The adjacency matrix of a graph as a data table: a geometric perspective

Chiaselotti, Giampiero; Gentile, Tommaso; Infusino, Federico; Oliverio, Paolo Antonio

doi:10.1007/s10231-016-0608-1

Cited by 30 publications

(19 citation statements)

References 83 publications

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“…the electronic journal of combinatorics 27(1) (2020), #P1. 19 To see that, it suffices to show that (14) implies (13), since the reverse implication is obvious. To this regard, let b ∈ M W (A) and u, u ∈ U S be such that u ≡ A u .…”

Section: Representation Theorem For Closure Systemsmentioning

confidence: 99%

“…However, due to the fact that in the present paper the above sets U , Ω and Λ have only a formal nature, we will use the more mathematical term pairing instead of the previous ones. Hence, in our specific mathematical viewpoint, we can speak of algebraic GrC methods on pairings (for further results on such topics see also [13,14,17]).…”

Section: Introductionmentioning

confidence: 99%

“…On the other hand, the relation ≡ A becomes a type of local symmetry relation with respect to a fixed vertex subset A ⊆ V (G) when one interprets a finite simple undirected graph G with vertex set V (G) as a pairing F, V (G), V (G), {0, 1} on V (G), where F (u, a) := 1 if the vertices u, a ∈ V (G) are adjacent and F (u, a) := 0 otherwise. In [13] such a local symmetry relation has been investigated for some basic graph families and, in particular, a classification theorem for the Petersen graph, concerning all subgraphs induced by the vertex subsets A that are minimal with respect to the property that ≡ A agrees with ≡ V (G) has been proved (these vertex subsets have been called symmetry bases of the graph). Again, in [18] the local symmetry relation ≡ A has been studied in its interrelations with a specific type of binary operation • defined on the power set P(V (G)) and whose automorphism group is isomorphic to a subgroup of Aut(G) (for works on similar topics see also [31,35]).…”

Section: Introductionmentioning

confidence: 99%

“…Again, in [18] the local symmetry relation ≡ A has been studied in its interrelations with a specific type of binary operation • defined on the power set P(V (G)) and whose automorphism group is isomorphic to a subgroup of Aut(G) (for works on similar topics see also [31,35]). Based on the particular graph interpretation introduced in [13], the same applies to the general case. Therefore, we call ≡ A the A-symmetry relation and π W (A) the A-symmetry partition of the pairing W. Moreover, we also consider the equivalence relation ≈ W on the power set P(Ω) defined by A ≈ W A : ⇐⇒ π W (A) = π W (A ),…”

Section: Introductionmentioning

confidence: 99%

“…Based on recent results obtained in the graph context [13,14] and on other general results linked to theoretical computer science [1,2,12,15,25,26,36,37,39], in this paper we continue the study of the basic property of the set system M(W) and use methods from GrC in relation to order-theory, limited to finite lattices.…”

Section: Introductionmentioning

confidence: 99%

See 4 more Smart Citations

Lattice Representations with Set Partitions Induced by Pairings

Chiaselotti¹,

Gentile²,

Infusino³

2020

Electron. J. Combin.

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We call a quadruple $\mathcal{W}:=\langle F,U,\Omega,\Lambda \rangle$, where $U$ and $\Omega$ are two given non-empty finite sets, $\Lambda$ is a non-empty set and $F$ is a map having domain $U\times \Omega$ and codomain $\Lambda$, a pairing on $\Omega$. With this structure we associate a set operator $M_{\mathcal{W}}$ by means of which it is possible to define a preorder $\ge_{\mathcal{W}}$ on the power set $\mathcal{P}(\Omega)$ preserving set-theoretical union. The main results of our paper are two representation theorems. In the first theorem we show that for any finite lattice $\mathbb{L}$ there exist a finite set $\Omega_{\mathbb{L}}$ and a pairing $\mathcal{W}$ on $\Omega_\mathbb{L}$ such that the quotient of the preordered set $(\mathcal{P}(\Omega_\mathbb{L}), \ge_\mathcal{W})$ with respect to its symmetrization is a lattice that is order-isomorphic to $\mathbb{L}$. In the second result, we prove that when the lattice $\mathbb{L}$ is endowed with an order-reversing involutory map $\psi: L \to L$ such that $\psi(\hat 0_{\mathbb{L}})=\hat 1_{\mathbb{L}}$, $\psi(\hat 1_{\mathbb{L}})=\hat 0_{\mathbb{L}}$, $\psi(\alpha) \wedge \alpha=\hat 0_{\mathbb{L}}$ and $\psi(\alpha) \vee \alpha=\hat 1_{\mathbb{L}}$, there exist a finite set $\Omega_{\mathbb{L},\psi}$ and a pairing on it inducing a specific poset which is order-isomorphic to $\mathbb{L}$.

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Section: Representation Theorem For Closure Systemsmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%