On the possible values of the entropy of undirected graphs

Gadouleau, Maximilien

doi:10.1002/jgt.22213

Cited by 3 publications

(4 citation statements)

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“…Let F be a graph in F 1 . By [9], the values of the asymptotic guessing number of graphs can be enumerated as (a i : i ∈ N) where a i < a i+1 for all i ∈ N. We have a 0 = 0, a 1 = 1, a 2 = 2, a 3 = 2.5, . .…”

Section: Guessing Number and Forbidden Subgraphsmentioning

confidence: 99%

“…Let A be a maximum independent set in F and B = V (F ) \ A. By Lemma 3.2 of [9], there exists a nonempty Corollary 34. For any a > 0 there exists a unique finite family of minimal forbidden subgraphs F a such that, for any graph G, gn(G) < a ⇔ G is F a -free.…”

Section: Guessing Number and Forbidden Subgraphsmentioning

confidence: 99%

“…It is also related to the problem of index coding with side information [2,10]. Gadouleau showed that this problem can also be recast in terms of fixed points of finite dynamical systems [9]. Christofides and Markström were the first to expand the study of guessing numbers of undirected graphs [6].…”

Section: Introductionmentioning

confidence: 99%

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Guessing Numbers and Extremal Graph Theory

Martin¹,

Rombach

2022

Electron. J. Combin.

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For a given number of colors, $s$, the guessing number of a graph is the (base $s$) logarithm of the cardinality of the largest family of colorings of the vertex set of the graph such that the color of each vertex can be determined from the colors of the vertices in its neighborhood. This quantity is related to problems in network coding, circuit complexity and graph entropy. We study the guessing number of graphs as a graph property in the context of classic extremal questions, and its relationship to the forbidden subgraph property. We find the extremal number with respect to the property of having guessing number $\leq a$, for fixed $a$. Furthermore, we find an upper bound on the saturation number for this property, and a method to construct further saturated graphs that lie between these two extremes. We show that, for a fixed number of colors, bounding the guessing number is equivalent to forbidding a finite set of subgraphs.

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Section: Guessing Number and Forbidden Subgraphsmentioning

confidence: 99%

Section: Guessing Number and Forbidden Subgraphsmentioning

confidence: 99%

See 1 more Smart Citation

Guessing Numbers and Extremal Graph Theory

Martin¹,

Rombach

2022

Electron. J. Combin.

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Section: Introductionmentioning

confidence: 99%

Guessing Numbers and Extremal Graph Theory

Martin¹,

Rombach²

2020

Preprint

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For a given number of colors, s, the guessing number of a graph is the (base s) logarithm of the cardinality of the largest family of colorings of the vertex set of the graph such that the color of each vertex can be determined from the colors of the vertices in its neighborhood. This quantity is related to problems in network coding, circuit complexity and graph entropy. We study the guessing number of graphs as a graph property in the context of classic extremal questions, and its relationship to the forbidden subgraph property. We find the extremal number with respect to the property of having guessing number ≤ a, for fixed a. Furthermore, we find an upper bound on the saturation number for this property, and a method to construct further saturated graphs that lie between these two extremes. We show that, for a fixed number of colors, bounding the guessing number is equivalent to forbidding a finite set of subgraphs.

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On the influence of the interaction graph on a finite dynamical system

Gadouleau

2019

Nat Comput

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A finite dynamical system (FDS) is a system of multivariate functions over a finite alphabet, that is typically used to model a network of interacting entities. The main feature of a finite dynamical system is its interaction graph, which indicates which local functions depend on which variables; the interaction graph is a qualitative representation of the interactions amongst entities on the network. As such, a major problem is to determine the effect of the interaction graph on the dynamics of the FDS. In this paper, we are interested in three main properties of an FDS: the number of images (the so-called rank), the number of periodic points (the so-called periodic rank) and the number of fixed points. In particular, we investigate the minimum, average, and maximum number of images (or periodic points, or fixed points) of FDSs with a prescribed interaction graph and a given alphabet size; thus yielding nine quantities to study. The paper is split into two parts. The first part considers the minimum rank, for which we derive the first meaningful results known so far. In particular, we show that the minimum rank decreases with the alphabet size, thus yielding the definition of an absolute minimum rank. We obtain lower and upper bounds on this absolute minimum rank, and we give classification results for graphs with very low (or highest) rank. The second part is a comprehensive survey of the results obtained on the nine quantities described above. We not only give a review of known results, but we also give a list of relevant open questions. *

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On the possible values of the entropy of undirected graphs

Cited by 3 publications

References 31 publications

Guessing Numbers and Extremal Graph Theory

Guessing Numbers and Extremal Graph Theory

Guessing Numbers and Extremal Graph Theory

On the influence of the interaction graph on a finite dynamical system

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