Spectrum, distance spectrum, and Wiener index of wreath products of complete graphs

Donno, Alfredo

doi:10.26493/1855-3974.1117.7ba

Cited by 11 publications

(9 citation statements)

References 33 publications

(38 reference statements)

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“…When the considered wreath product is vertex-transitive (as in the cases studied in [4,5,13]), the graph is distance-balanced and, according to Proposition 6, the total distance for every vertex is constant and can be easily deduced from the Wiener index by Equation (2). It is possible to check that, in these special cases, the value of the total distance via the Wiener index computation in [7] coincides with the one provided by Equation (3) in Theorem 17.…”

Section: Distances In a Wreath Productmentioning

confidence: 99%

“…Moreover, we mention that in [22] the notion of strongly distance-balanced graph has been introduced, and it has been investigated in [2,22] for graph compositions. The Wiener index of a wreath product has been studied in [7], and in [4,13] for some special families of graphs. Spectral computations have been performed in [5,13].…”

Section: Introductionmentioning

confidence: 99%

“…The Wiener index of a wreath product has been studied in [7], and in [4,13] for some special families of graphs. Spectral computations have been performed in [5,13]. However, to the best of our knowledge, the total distance, the opportunity index and distance-balanced property have never been investigated for wreath products.…”

Section: Introductionmentioning

confidence: 99%

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Total Distance, Wiener Index and Opportunity Index in Wreath Products of Star Graphs

Cavaleri

Donno

Scozzari

2019

Electron. J. Combin.

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In the last decades much attention has turned towards centrality measures on graphs. The Wiener index and the total distance are key tools to investigate the median vertices, the distance-balanced property and the opportunity index of a graph. This interest has recently been addressed to graphs obtained via classical graph products like the Cartesian, the direct, the strong and the lexicographic product. We extend this study to a relatively new graph product, that is, the wreath product. In this paper, we compute the total distance for the vertices of an arbitrary wreath product graph $G\wr H$ in terms of the total distances in $H$ and of some distance-based indices of $G$. We explicitly compute these indices for the star graph $S_n$, providing a closed formula for the total distances in $S_n\wr H$ when $H$ is complete or a star. As a consequence, we obtain the Wiener index of these graphs, we characterize the median and the central vertices, and finally we give an upper and a lower bound for the opportunity index of $S_n\wr S_m$ in terms of tail conditional expectations of an associated binomial distribution.

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Section: Distances In a Wreath Productmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

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Total Distance, Wiener Index and Opportunity Index in Wreath Products of Star Graphs

Cavaleri

Donno

Scozzari

2019

Electron. J. Combin.

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“…Moreover, we want to mention that in [8] a matrix operation has been introduced, called wreath product of matrices, which is a matrix analogue of the wreath product of graphs, since it provides the adjacency matrix of the wreath product of two graphs, when applied to the adjacency matrices of the factors. In the paper [14] the wreath product of two complete graphs has been investigated: an explicit computation of the spectrum and of the Wiener index has been performed. In the paper [5], jointly with F. Belardo, we have considered the wreath product of a complete graph with a Cocktail Party graph, and we have described its spectrum, together with the Zagreb indices and the Randić index.…”

Section: Introductionmentioning

confidence: 99%

Some degree and distance-based invariants of wreath products of graphs

Cavaleri

Donno

2020

Discrete Applied Mathematics

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The wreath product of graphs is a graph composition inspired by the notion of wreath product of groups, with interesting connections with Geometric Group Theory and Probability. This paper is devoted to the description of some degree and distancebased invariants, of large interest in Chemical Graph Theory, for a wreath product of graphs. An explicit formula is obtained for the Zagreb indices, in terms of the Zagreb indices of the factor graphs. A detailed analysis of distances in a wreath product is performed, allowing to describe the antipodal graph and to provide a formula for the Wiener index. Finally, a formula for the Szeged index is obtained. Several explicit examples are given. (2010): 05C07, 05C12, 05C40, 05C76. Mathematics Subject Classification

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“…In [10], this correspondence is proven in the more general context of generalized wreath products of graphs, inspired by the construction introduced in [1] for permutation groups. Also, observe that in [9] the wreath product of matrices has been defined, in order to describe the adjacency matrix of the wreath product of two graphs: spectral computations using this matrix representation have been developed for some infinite families of wreath products in [5,4,11].…”

Section: Introductionmentioning

confidence: 99%

Distance-balanced graphs and travelling salesman problems

Cavaleri¹,

Donno²

2020

Ars Math. Contemp.

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For every probability p ∈ [0, 1] we define a distance-based graph property, the pTSdistance-balancedness, that in the case p = 0 coincides with the standard property of distance-balancedness, and in the case p = 1 is related to the Hamiltonian-connectedness. In analogy with the classical case, where the distance-balancedness of a graph is equivalent to the property of being self-median, we characterize the class of pTS-distance-balanced graphs in terms of their equity with respect to certain probabilistic centrality measures, inspired by the Travelling Salesman Problem. We prove that it is possible to detect this property looking at the classical distance-balancedness (and therefore looking at the classical centrality problems) of a suitable graph composition, namely the wreath product of graphs. More precisely, we characterize the distance-balancedness of a wreath product of two graphs in terms of the pTS-distance-balancedness of the factors.

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Spectrum, distance spectrum, and Wiener index of wreath products of complete graphs

Cited by 11 publications

References 33 publications

Total Distance, Wiener Index and Opportunity Index in Wreath Products of Star Graphs

Total Distance, Wiener Index and Opportunity Index in Wreath Products of Star Graphs

Some degree and distance-based invariants of wreath products of graphs

Distance-balanced graphs and travelling salesman problems

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