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.

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.

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.

The total irregularity of a graph G is defined as irrIn this paper we give (sharp) upper bounds on the total irregularity of graphs under several graph operations including join, lexicographic product, Cartesian product, strong product, direct product, corona product, disjunction and symmetric difference.

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