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Let $G$ be a finite group. The prime graph of $G$ is a graph whose vertex set is the set of prime divisors of $| G |$ and two distinct primes $p$ and $q$ are joined by an edge, whenever $G$ contains an element of order $p q$ . The prime graph of $G$ is denoted by $Γ (G)$ . It is proved that some finite groups are uniquely determined by their prime graph. In this paper, we show that if $G$ is a finite group such that $Γ (G) = Γ (B_{n} (5))$ , where $n \geq 6$ , then $G$ has a unique nonabelian composition factor isomorphic to $B_{n} (5)$ or $C_{n} (5)$ .

On the Corona of Two Graphs.

Frank Harary, Roberto Frucht (1970)

Aequationes mathematicae

On the Corona of Two Graphs. (Short Communication).

Frank Harary, Roberto Frucht (1970)

Aequationes mathematicae

On the determining number and the metric dimension of graphs.

Cáceres, José, Garijo, Delia, Puertas, María Luz, Seara, Carlos (2010)

The Electronic Journal of Combinatorics [electronic only]

On the diameter of the intersection graph of a finite simple group

Xuanlong Ma (2016)

Czechoslovak Mathematical Journal

Let $G$ be a finite group. The intersection graph $Δ_{G}$ of $G$ is an undirected graph without loops and multiple edges defined as follows: the vertex set is the set of all proper nontrivial subgroups of $G$ , and two distinct vertices $X$ and $Y$ are adjacent if $X \cap Y \neq 1$ , where $1$ denotes the trivial subgroup of order $1$ . A question was posed by Shen (2010) whether the diameters of intersection graphs of finite non-abelian simple groups have an upper bound. We answer the question and show that the diameters of intersection...

On the energy of unitary Cayley graphs.

Ramaswamy, H.N., Veena, C.R. (2009)

The Electronic Journal of Combinatorics [electronic only]

On the function “sandwiched" between $α (G)$ and $\bar{χ} (G)$ .

Dobrynin, V.Y. (1997)

The Electronic Journal of Combinatorics [electronic only]

On the graphs of Hoffman-Singleton and Higman-Sims.

Hafner, Paul R. (2004)

The Electronic Journal of Combinatorics [electronic only]

On the intersection graph of a commutative distributive groupoid

Bedřich Pondělíček (1979)

Mathematica Slovaca

On the intersection graph of a finite group

Hossein Shahsavari, Behrooz Khosravi (2017)

Czechoslovak Mathematical Journal

For a finite group $G$ , the intersection graph of $G$ which is denoted by $Γ (G)$ is an undirected graph such that its vertices are all nontrivial proper subgroups of $G$ and two distinct vertices $H$ and $K$ are adjacent when $H \cap K \neq 1$ . In this paper we classify all finite groups whose intersection graphs are regular. Also, we find some results on the intersection graphs of simple groups and finally we study the structure of $Aut (Γ (G))$ .

On the intersection graphs of ideals of direct product of rings

Nader Jafari Rad, Sayyed Heidar Jafari, Shamik Ghosh (2014)

Discussiones Mathematicae - General Algebra and Applications

In this paper we first calculate the number of vertices and edges of the intersection graph of ideals of direct product of rings and fields. Then we study Eulerianity and Hamiltonicity in the intersection graph of ideals of direct product of commutative rings.

On the intersection of subgroups of a free group.

Gábor Tardos (1992)

Inventiones mathematicae

On the isoperimetry of graphs with many ends

Christophe Pittet (1998)

Colloquium Mathematicae

Let X be a connected graph with uniformly bounded degree. We show that if there is a radius r such that, by removing from X any ball of radius r, we get at least three unbounded connected components, then X satisfies a strong isoperimetric inequality. In particular, the non-reduced $l^{2}$ -cohomology of X coincides with the reduced $l^{2}$ -cohomology of X and is of uncountable dimension. (Those facts are well known when X is the Cayley graph of a finitely generated group with infinitely many ends.)

On the maximum automorphism group of self-complementary graphs

Tomasz Łuczak (2000)

Mathematica Slovaca

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