and Sudakov [J. London Math. Soc., 70 (2004) 273–288], we determine F (n, 4, 3) and F (n, 4, 4) and describe the extremal graphs for all large n.

If G is a graph, and if for e = uv ? E(G) the number of vertices closer to u than v denoted by nu, then Mo(G) uv?E(G) |nu-nv| Mostar index G. In this paper, studied on trees graph products. Lower upper bounds are given difference between indices tree obtained contraction one its edges corresponding extremal characterized. An bound class all but stars proved. Extremal also determined (k+1)-th la...

Compared to binary self-dual codes, few methods are known to construct ternary self-dual codes. In this paper, a construction method for ternary self-dual codes is presented. Using this method, a number of new extremal ternary self-dual codes are obtained from weighing matrices. In addition, a classification is given for extremal ternary self-dual codes of length 40 constructed from Hadamard ma...

X iv :0 81 0. 19 24 v3 [ m at h. A G ] 7 F eb 2 00 9 EXTREMAL METRICS ON DEL PEZZO THREEFOLDS IVAN CHELTSOV AND CONSTANTIN SHRAMOV In memory of Vasily Iskovskikh Abstract. We prove the existence of Kähler–Einstein metrics on a nonsingular section of the Grassmannian Gr(2, 5) ⊂ P by a linear subspace of codimension 3, and the Fermat hypersurface of degree 6 in P(1, 1, 1, 2, 3). We also show that...

In this note I prove the existence of extremal Sasakian metrics on S3-bundles over S2. These occur in a collection of open cones that I call a

We consider weighted graphs, where the edge weights are positive definite matrices. The eigenvalues of a graph are the eigenvalues of its adjacency matrix. We obtain a lower bound and an upper bound on the spectral radius of the adjacency matrix of weighted graphs and characterize graphs for which the bounds are attained. 2011 Elsevier Inc. All rights reserved.

Every pseudocompact Abelian group of uncountable weight has both a proper dense pseudocompact subgroup and a strictly finer pseudocompact group topology.