A fullerene graph is a cubic 3-connected plane graph with (exactly 12) pentagonal faces and hexagonal faces. Let Fn be a fullerene graph with n vertices. A set H of mutually disjoint hexagons of Fn is a sextet pattern if Fn has a perfect matching which alternates on and off each hexagon in H. The maximum cardinality of sextet patterns of Fn is the Clar number of Fn. It was shown that the Clar n...

The thesis consists of two parts. In both parts, the problems studied are of significant interest, but are either NP-hard or unknown to be polynomially decidable. Realistically, this forces us to relax the objective of optimality or restrict the problem. As projected by the title, the chosen tool of this thesis is an extremal type approach. The lesson drawn by the theorems proved in the thesis ...

Graham and Lovász conjectured in 1978 that the sequence of normalized coefficients distance characteristic polynomial a tree order n is unimodal with maximum value occurring at ⌊n2⌋. In this paper we investigate problem for block graphs. particular, prove unimodality part establish peak several extremal cases uniform graphs small diameter.

We consider the problem of determining inducibility (maximum possible asymptotic density induced copies) oriented graphs on four vertices. provide exact values for more than half graphs, and very close lower upper bounds all remaining ones. It occurs that, some structure extremal constructions maximizing its copies is sophisticated complex.

The uniting feature of combinatorial optimization and extremal graph theory is that in both areas one should find extrema of a function defined in most cases on a finite set. While in combinatorial optimization the point is in developing efficient algorithms and heuristics for solving specified types of problems, the extremal graph theory deals with finding bounds for various graph invariants u...

Graphical designs are an extension of spherical to functions on graphs. We connect linear codes graphical cube graphs, and show that the Hamming code in particular is a highly effective design. even structured distinct from related concepts extremal designs, maximum stable sets distance t-designs association schemes.

The general sum-connectivity index of a graph G is χα(G) = ∑ uv∈E(G) (d(u)+d(v)), where d(u) denotes the degree of vertex u ∈ V (G), and α is a real number. In this paper, we show that in the class of graphs G of order n ≥ 3 and minimum degree δ(G) ≥ 2, the unique graph G having minimum χα(G) is K2 + Kn−2 if −1 ≤ α < α0 ≈ −0.867. Similarly, if we impose the supplementary condition for G to be t...

We shall prove that if L is a 3-chromatic (so called "forbidden") graph, and -R" is a random graph on n vertices, whose edges are chosen indepen-6" is a bipartite subgraph of R" of maximum size, -F" is an L-free subgraph of R" of maximum size, dently, with probability p, and then (in some sense) F" and 6" are very near to each other: almost surely they have almost the same number of edges, and ...

Let d and S? be two intersecting families of k-subsets of an n-element set. It is proven that l~.JuS?l <(;:i)+(;::) holds for n>f(3+,/?)k, and equality holds only if there exist two points a, b such that {a, b} n F# 0 for all FE d u g, For n=2k+o(Jj;) an example showing that in this case max 1 d u B 1 = (1 o( 1 ))( ;) is given. This disproves an old conjecture of Erdiis [7]. In the second part ...

We say that a set system F ⊆ 2[n] shatters a given set S ⊆ [n] if 2S = {F ∩ S : F ∈ F}. The Sauer inequality states that in general, a set system F shatters at least |F| sets. Here we concentrate on the case of equality. A set system is called shattering-extremal if it shatters exactly |F| sets. In this paper we characterize shattering-extremal set systems of Vapnik-Chervonenkis dimension 2 in ...