Extremal digraphs with given clique number

On the Spectral Moment of Graphs with given Clique Number

Let Ln,t be the set of all n-vertex connected graphs with clique number t (2 ≤ t ≤ n). For nvertex connected graphs with given clique number, lexicographic ordering by spectral moments (S-order) is discussed in this paper. The first ∑⌊(n−t−1)/3⌋ i=1 (n − t − 3i) + 1 graphs with 3 ≤ t ≤ n − 4, and the last few graphs, in the Sorder, among Ln,t are characterized. In addition, all graphs in Ln,n ∪...

Extremal Graphs With a Given Number of Perfect Matchings

Let f(n, p) denote the maximum number of edges in a graph having n vertices and exactly p perfect matchings. For fixed p, Dudek and Schmitt showed that f(n, p) = n2/4 + cp for some constant cp when n is at least some constant np. For p ≤ 6, they also determined cp and np. For fixed p, we show that the extremal graphs for all n are determined by those with O( √ p) vertices. As a corollary, a com...

The asymptotic number of labelled weakly-connected digraphs with a given number of vertices and edges

Edward A. Benderl Depariment of Mathel1wtics, University of California, San Diego, La Jolla, CA 92093, USA E. Rodney Canfieldl Department of Computer Science, University of Georgia, Athens, GA 30602, USA Brendan D. McKay2 Computer Science Department, Australian National University, GPO Box 4, ACT 2601., Australia In a recent paper, we determined the asymptotic number of labelled connected graph...

Extremal Problems in Digraphs

Let G be a finite simple directed graph on n vertices. Say G is m-free if it has no directed cycles of length at most m. In 1978, Caccetta and Häggkvist [3] conjectured that if G has minimum out-degree at least r, then G is not dn/re-free. Finding upper bounds on the minimum out-degree in 3-free digraphs has been of particular interest in recent research. In this thesis, we present new results ...

Supereulerian digraphs with given local structures