The Waring’s problem over finite fields through generalized Paley graphs

Podestá, Ricardo A.; Videla, Denis E.

doi:10.1016/j.disc.2021.112324

Cited by 9 publications

(5 citation statements)

References 35 publications

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“…As in proof of Theorem 3.1 of [23], we can show that (e , p) = 1 and e ≡ 1 (mod 4). Indeed, notice that (20) implies that e +1 ≡ e (mod 4) and by hypothesis e 1 ≡ 1 (mod 4) and so e ≡ 1 (mod 4) for all ≥ 1. On the other hand, we have that (e 1 , p) = 1 by hypothesis.…”

Section: The Numbermentioning

confidence: 98%

“…As in the proof of Theorem 3.2, we can put z e 1 ,f 1 =: z e ,f , where e , f are defined recursively as follows (20) e +1 = e 1 e − 4f 1 f and…”

Section: The Numbermentioning

confidence: 99%

“…The graphs Γ(3, q) and Γ(4, q) are also of interest (see [23]). The GP-graphs have been extensively studied in the few past years (see for instance [2,3,17,20,22,[25][26][27]. )…”

Section: Introductionmentioning

confidence: 99%

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Number of cliques of Paley-type graphs over finite commutative local rings

Gallo,

Videla

2024

Discuss. Math. Graph Theory

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In this work, given (R, m) a finite commutative local ring with identity and k ∈ N with (k, |R|) = 1, we study the number of cliques of any size in theUsing the known fact that the graph G R (k) can be obtained by blowing-up the vertices of G Fq (k) a number |m| of times, we reduce the study of the number of cliques in G R (k) over the local ring R to the computation of the number of cliques of G R/m (k) over the finite residue field R/m F q . In this way, using known numbers of -cliques of generalized Paley graphs (k = 2, 3, 4 and = 3, 4), we obtain several explicit results for the number of -cliques over finite commutative local rings with identity.

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Section: The Numbermentioning

confidence: 98%

“…As in the proof of Theorem 3.2, we can put z e 1 ,f 1 =: z e ,f , where e , f are defined recursively as follows (20) e +1 = e 1 e − 4f 1 f and…”

Section: The Numbermentioning

confidence: 99%

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Number of cliques of Paley-type graphs over finite commutative local rings

Gallo,

Videla

2024

Discuss. Math. Graph Theory

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“…The number of walks in GP-graphs are related with the number of solutions of diagonal equations over finite fields ([31]). Moreover, the diameter of Γ(k, q), if it exists, coincides with the Waring number g(k, q) (see [25], [26]). Under some mild restrictions, the spectrum of GP-graphs determines the weight distribution of their associated irreducible codes ( [21], [23]).…”

Section: Introductionmentioning

confidence: 99%

Generalized Paley graphs equienergetic with their complements

Podestá¹,

Videla²

2022

Preprint

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We consider generalized Paley graphs Γ(k, q), generalized Paley sum graphs Γ + (k, q), and their corresponding complements Γ(k, q) and Γ+ (k, q), for k = 3, 4. Denote by Γ = Γ * (k, q) either Γ(k, q) or Γ + (k, q). We compute the spectra of Γ(3, q) and Γ(4, q) and from them we obtain the spectra of Γ + (3, q) and Γ + (4, q) also. Then we show that, in the non-semiprimitive case, the spectrum of Γ(3, p 3ℓ ) and Γ(4, p 4ℓ ) with p prime can be recursively obtained, under certain arithmetic conditions, from the spectrum of the graphs Γ(3, p) and Γ(4, p) for any ℓ ∈ N, respectively. Using the spectra of these graphs we give necessary and sufficient conditions on the spectrum of Γ * (k, q) such that Γ * (k, q) and Γ * (k, q) are equienergetic for k = 3, 4. In a previous work we have classified all bipartite regular graphs Γ bip and all strongly regular graphs Γsrg which are complementary equienergetic, i.e. {Γ bip , Γbip } and {Γsrg, Γsrg} are equienergetic pairs of graphs. Here we construct infinite pairs of equienergetic non-isospectral regular graphs {Γ, Γ} which are neither bipartite nor strongly regular.

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“…The graphs Γ(3, q) and Γ(4, q) are also of interest (see [19]). The GPgraphs have been extensively studied in the few past years (see for instance [12], [15], [16], [18], [21]. [22], [23], [24].…”

Section: Introductionmentioning

confidence: 99%

Number of cliques of Paley-type graphs over finite commutative local rings

Gallo¹,

Videla²

2023

Preprint

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In this work, given (R, m) a finite commutative local ring with identity and k ∈ N with (k, |R|) = 1, we study the number of cliques of any size in the Cayley graphUsing the known fact that the graph G R (k) can be obtained by blowing-up the vertices of G Fq (k) a number |m| of times, with independence sets the cosets of m, where q is the size of the residue field R/m (see [20]). Then, by using the above blowing-up, we reduce the study of the number of cliques in G R (k) over the local ring R to the computation of the number of cliques of G R/m (k) over the finite residue field R/m ≃ F q . In this way, using known numbers of cliques of generalized Paley graphs (k = 2, 3, 4 and ℓ = 3, 4), we obtain several explicit results for the number of cliques over finite commutative local rings with identity.

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The Waring’s problem over finite fields through generalized Paley graphs

Cited by 9 publications

References 35 publications

Number of cliques of Paley-type graphs over finite commutative local rings

Number of cliques of Paley-type graphs over finite commutative local rings

Generalized Paley graphs equienergetic with their complements

Number of cliques of Paley-type graphs over finite commutative local rings

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