A remark on self-centroidal graphs

Centroidal bases in graphs

We introduce the notion of a centroidal locating set of a graph G, that is, a set L of vertices such that all vertices in G are uniquely determined by their relative distances to the vertices of L. A centroidal locating set of G of minimum size is called a centroidal basis, and its size is the centroidal dimension CD(G). This notion, which is related to previous concepts, gives a new way of ide...

A Remark on Total Domination Critical Graphs

A graph G with no isolated vertex is total domination vertex critical if for any vertex v of G that is not adjacent to a vertex of degree one, the total domination number of G− v is less than the total domination number of G. We call these graphs γt-critical. In this paper, we disprove a conjecture posed in a recent paper(On an open problem concerning total domination critical graphs, Expo. Mat...

A remark on the connectedness of spheres in Cayley graphs

The aim of this note is to prove an elementary yet useful property of nitely presented groups. This property is called connected spheres in Blachère's work [2] (where he shows that the Heisenberg group has this property). Filimonov & Kleptsyn [6] use this remark to get some nice results on certain groups of di eomorphisms of the circle. Recall that, for a nitely generated group Γ and S ⊂ Γ a ni...

Remark on Forgotten Topological Index of a Line Graphs

Let G be a simple connected graph with n vertices and m edges and let d(e1) > d(e2) > · · · > d(em) be edge degree sequence of graph G. Denote by EF = ∑m i=1 d(ei) 3 reformulated forgotten index of G. Lower and upper bounds for the invariant EF are obtained.

A Remark on a Remark by Macaulay or Enhancing Lazard Structural Theorem