Regular character degree graphs

Zuccari, Claudio Paolo Morresi

doi:10.1016/j.jalgebra.2014.03.042

Cited by 12 publications

(4 citation statements)

References 14 publications

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“…Morresi-Zuccari showed that Tong-viet's conjecture is true for solvable groups [25,Theorem A]. Also in [27, Theorem A], Sayanjaly et al proved that a regular character-graph ∆(G) with odd order, of a group G, is a complete graph, which confirm the Tong-viet's conjecture in some cases.…”

Section: Introductionmentioning

confidence: 83%

Regular character-graphs whose eigenvalues are greater than or equal to -2

Ebrahimi¹,

Khatami²,

Mirzaei³

2021

Preprint

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Let G be a finite group and Irr(G) be the set of all complex irreducible characters of G. The character-graph ∆(G) associated to G, is a graph whose vertex set is the set of primes which divide the degrees of some characters in Irr(G) and two distinct primes p and q are adjacent in ∆(G) if the product pq divides χ(1), for some χ ∈ Irr(G). Tong-viet posed the conjecture that if ∆(G) is k-regular for some integer k 2, then ∆(G) is either a complete graph or a cocktail party graph. In this paper, we show that his conjecture is true for all regular character-graphs whose eigenvalues are in the interval [−2, ∞).

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Section: Introductionmentioning

confidence: 83%

Regular character-graphs whose eigenvalues are greater than or equal to -2

Ebrahimi¹,

Khatami²,

Mirzaei³

2021

Preprint

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“…Now, we prove that ∆(G) has 3 distinct eigenvalues if and only if (i) holds. Since G is a solvable group, with respect to the results in [21], regularity of ∆(G) implies that it is either a complete or (n − 2)-regular graph (i.e. ∆(G) = K n or K n − M for a perfect matching M ).…”

Section: Theorem 5 Let G Be a Solvable Group The Number Of Distinct E...mentioning

confidence: 99%

The influence of cut vertices and eigenvalues on character graphs of solvable groups

Hafezieh,

Hosseinzadeh,

Hossein-Zadeh

et al. 2019

Preprint

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Given a finite group G, the character graph, denoted by ∆(G), for its irreducible character degrees is a graph with vertex set ρ(G) which is the set of prime numbers that divide the irreducible character degrees of G, and with {p, q} being an edge if there exist a nonlinear χ ∈ Irr(G) whose degree is divisible by pq. In this paper, we discuss the influences of cut vertices and eigenvalues of ∆(G) on the group structure of G. Recently, Lewis and Meng proved the character graph of each solvable group has at most one cut vertex. Now, we determine the structure of character graphs of solvable groups with a cut vertex and diameter 3. Furthermore, we study solvable groups whose character graphs have at most two distinct eigenvalues. Moreover, we investigate the solvable groups whose character graphs are regular with three distinct eigenvalues. In addition, we give some lower bounds for the number of edges of ∆(G).

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“…This graph, which is called the prime graph of G, has the primes dividing degrees in cd(G) as its vertices, and there is an edge between p and q if pq divides some degree a ∈ cd(G). The regularity of the prime graph Δ(G) of a finite group G has been studied in [18,21]. In [21], it has been shown that Δ(G) is 3-regular if and only if it is a complete graph with four vertices and then in [18], the question is answered completely in the case of solvable groups.…”

Section: Introductionmentioning

confidence: 99%

“…The regularity of the prime graph Δ(G) of a finite group G has been studied in [18,21]. In [21], it has been shown that Δ(G) is 3-regular if and only if it is a complete graph with four vertices and then in [18], the question is answered completely in the case of solvable groups. In fact it has been shown that if G is a solvable group with a regular prime graph of order k, then Δ(G) is either complete or (k − 2)-regular.…”

Section: Introductionmentioning

confidence: 99%

Finite Groups Whose Common-Divisor Graph is Regular

Ghaffarzadeh¹,

Ghasemi²,

Lewis³

2018

Proceedings of the Edinburgh Mathematical Society

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Let G be a finite group, and write cd (G) for the set of degrees of irreducible characters of G. The common-divisor graph Γ(G) associated with G is the graph whose vertex set is cd (G)∖{1} and there is an edge between distinct vertices a and b, if (a, b) > 1. In this paper we prove that if Γ(G) is a k-regular graph for some k ⩾ 0, then for the solvable groups, either Γ(G) is a complete graph of order k + 1 or Γ(G) has two connected components which are complete of the same order and for the non-solvable groups, either k = 0 and cd(G) = cd(PSL2(2f)), where f ⩾ 2 or Γ(G) is a 4-regular graph with six vertices and cd(G) = cd(Alt7) or cd(Sym7).

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Regular character degree graphs

Cited by 12 publications

References 14 publications

Regular character-graphs whose eigenvalues are greater than or equal to -2

Regular character-graphs whose eigenvalues are greater than or equal to -2

The influence of cut vertices and eigenvalues on character graphs of solvable groups

Finite Groups Whose Common-Divisor Graph is Regular

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