Graphs with chromatic roots in the interval .
Graphs with convex domination number close to their order
For a connected graph G = (V,E), a set D ⊆ V(G) is a dominating set of G if every vertex in V(G)-D has at least one neighbour in D. The distance between two vertices u and v is the length of a shortest (u-v) path in G. An (u-v) path of length is called an (u-v)-geodesic. A set X ⊆ V(G) is convex in G if vertices from all (a-b)-geodesics belong to X for any two vertices a,b ∈ X. A set X is a convex dominating set if it is convex and dominating. The convex domination number of a graph G is the...
Graphs with disjoint dominating and paired-dominating sets
A dominating set of a graph is a set of vertices such that every vertex not in the set is adjacent to a vertex in the set, while a paired-dominating set of a graph is a dominating set such that the subgraph induced by the dominating set contains a perfect matching. In this paper, we show that no minimum degree is sufficient to guarantee the existence of a disjoint dominating set and a paired-dominating set. However, we prove that the vertex set of every cubic graph can be partitioned into a dominating...
Graphs with equal domination and 2-distance domination numbers
Let G = (V,E) be a graph. The distance between two vertices u and v in a connected graph G is the length of the shortest (u-v) path in G. A set D ⊆ V(G) is a dominating set if every vertex of G is at distance at most 1 from an element of D. The domination number of G is the minimum cardinality of a dominating set of G. A set D ⊆ V(G) is a 2-distance dominating set if every vertex of G is at distance at most 2 from an element of D. The 2-distance domination number of G is the minimum cardinality...
Graphs with extremal Randić index when the minimum degree of vertices is two
Graphs with four boundary vertices.
Graphs with given degree sequence and maximal spectral radius.
Graphs with given subgraphs represent all categories
Graphs with given subgraphs represent all categories. II.
Graphs with greatest number of matchings.
Graphs with large double domination numbers
In a graph G, a vertex dominates itself and its neighbors. A subset S ⊆ V(G) is a double dominating set of G if S dominates every vertex of G at least twice. The minimum cardinality of a double dominating set of G is the double domination number . If G ≠ C₅ is a connected graph of order n with minimum degree at least 2, then we show that and we characterize those graphs achieving equality.
Graphs with Large Generalized (Edge-)Connectivity
The generalized k-connectivity κk(G) of a graph G, introduced by Hager in 1985, is a nice generalization of the classical connectivity. Recently, as a natural counterpart, we proposed the concept of generalized k-edge-connectivity λk(G). In this paper, graphs of order n such that [...] for even k are characterized.
Graphs with least eigenvalue -2 attaining a convex quadratic upper bound for the stability number
Graphs with Least Eigenvalue at Least -sqrt(3)
Graphs with many copies of a given subgraph.
Graphs with maximum and minimum independence numbers.
Graphs with nonisomorphic vertex neighbourhoods of the first and second types
Graphs with prescribed maximal subgraphs and critical chromatic graphs
Graphs with prescribed neighbourhood graphs
Graphs with prescribed size and vertex parities