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The atom-bond connectivity (ABC) index is one of the recently most investigated degree-based molecular structure descriptors, that have applications in chemistry. For a graph G, the ABC index is defined as uv∈E(G), where d(u) is the degree of vertex u in G and E(G) is the set of edges of G. Despite many attempts in the last few years, it is still an open problem to characterize trees with minimal ABC index. In this paper, we present an efficient approach of computing trees with minimal ABC index, by considering the degree sequences of trees and some known properties of trees with minimal ABC index. The obtained results disprove some existing conjectures and suggest new ones to be set.

The atom-bond connectivity (ABC) index is a degree-based molecular descriptor, that found chemical applications. It is well known that among all connected graphs, the graphs with minimal ABC index are trees. A complete characterization of trees with minimal ABC index is still an open problem. In this paper, we present new structural properties of trees with minimal ABC index. Our main results reveal that trees with minimal ABC index do not contain so-called B k -branches, with k ≥ 5, and that they do not have more than four B 4 -branches.

Albertson [4] has defined the irregularity of a simple undirected graph G as irr(G) = uv∈E(G) |d G (u) − d G (v)| , where d G (u) denotes the degree of a vertex u ∈ V (G). Recently, in [1] a new measure of irregularity of a graph, so-called the total irregularity, was defined as irr tHere, we compare the irregularity and the total irregularity of graphs. For a connected graph G with n vertices, we show that irr t (G) ≤ n 2 irr(G)/4. Moreover, if G is a tree, then irr t (G) ≤ (n − 2)irr(G).

The problem of data transmission in communication network can be transformed into the problem of fractional factor existing in graph theory. In recent years, the data transmission problem in the specific network conditions has received a great deal of attention, and it raises new demands to the corresponding mathematical model. Under this background, many advanced results are presented on fractional critical deleted graphs and fractional ID deleted graphs. In this paper, we determine that G is a fractional (g, f, n , m)-, and G). Furthermore, the independent set neighborhood union condition for a graph to be fractional ID-(g, f, m)-deleted is raised. Some examples will be manifested to show the sharpness of independent set neighborhood union conditions.

Albertson [3] has defined the irregularity of a simple undirected graph G = (V, E) as irr(G) = uv∈E |d G (u) − d G (v)| , where d G (u) denotes the degree of a vertex u ∈ V. Recently, this graph invariant gained interest in the chemical graph theory, where it occured in some bounds on the first and the second Zagreb index, and was named the third Zagreb index [12]. For general graphs with n vertices, Albertson has obtained an asymptotically tight upper bound on the irregularity of 4n 3 /27. Here, by exploiting a different approach than in [3], we show that for general graphs with n vertices the upper bound n 3 2n 3 2n 3 − 1 is sharp. We also present lower bounds on the maximal irregularity of graphs with fixed minimal and/or maximal vertex degrees, and consider an approximate computation of the irregularity of a graph.

Graph Theory International audience In this note a new measure of irregularity of a graph G is introduced. It is named the total irregularity of a graph and is defined as irr(t)(G) - 1/2 Sigma(u, v is an element of V(G)) vertical bar d(G)(u) - d(G)(v)vertical bar, where d(G)(u) denotes the degree of a vertex u is an element of V(G). All graphs with maximal total irregularity are determined. It is also shown that among all trees of the same order the star has the maximal total irregularity.