The monoid of strong endomorphisms of a graph.
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 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 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.
The Nonexistence of Certain Moufang Polygons.
The nonnegative -matrix completion problem.
The NP-completeness of automorphic colorings
Given a graph G, an automorphic edge(vertex)-coloring of G is a proper edge(vertex)-coloring such that each automorphism of the graph preserves the coloring. The automorphic chromatic index (number) is the least integer k for which G admits an automorphic edge(vertex)-coloring with k colors. We show that it is NP-complete to determine the automorphic chromatic index and the automorphic chromatic number of an arbitrary graph.
The number of 1-factors in some polyhexes.
The number of archiral trees.
The number of crossings in a regular drawing of the complete bipartite graph.
The number of edges in critical graphs.
The number of edges on generalizations of Paley graphs.
The number of eigenvalues greater than two in the Laplacian spectrum of a graph