On the number of matchings in regular graphs.
On the number of matroids on a finite set.
On the Number of Minimal 1-Steiner Trees.
On the number of monotonic functions from two-valued logic to -valued logic
On the number of m-term zero-sum subsequences
On the number of non-isomorphic Steiner triples
On the number of orthogonal systems in vector spaces over finite fields.
On the number of pairs of partitions of n without common subsums
On the number of perfect matchings and Hamilton cycles in -regular non-bipartite graphs.
On the number of permutations admitting an th root.
On the number of permutations arising from a problem in cell-biology.
On the number of planar orientations with prescribed degrees.
On the number of possible row and column sums of -matrices.
On the number of prime divisors of a binomial coefficient.
On the number of representations of an element in a polygonal Cayley graph
We compute explicitly the number of paths of given length joining two vertices of the Cayley graph of the free product of cyclic groups of order k.
On the number of rim hook tableaux
On the number of Russell’s socks or
The following question is analyzed under the assumption that the Axiom of Choice fails badly: Given a countable number of pairs of socks, then how many socks are there? Surprisingly this number is not uniquely determined by the above information, thus giving rise to the concept of Russell-cardinals. It will be shown that: • some Russell-cardinals are even, but others fail to be so; • no Russell-cardinal is odd; • no Russell-cardinal is comparable with any cardinal of the form or ; • finite sums...
On the number of spanning trees, the Laplacian eigenvalues, and the Laplacian Estrada index of subdivided-line graphs
As a generalization of the Sierpiński-like graphs, the subdivided-line graph Г(G) of a simple connected graph G is defined to be the line graph of the barycentric subdivision of G. In this paper we obtain a closed-form formula for the enumeration of spanning trees in Г(G), employing the theory of electrical networks. We present bounds for the largest and second smallest Laplacian eigenvalues of Г(G) in terms of the maximum degree, the number of edges, and the first Zagreb index of G. In addition,...
On the number of subsets relatively prime to an integer.
On the Numbers of Cut-Vertices and End-Blocks in 4-Regular Graphs
A cut-vertex in a graph G is a vertex whose removal increases the number of connected components of G. An end-block of G is a block with a single cut-vertex. In this paper we establish upper bounds on the numbers of end-blocks and cut-vertices in a 4-regular graph G and claw-free 4-regular graphs. We characterize the extremal graphs achieving these bounds.