The minor crossing number of graphs with an excluded minor.
The Möbius function of the consecutive pattern poset.
The modified negative decision number in graphs.
The moments of partitions, I
The monadic second-order logic of graphs III : tree-decompositions, minors and complexity issues
The monoid of ordered partitions of a natural number.
The monoid of strong endomorphisms of a graph.
The monotone cumulants
In the present paper we define the notion of generalized cumulants which gives a universal framework for commutative, free, Boolean and especially, monotone probability theories. The uniqueness of generalized cumulants holds for each independence, and hence, generalized cumulants are equal to the usual cumulants in the commutative, free and Boolean cases. The way we define (generalized) cumulants needs neither partition lattices nor generating functions and then will give a new viewpoint to cumulants....
The Moore-Penrose inverse of a free matrix.
The Morse landscape of a riemannian disk
We study upper bounds on the length functional along contractions of loops in Riemannian disks of bounded diameter and circumference. By constructing metrics adapted to imbedded trees of increasing complexity, we reduce the nonexistence of such upper bounds to the study of a topological invariant of imbedded finite trees. This invariant is related to the complexity of the binary representation of integers. It is also related to lower bounds on the number of points in level sets of a real-valued...
The multiplicities of a dual-thin -polynomial association scheme.
The multiplicity of the Lyashko-Looijenga mapping on the discriminant strata of even and odd polynomials
The multiset chromatic number of a graph
A vertex coloring of a graph is a multiset coloring if the multisets of colors of the neighbors of every two adjacent vertices are different. The minimum for which has a multiset -coloring is the multiset chromatic number of . For every graph , is bounded above by its chromatic number . The multiset chromatic number is determined for every complete multipartite graph as well as for cycles and their squares, cubes, and fourth powers. It is conjectured that for each , there exist sufficiently...
The Mycielskian of a Graph
Let ω(G) and χ(G) be the clique number and the chromatic number of a graph G. Mycielski [11] presented a construction that for any n creates a graph Mn which is triangle-free (ω(G) = 2) with χ(G) > n. The starting point is the complete graph of two vertices (K2). M(n+1) is obtained from Mn through the operation μ(G) called the Mycielskian of a graph G.We first define the operation μ(G) and then show that ω(μ(G)) = ω(G) and χ(μ(G)) = χ(G) + 1. This is done for arbitrary graph G, see also [10]....
The neighborhood characteristic parameter for graphs.
The neighborhood complex of an infinite graph.
The Nekrasov-Okounkov hook length formula: refinement, elementary proof, extension and applications
The paper is devoted to the derivation of the expansion formula for the powers of the Euler Product in terms of partition hook lengths, discovered by Nekrasov and Okounkov in their study of the Seiberg-Witten Theory. We provide a refinement based on a new property of -cores, and give an elementary proof by using the Macdonald identities. We also obtain an extension by adding two more parameters, which appears to be a discrete interpolation between the Macdonald identities and the generating function...
The niche graphs of interval orders
The niche graph of a digraph D is the (simple undirected) graph which has the same vertex set as D and has an edge between two distinct vertices x and y if and only if N+D(x) ∩ N+D(y) ≠ ∅ or N−D(x) ∩ N−D(y) ≠ ∅, where N+D(x) (resp. N−D(x)) is the set of out-neighbors (resp. in-neighbors) of x in D. A digraph D = (V,A) is called a semiorder (or a unit interval order ) if there exist a real-valued function f : V → R on the set V and a positive real number δ ∈ R such that (x, y) ∈ A if and only if...
The n-minimal chromatic multiplicity of a graph
The non-crossing graph.