The robustness of networks is an important problem that has been studied in a variety of situations, see for instance [3] or the survey [5] and the references therein. A milestone in the study of this problem was the discovery by Albert et al. of the dramatic effect of targeted attacks on networks whose vertex degrees follow a power-law distribution, like the Internet and the World Wide Web [1]...

In 1968, Vizing made the following two conjectures for graphs which are critical with respect to the chromatic index: (1) every critical graph has a 2-factor, and (2) every independent vertex set in a critical graph contains at most half of the vertices. We prove both conjectures for critical graphs with many edges, and determine upper bounds for the size of independent vertex sets in those gra...

We consider a strict k-colouring of a multigraph G as a surjection f from the vertex set of G into a set of colours {1,2,. . . ,k} such that, for every non-pendant vertex x of G, there exist at least two edges incident to x and coloured by the same colour. The maximum number of colours in a strict edge colouring of G is called the upper chromatic index of G and is denoted by χ(G). In this paper...

the rings considered in this article are  commutative  with identity which admit at least two  nonzero annihilating ideals. let $r$ be a ring. let $mathbb{a}(r)$ denote the set of all annihilating ideals of $r$ and let $mathbb{a}(r)^{*} = mathbb{a}(r)backslash {(0)}$. the annihilating-ideal graph of $r$, denoted by $mathbb{ag}(r)$  is an undirected simple graph whose vertex set is $mathbb{a}(r)...

The first Zagreb index of a graph G, with vertex set V (G) and edge set E(G), is defined as M1(G) = ∑ u∈V (G) d(u) 2 where d(u) denotes the degree of the vertex v. An alternative expression for M1(G) is ∑ uv∈E(G)[d(u) + d(v)]. We consider a multiplicative version of M1 defined as Π∗1(G) = ∏ uv∈E(G)[d(u) + d(v)]. We prove that among all connected graphs with a given number of vertices, the path ...