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Structure of Fibonacci cubes: a survey
Sandi Klavžar,Sandi Klavžar +1 more
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A survey on Fibonacci cubes is given with an emphasis on their structure, including representations, recursive construction, hamiltonicity, degree sequence and other enumeration results, and their median nature that leads to a fast recognition algorithm is discussed.Abstract:
The Fibonacci cube Γ
n
is the subgraph of the n-cube induced by the binary strings that contain no two consecutive 1s. These graphs are applicable as interconnection networks and in theoretical chemistry, and lead to the Fibonacci dimension of a graph. In this paper a survey on Fibonacci cubes is given with an emphasis on their structure, including representations, recursive construction, hamiltonicity, degree sequence and other enumeration results. Their median nature that leads to a fast recognition algorithm is discussed. The Fibonacci dimension of a graph, studies of graph invariants on Fibonacci cubes, and related classes of graphs are also presented. Along the way some new short proofs are given.read more
Citations
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Journal ArticleDOI
Equal opportunity networks, distance-balanced graphs, and Wiener game
Kannan Balakrishnan,Boštjan Brešar,Boštjan Brešar,Manoj Changat,Sandi Klavžar,Sandi Klavžar,Sandi Klavžar,Aleksander Vesel,Aleksander Vesel,Petra Žigert Pleteršek,Petra Žigert Pleteršek +10 more
TL;DR: In this note it is proved that a graph G of even order is an equal opportunity graph if and only if it is a distance-balanced graph.
Journal ArticleDOI
On maximum Wiener index of trees and graphs with given radius
TL;DR: An upper bound on Wiener index of trees and graphs in terms of number of vertices n, radius r, and maximum degree is given and the extremal graphs are characterized.
Wiener Index and Hosoya Polynomial of Fibonacci and Lucas Cubes
Sandi Klavzar,Michel Mollard +1 more
TL;DR: In this paper, it was proved that the Wiener index of Fibonacci cubes can be written as the sum of products of four Fibonca numbers, which in turn yields a closed formula for the index.
Journal ArticleDOI
q-cube enumerator polynomial of Fibonacci cubes
Elif Saygı,Ömer Eğecioğlu +1 more
TL;DR: These bivariate polynomials satisfy a recurrence relation similar to the standard one and refine the count of the number of hypercubes of a given dimension in Fibonacci cubes by keeping track of the distances of thehypercubes to the all 0 vertex.
Journal ArticleDOI
The eccentricity sequences of Fibonacci and Lucas cubes
Aline Castro,Michel Mollard +1 more
TL;DR: The generating functions of the eccentricity sequences of @C"n and @L"n are obtained and the number of vertices of a given eccentricity is deduced.
References
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Journal Article
The On-Line Encyclopedia of Integer Sequences.
TL;DR: The On-Line Encyclopedia of Integer Sequences (OEIS) as mentioned in this paper is a database of 13,000 number sequences and is freely available on the Web (http://www.att.com/~njas/sequences/) and is widely used.
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Distance-preserving subgraphs of hypercubes
TL;DR: In this article, the authors give a characterization of connected subgraphs G of hypercubes H such that the distance in G between any two vertices a, b ϵ G is the same as their distance in H.
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Fibonacci cubes-a new interconnection Topology
TL;DR: A novel interconnection topology called the Fibonacci cube is shown to possess attractive recurrent structures in spite of its asymmetric and relatively sparse interconnections.
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Isometric embedding in products of complete graphs
TL;DR: It is shown that any two such embeddings of the same graph G are essentially the same, and a polynomial-time algorithm is given which will find such an embedding if it exists.
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The structure of median graphs
TL;DR: It is proved that a median graph can be obtained from a one-vertex graph by an expansion procedure, and from this characterization some nice properties are derived.