When is NEPS of graphs connected?

On the components of NEPS of connected bipartite graphs

Back in 1983, D. Cvetković posed the conjecture that the components of NEPS of connected bipartite graphs are almost cospectral. In 2000, we showed that this conjecture does not hold for infinitely many bases of NEPS, and we posed a necessary condition on the base of NEPS for NEPS to have almost cospectral components. At the same time, D. Cvetković posed weaker version of his original conjectur...

Comparing the Zagreb indices of the NEPS of graphs

The first and the second Zagreb indices of a graph G = (V,E) are defined as M1(G) = ∑ u∈V dG(u) 2 and M2(G) = ∑ uv ∈ EdG(u)dG(v), where dG(u) denotes the degree of a vertex u in G. It has recently been conjectured that M1(G)/|V | ≤ M2(G)/|E|. Although some counterexamples have already been found, the question of characterizing graphs for which the inequality holds is left open. We show that thi...

The poset on connected graphs is Sperner

The maximal total irregularity of some connected graphs

The total irregularity of a graph G is defined as 〖irr〗_t (G)=1/2 ∑_(u,v∈V(G))▒〖|d_u-d_v |〗, where d_u denotes the degree of a vertex u∈V(G). In this paper by using the Gini index, we obtain the ordering of the total irregularity index for some classes of connected graphs, with the same number of vertices.

When every $P$-flat ideal is flat

In this paper‎, ‎we study the class of rings in which every $P$-flat‎ ‎ideal is flat and which will be called $PFF$-rings‎. ‎In particular‎, ‎Von Neumann regular rings‎, ‎hereditary rings‎, ‎semi-hereditary ring‎, ‎PID and arithmetical rings are examples of $PFF$-rings‎. ‎In the context domain‎, ‎this notion coincide with‎ ‎Pr"{u}fer domain‎. ‎We provide necessary and sufficient conditions for‎...