On graphs whose spectral spread does not exceed 4.
On graphs with a unique minimum hull set
We show that for every integer k ≥ 2 and every k graphs G₁,G₂,...,Gₖ, there exists a hull graph with k hull vertices v₁,v₂,...,vₖ such that link for 1 ≤ i ≤ k. Moreover, every pair a, b of integers with 2 ≤ a ≤ b is realizable as the hull number and geodetic number (or upper geodetic number) of a hull graph. We also show that every pair a,b of integers with a ≥ 2 and b ≥ 0 is realizable as the hull number and forcing geodetic number of a hull graph.
On graphs with cyclic defect or excess.
On Graphs with Disjoint Dominating and 2-Dominating Sets
A DD2-pair of a graph G is a pair (D,D2) of disjoint sets of vertices of G such that D is a dominating set and D2 is a 2-dominating set of G. Although there are infinitely many graphs that do not contain a DD2-pair, we show that every graph with minimum degree at least two has a DD2-pair. We provide a constructive characterization of trees that have a DD2-pair and show that K3,3 is the only connected graph with minimum degree at least three for which D ∪ D2 necessarily contains all vertices of the...
On graphs with given neighbourhoods
On graphs with isomorphic, non-isomorphic and connected -neighbourhoods
On graphs with maximum size in their switching classes
In his PhD thesis [Structural aspects of switching classes, Leiden Institute of Advanced Computer Science, 2001] Hage posed the following problem: “characterize the maximum size graphs in switching classes”. These are called s-maximal graphs. In this paper, we study the properties of such graphs. In particular, we show that any graph with sufficiently large minimum degree is s-maximal, we prove that join of two s-maximal graphs is also an s-maximal graph, we give complete characterization of triangle-free...
On graphs with nonisomorphic -neighbourhoods
On graphs with prescribed edge neighbourhoods
On graphs with prescribed subgraphs of order , and a theorem of Kelly and Merriell
On graphs with restricted link graphs and the chromatic number at most
On graphs with the largest Laplacian index
Let be a connected simple graph on vertices. The Laplacian index of , namely, the greatest Laplacian eigenvalue of , is well known to be bounded above by . In this paper, we give structural characterizations for graphs with the largest Laplacian index . Regular graphs, Hamiltonian graphs and planar graphs with the largest Laplacian index are investigated. We present a necessary and sufficient condition on and for the existence of a -regular graph of order with the largest Laplacian...
On Graver's Conjecture Concerning the Rigidity Problem of Graphs.
On Groups with Several Doubly-Transitive Permuation Representations.
On growth rates of closed permutation classes.
On growth rates of permutations, set partitions, ordered graphs and other objects.
On -closed graphs
On ()-Decompositions of a Complete Graph
On Hadwiger number of a graph
On Hales-Jewett's theorem.