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On the structure of compact graphs

Nikandish R., Shaveisi F.

Opuscula Mathematica

2017

Vol. 37, no. 6

875--886

A simple graph G is called a compact graph if G contains no isolated vertices and for each pair x, y of non-adjacent vertices of G, there is a vertex z with N(x) ∪ N(y) ⊆ N(z), where N(v) is the neighborhood of v, for every vertex v of G. In this paper, compact graphs with sufficient number of edges are studied. Also, it is proved that every regular compact graph is strongly regular. Some results about cycles in compact graphs are proved, too. Among other results, it is proved that if the ascending chain condition holds for the set of neighbors of a compact graph G, then the descending chain condition holds for the set of neighbors of G.

On the multiplicative Zagreb coindex of graphs

Xu K., Das K. Ch., Tang K. K.

Opuscula Mathematica

2013

Vol. 33, no. 1

191-204

For a (molecular) graph G with vertex set V (G) and edge set E(G), the first and second Zagreb indices of G are defined as [formula] and [formula] respectively, where dG(v) is the degree of vertex v in G. The alternative expression of M1 (G) is [formula]. Recently Ashrafi, Doslic and Hamzeh introduced two related graphical invariants [formula] and [formula] named as first Zagreb coindex and second Zagreb coindex, respectively. Here we define two new graphical invariants [formula] and [formula] as the respective multiplicative versions of [formula]. In this paper, we have reported some properties, especially upper and lower bounds, for these two graph invariants of connected (molecular) graphs. Moreover, some corresponding extremal graphs have been characterized with respect to these two indices.

Vertices of small degrees in random recursive dags

Dondajewski M., Szymański J.

Fasciculi Mathematici

1998

Nr 28

31-41

In this paper random recursive dags ( directed acyclic graphs) are considered. For such combinatorial structures the expected number of vertices of small outdegrees as well as the degree of a given vertex are studied.