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On rational radii coin representations of the wheel graph

Geir Agnarsson, Jill Bigley Dunham (2013)

Discussiones Mathematicae - General Algebra and Applications

A flower is a coin graph representation of the wheel graph. A petal of a flower is an outer coin connected to the center coin. The results of this paper are twofold. First we derive a parametrization of all the rational (and hence integer) radii coins of the 3-petal flower, also known as Apollonian circles or Soddy circles. Secondly we consider a general n-petal flower and show there is a unique irreducible polynomial Pₙ in n variables over the rationals ℚ, the affine variety of which contains the...

On semiregular digraphs of the congruence x k ≡ y ( mod n )

Lawrence Somer, Michal Křížek (2007)

Commentationes Mathematicae Universitatis Carolinae

We assign to each pair of positive integers n and k ≥ 2 a digraph G ( n , k ) whose set of vertices is H = { 0 , 1 , ⋯ , n - 1 } and for which there is a directed edge from a ∈ H to b ∈ H if a k ≡ b ( mod n ) . The digraph G ( n , k ) is semiregular if there exists a positive integer d such that each vertex of the digraph has indegree d or 0. Generalizing earlier results of the authors for the case in which k = 2 , we characterize all semiregular digraphs G ( n , k ) when k ≥ 2 is arbitrary.

On some characterizations of strong power graphs of finite groups

A. K. Bhuniya, Sudip Bera (2016)

Special Matrices

Let G be a finite group of order n. The strong power graph Ps(G) of G is the undirected graph whose vertices are the elements of G such that two distinct vertices a and b are adjacent if am1=bm2 for some positive integers m1, m2 < n. In this article we classify all groups G for which Ps(G) is a line graph. Spectrum and permanent of the Laplacian matrix of the strong power graph Ps(G) are found for any finite group G.

On the composition factors of a group with the same prime graph as B n ( 5 )

Azam Babai, Behrooz Khosravi (2012)

Czechoslovak Mathematical Journal

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 ≥ 6 , then G has a unique nonabelian composition factor isomorphic to B n ( 5 ) or C n ( 5 ) .

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 ∩ Y ≠ 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...

Currently displaying 241 – 260 of 430