A nonzero element x of a Lie algebra L over a field F is called extremal if [x, [x,L]] ⊆ Fx. Extremal elements are a well-studied class of elements in simple finite-dimensional Lie algebras of Chevalley type: they are the long root elements. In [CSUW01], Cohen, Steinbach, Ushirobira and Wales have studied Lie algebras generated by extremal elements, in particular those of Chevalley type. The au...

In this paper we discuss the bounds of and relations among various kinds of intersection numbers of graphs. Especially, we address extremal graphs with respect to the established bounds. The uniqueness of the minimum-size intersection representations for some graphs is also studied. In the course of this work, we introduce a superclass of chordal graphs, defined in terms of a generalization of ...

In this paper, we present three improved upper bounds for the Laplacian spectral radius of graphs. Moreover, we determine all extremal graphs which achieve these upper bounds. Finally, some examples illustrate that the results are best in all known upper bounds in some sense.

In arrangements of pseudocircles (i.e., Jordan curves) the weight of a vertex (i.e., an intersection point) is the number of pseudocircles that contain the vertex in its interior. We show that in complete arrangements (in which each two pseudocircles intersect) 2n−1 vertices of weight 0 force an α-subarrangement, a certain arrangement of three pseudocircles. Similarly, 4n−5 vertices of weight 0...

Let F2k,k2 consist of all simple graphs on 2k vertices and k2 edges. For a simple graph G and a positive integer λ, let PG(λ) denote the number of proper vertex colorings of G in at most λ colors, and let f(2k, k2, λ) = max{PG(λ) : G ∈ F2k,k2}. We prove that f(2k, k2, 3) = PKk,k(3) and Kk,k is the only extremal graph. We also prove that f(2k, k2, 4) = (6 + o(1))4k as k →∞.

The Fibonacci index of a graph is the number of its stable sets. This parameter is widely studied and has applications in chemical graph theory. In this paper, we establish tight upper bounds for the Fibonacci index in terms of the stability number and the order of general graphs and connected graphs. Turán graphs frequently appear in extremal graph theory. We show that Turán graphs and a conne...

The Turán number of a graph H, ex(n,H), is the maximum number of edges in a graph on n vertices which does not have H as a subgraph. We determine the Turán number and find the unique extremal graph for forests consisting of paths when n is sufficiently large. This generalizes a result of Bushaw and Kettle [Combinatorics, Probability and Computing 20:837–853, 2011]. We also determine the Turán n...

What you see below are notes related to a course that I have given several times in Extremal Graph Theory. I guarantee no accuracy with respect to these notes and I certainly do not guarantee completeness or proper attribution. This is an early draft and, with any luck and copious funding, some of this can be made into a publishable work and some will just remain as notes. Please do not distrib...

Edwards showed that every graph of size m ≥ 1 has a bipartite subgraph of size at least m/2 + √ m/8 + 1/64− 1/8. We show that every graph of size m ≥ 1 has a bipartition in which the Edwards bound holds, and in addition each vertex class contains at most m/4 + √ m/32 + 1/256 − 1/16 edges. This is exact for complete graphs of odd order, which we show are the only extremal graphs without isolated...

We find the maximum number of maximal independent sets in two families of graphs: all graphs with n vertices and at most r cycles, and all such graphs that are also connected. In addition, we characterize the extremal graphs. This proves a strengthening of a conjecture of Goh and Koh [3]. We do the same for the maximum number of maximum independent sets, generalizing a theorem of Jou and Chang ...