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 2003 Vladimir Nikiforov [7] began a line of research whose aim was to build an extremal theory of graphs based on spectral theory. We will discuss some of his results and in particular we will focus on a result of Babai and Guiduli [1] that gives a Kövari-Sós-Turán type upper bound on the largest eigenvalue of the adjacency matrix of a Ks,t-free graph. This spectral approach sheds new light ...

We study two extremal problems about subgraphs excluding a family F of graphs. i) Among all graphs with m edges, what is the smallest size f(m,F) of a largest F–free subgraph? ii) Among all graphs with minimum degree δ and maximum degree ∆, what is the smallest minimum degree h(δ,∆,F) of a spanning F– free subgraph with largest minimum degree? These questions are easy to answer for families not...

Here we overview some of the methods and results of extremal graph and hypergraph theory. A few geometric applications are also given.

Symmetric Division Deg index is one of the 148 discrete Adriatic indices that showed good predictive properties on the testing sets provided by International Academy of Mathematical Chemistry. Symmetric Division Deg index is defined by $$ SDD(G) = sumE left( frac{min{d_u,d_v}}{max{d_u,d_v}} + frac{max{d_u,d_v}}{min{d_u,d_v}} right), $$ where $d_i$ is the degree of vertex $i$ in graph $G$. In th...

A system F of functions f1; 2 most d if no (d + 1)-element subset A X is 2-shattered. A is 2-shattered if for each x 2 A there is a 2-element set V x f1; 2; : : :; kg such that for any choice of elements c x 2 V x , a function f 2 F exists with f(x) = c x for all x 2 A. We improve a lower bound of c d k d n d (due to Haussler and Long) for the maximum size of F of Natarajan dimension at most d ...

For a graph G = (V,E), the modified Schultz index of G is defined as S∗(G) = ∑ {u,v}⊆V (G) (dG(u) · dG(v))dG(u, v) where dG(u) (or d(u)) is the degree of the vertex u of G, and dG(u, v) is the distance between u and v. Let B(n) be the set of bicyclic graph with n vertices. In this paper, we study the modified Schultz index of B(n), graphs in B(n) with the smallest modified Schultz index S∗(G) a...

The Ear Search program implements isomorph-free generation of 2-connected graphs by ear augmentations. This document describes the interfaces used for customized searches, as well as describes three example searches: unique saturation, edge reconstruction, and extremal graphs with a fixed number of perfect matchings.

A function f bounds graphs from above if there exists an infinite family of graph G, such that if G ∈ G then f(|VG|) = |EG| and for all nonempty subgraphs H of G we have that F (|VH |) ≥ |EH |. This paper considers the question: which functions bound graphs? keywords graphs, extremal graph theory AMS subject classification 05C99

This paper studies the following question: Given a surface Σ and an integer n, what is the maximum number of cliques in an n-vertex graph embeddable in Σ? We characterise the extremal graphs for this question, and prove that the answer is between 8(n− ω)+ 2 and 8n + 3 2 2 + o(2), where ω is the maximum integer such that the complete graph Kω embeds in Σ. For the surfaces S0, S1, S2, N1, N2, N3 ...