On the maximal energy tree with two maximum degree vertices

Cycles containing all vertices of maximum degree

For a graph Gand an integer k, denote by Vk the set {u E V(G) I d(u) 2 k}. Veldman proved that if G is a 2-connected graph of order n with n 5 3k 2 and IVkl 5 k, then G has a cycle containing all vertices of Vk. It is shown that the upper bound k on IVkl is close to best possible in general. For the special case k = A(G), it is conjectured that the condition lVkl I k can be omitted. Using a var...

Star multigraphs with three vertices of maximum degree

The graphs we consider here are either simple graphs, that is they have no loops or multiple edges, or are multigraphs, that is they may have more than one edge joining a pair of vertices, but again have no loops. In particular we shall consider a special kind of multigraph, called a star-multigraph: this is a multigraph which contains a vertex v*, called the star-centre, which is incident with...

Reducing the maximum degree of a graph by deleting vertices

We investigate the smallest number λ(G) of vertices that need to be removed from a non-empty graph G so that the resulting graph has a smaller maximum degree. We prove that if n is the number of vertices, k is the maximum degree, and t is the number of vertices of degree k, then λ(G) ≤ n+(k−1)t 2k . We also show that λ(G) ≤ n k+1 if G is a tree. These bounds are sharp. We provide other bounds t...

Vertices of high degree in the preferential attachment tree ∗

We study the basic preferential attachment process, which generates a sequence of random trees, each obtained from the previous one by introducing a new vertex and joining it to one existing vertex, chosen with probability proportional to its degree. We investigate the number Dt(`) of vertices of each degree ` at each time t, focussing particularly on the case where ` is a growing function of t...

On the Maximal Degree of the K-Step Obrechkoff’s Method