The general sum-connectivity index of a graph $G$ is defined as $chi_{alpha}(G)= sum_{uvin E(G)} (d_u + d_{v})^{alpha}$ where $d_{u}$ is degree of the vertex $uin V(G)$, $alpha$ is a real number different from $0$ and $uv$ is the edge connecting the vertices $u,v$. In this note, the problem of characterizing the graphs having extremum $chi_{alpha}$ values from a certain collection of polyomino ...

This paper generalises the concept of vertex pancyclic graphs. We define a graph as set-pancyclic if for every set S of vertices there is a cycle of every possible length containing S. We show that if the minimum degree of a graph exceeds half its order then the graph is set-pancyclic. We define a graph as k-ordered-pancyclic if, for every set S of cardinality k and every cyclic ordering of S, ...

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...

Let Hn be the class of unlabeled trees with n vertices, and denote by Hn a tree that is drawn uniformly at random from this set. The asymptotic behavior of the random variable degk(Hn) that counts vertices of degree k in Hn was studied, among others, by Drmota and Gittenberger in [3], who showed that this quantity satisfies a central limit theorem. This result provides a very precise characteri...

We prove that every minimal brick on n vertices has at least n/9 vertices of degree at most 4.

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...

Let Tn be a random recursive tree with n nodes. List vertices of Tn in decreasing order of degree as v1, . . . , vn, and write di and hi for the degree of vi and the distance of vi from the root, respectively. We prove that, as n→∞ along suitable subsequences, ( d − blog2 nc, hi − μ lnn √ σ2 lnn ) → ((Pi, i ≥ 1), (Ni, i ≥ 1)) , where μ = 1 − (log2 e)/2, σ2 = 1 − (log2 e)/4, (Pi, i ≥ 1) is a Poi...

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...