The generating graph $\Gamma(G)$ of a group $G$ is the graph whose vertex set is $G$, where two distinct vertices are adjacent if and only if they generate $G$. In this paper, we systematically study the structure of generating graphs of finite abelian groups (non-cyclic) and determine the set of all generating pairs. Moreover, we give some structural characterizations, in particular, we determine conditions under which $\Gamma(G)$ is regular, characterize when the isolated vertices form a subgroup, and establish necessary and sufficient conditions for two non-isomorphic finite abelian groups $G$ and $H$ to satisfy $\Gamma(G)\cong \Gamma(H)$. Furthermore, we compute the spectra of the adjacency and Laplacian matrices of these graphs.
The generating graph $\Gamma(G)$ of a finite group $G$ has vertex set $G\setminus\{1\}$, and two distinct vertices are adjacent if and only if they generate $G$. Breuer, Guralnick, Lucchini, Maroti and Nagy [Bull. Lond. Math. Soc. 42 (2010), 621--633] conjectured that, for every finite group $G$ with at least four elem...
The difference graph $D(G)$ of a finite group $G$ is obtained from the edge difference between its intersection power graph and power graph, after deleting isolated vertices. This graph has already been studied, with sufficient conditions for connectedness and a diameter bound $6$ for finite groups satisfying those con...
For a finite graph $G$ on $n$ vertices, let $\eta(G)$ denote the least order of a finite abelian group $\Gamma$ for which $G$ is an induced subgraph of some Cayley graph of $\Gamma$. Babai and S\'os (1985) settled the worst-case order of magnitude: it is $\Theta(n^2)$. We treat $\eta$ instead as an invariant of the ind...
Let $G$ be a finite group. The co-intersection graph $\Delta_G ^c$ of $G$ has as its vertices the nontrivial proper subgroups of $G$, with edges joining those pairs of subgroups which intersect trivially. It is clear that every connected component of $\Delta_G ^c$ has diameter at most three. In this Note, we show that...
The essential graph of an abelian group $G$, denoted by $\varepsilon(G)$, is an undirected graph with vertex set as the collection of all non-trivial proper subgroups of $G$ and any two distinct vertices $A$ and $B$ are adjacent if and only if $AB$ is an essential subgroup of $G$. We discuss connectedness, traversabili...
N. Hazarika, K. Rajkhowa, Bikash Barman· International Electronic Jou...· 0 citations
The prime graph of a finite group $G$ is the graph $\Gamma(G)$ with vertex set the set of prime divisors $\pi(G)$ of $|G|$ and an edge between vertices $p, q\in\pi(G)$ if and only if there exists an element $g\in G$ with order $o(g) = pq$. Given a finite nonabelian simple group $T$, a group $G$ is $T$-solvable if there...
Alexa Renner· 0 citations
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