Classification by girth of three-dimensional algebraically defined monomial graphs over the real numbers

Kodess, Alex; Kronenthal, Brian G.; Manzano-Ruiz, Diego; Noe, Ethan

doi:10.1016/j.disc.2020.112286

Cited by 4 publications

(7 citation statements)

References 10 publications

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Section: Introductionmentioning

confidence: 67%

“…While Theorems 5 and 8 subsume the classification obtained in [12], those results are still useful. Indeed, since the results of [12] deal specifically with monomials, [12] features a classification in which the girth of any monomial graph can be instantly determined by examination. In contrast, the results of this paper are of importance because they apply to a substantially broader family of graphs, but due to this generality, determining which category a given graph falls under may require a bit of analysis.…”

Section: Introductionmentioning

confidence: 67%

“…In addition, Kronenthal and Lazebnik [17] and Kronenthal, Lazebnik, and Williford [18] studied families of polynomial graphs over algebraically closed fields of characteristic zero and applied some of their techniques to graphs over finite fields; these results were recently extended by Cheng, Tang, and Xu [2]. Moreover, Kodess, Kronenthal, Manzano-Ruiz, and Noe [12] classified monomial graphs over the real numbers, and Ganger, Golden, Kronenthal, and Lyons [6] studied a two-dimensional analogue. A number of questions related to connectivity, diameter, and isomorphisms of similarly constructed directed graphs, as well as a peculiar result on the number of roots of certain polynomials in finite fields, were considered by Kodess [11], Kodess and Lazebnik [13,14], Kodess, Lazebnik, Smith, and Sporre [15], and Coulter, De Winter, Kodess, and Lazebnik [3].…”

Section: Introductionmentioning

confidence: 99%

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On the Girth of Three-Dimensional Algebraically Defined Graphs with Multiplicatively Separable Functions

Kodess

Kronenthal

Wong

2022

Electron. J. Combin.

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For a field $\mathbb{F}$ and functions $f,g,h,j\colon\mathbb{F}\to \mathbb{F}$, we define $\Gamma_\mathbb{F}(f(X)h(Y),g(X) j(Y))$ to be a bipartite graph where each partite set is a copy of $\mathbb{F}^3$, and a vertex $(a,a_2,a_3)$ in the first partite set is adjacent to a vertex $[x,x_2,x_3]$ in the second partite set if and only if \[a_2+x_2=f(a)h(x) \quad \text{and} \quad a_3+x_3=g(a)j(x).\] In this paper, we completely classify all such graphs by girth in the case $h=j$ (subject to some mild restrictions on $h$). We also present a partial classification when $h\neq j$ and provide some applications.

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Section: Introductionmentioning

confidence: 67%

Section: Introductionmentioning

confidence: 67%

Section: Introductionmentioning

confidence: 99%

See 1 more Smart Citation

On the Girth of Three-Dimensional Algebraically Defined Graphs with Multiplicatively Separable Functions

Kodess

Kronenthal

Wong

2022

Electron. J. Combin.

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“…We will use equality (4). Note that the system ((bc )), when a runs over all elements of F except b.…”

Section: Case 1 (I ): Supposementioning

confidence: 99%

A New Family of Algebraically Defined Graphs With Small Automorphism Group

Lazebnik¹,

Taranchuk²

2021

Preprint

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Let p be an odd prime, q = p e , e ≥ 1, and F = F q denote the finite field of q elements. Let f : F 2 → F and g : F 3 → F be functions, and let P and L be two copies of the 3-dimensional vector space F 3 . Consider a bipartite graph Γ F (f, g) with vertex partitions P and L and with edges defined as follows: for everyThe following question appeared in Nassau [9]: Given Γ F (f, g), is it always possible to find a function h : F 2 → F such that the graph Γ F (f, h) with the same vertex set as Γ F (f, g) and with edges (p)[l] defined in a similar way by the system p 2 + l 2 = f (p 1 , l 1 ) and p 3 + l 3 = h(p 1 , l 1 ), is isomorphic to Γ F (f, g) for infinitely many q? In this paper we show that the answer to the question is negative and the graphs Γ Fp (p 1 ℓ 1 , p 1 ℓ 1 p 2 (p 1 + p 2 + p 1 p 2 )) provide such an example for p ≡ 1 (mod 3). Our argument is based on proving that the automorphism group of these graphs has order p, which is the smallest possible order of the automorphism group of graphs of the form Γ F (f, g).

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“…For the details of this connection, generalizations of these graphs to other dimensions, and applications in finite geometries, extremal graph theory, and cryptography, see surveys by Lazebnik and Woldar [5], a more recent one by Lazebnik, Sun and Wang [6], and many references therein. For recent results not discussed in [6], see Nassau [9]; Kodess, Kronenthal, Manzano-Ruiz and Noe [4]; Xu, Cheng, and Tang [12].…”

Section: Introductionmentioning

confidence: 99%

A New Family of Algebraically Defined Graphs with Small Automorphism Group

Lazebnik

Taranchuk²

2022

Electron. J. Combin.

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Let $p$ be an odd prime, $q=p^e$, $e \geq 1$, and $\mathbb{F} = \mathbb{F}_q$ denote the finite field of $q$ elements. Let $f: \mathbb{F}^2\to \mathbb{F}$ and $g: \mathbb{F}^3\to \mathbb{F}$ be functions, and let $P$ and $L$ be two copies of the 3-dimensional vector space $\mathbb{F}^3$. Consider a bipartite graph $\Gamma_\mathbb{F} (f, g)$ with vertex partitions $P$ and $L$ and with edges defined as follows: for every $(p)=(p_1,p_2,p_3)\in P$ and every $[l]= [l_1,l_2,l_3]\in L$, $\{(p), [l]\} = (p)[l]$ is an edge in $\Gamma_\mathbb{F} (f, g)$ if $$p_2+l_2 =f(p_1,l_1) \;\;\;\text{and}\;\;\; p_3 + l_3 = g(p_1,p_2,l_1).$$The following question appeared in Nassau: Given $\Gamma_\mathbb{F} (f, g)$, is it always possible to find a function $h:\mathbb{F}^2\to \mathbb{F}$ such that the graph $\Gamma_\mathbb{F} (f, h)$ with the same vertex set as $\Gamma_\mathbb{F} (f, g)$ and with edges $(p)[l]$ defined in a similar way by the system $$p_2+l_2 =f(p_1,l_1) \;\;\;\text{and}\;\;\; p_3 + l_3 = h(p_1,l_1),$$ is isomorphic to $\Gamma_\mathbb{F} (f, g)$ for infinitely many $q$? In this paper we show that the answer to the question is negative and the graphs $\Gamma_{\mathbb{F}_p}(p_1\ell_1, p_1\ell_1p_2(p_1 + p_2 + p_1p_2))$ provide such an example for $p \equiv 1 \pmod{3}$. Our argument is based on proving that the automorphism group of these graphs has order $p$, which is the smallest possible order of the automorphism group of graphs of the form $\Gamma_{\mathbb{F}}(f, g)$.

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Classification by girth of three-dimensional algebraically defined monomial graphs over the real numbers

Cited by 4 publications

References 10 publications

On the Girth of Three-Dimensional Algebraically Defined Graphs with Multiplicatively Separable Functions

On the Girth of Three-Dimensional Algebraically Defined Graphs with Multiplicatively Separable Functions

A New Family of Algebraically Defined Graphs With Small Automorphism Group

A New Family of Algebraically Defined Graphs with Small Automorphism Group

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