Group-Annihilator Graphs Realised by Finite Abelian Groups and Its Properties
Tóm tắt
Let G be a finite abelian group viewed a
$${\mathbb {Z}}$$
-module and let
$${\mathcal {G}} = (V, E)$$
be a simple graph. In this paper, we consider a graph
$$\Gamma (G)$$
called as a group-annihilator graph. The vertices of
$$\Gamma (G)$$
are all elements of G and two distinct vertices x and y are adjacent in
$$\Gamma (G)$$
if and only if
$$[x : G][y : G]G = \{0\}$$
, where
$$x, y\in G$$
and
$$[x : G] = \{r\in {\mathbb {Z}} : rG \subseteq {\mathbb {Z}}x\}$$
is an ideal of a ring
$${\mathbb {Z}}$$
. We discuss in detail the graph structure realised by a group G. Moreover, we study the creation sequence, hyperenergeticity and hypoenergeticity of group-annihilator graphs. Finally, we conclude the paper with a discussion on Laplacian eigen values of the group-annihilator graph. We show that the Laplacian eigen values are representatives of orbits of the group action:
$$Aut(\Gamma (G)) \times G \rightarrow G$$
.
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