Counting unlabeled k-trees

Abstract

We count unlabeled k-trees by properly coloring them in k + 1 colors and then counting orbits of these colorings under the action of the symmetric group on the colors.

Extremal properties of distance-based graph invariants for $k$-trees

“…The notion of the n-vertex k-tree T k n for k 1 was first introduced by Beineke and Pippert [6] in 1969 (its definition is given in the following). The k-tree has been considered fully in mathematical literature, one may be referred to [3], [29], [33], [35], [42] and their references. In particular, Estes and Wei [12] determined sharp bounds on the sum of squares of the vertex degrees and the sum of the products of degrees of pairs of adjacent vertices of n vertex k-trees.…”

Graph limits of random unlabelled k-trees

“…However, counting general unlabelled k-trees (k 3) was a long-standing open problem, which was only recently solved by Gainer-Dewar [13] using -species. A simpler proof that combines front-colourings with hedronlabellings was later discovered by Gainer-Dewar and Gessel [14]. The advantage of this approach is that front-colouring breaks the symmetry of unlabelled k-trees and avoids the use of compatible cyclic orientation of each (k + 1)-clique in a k-tree.…”

“…The advantage of this approach is that front-colouring breaks the symmetry of unlabelled k-trees and avoids the use of compatible cyclic orientation of each (k + 1)-clique in a k-tree. Based on the simplified generating functions from [14], Drmota and the first author [7] have undertaken a systematic asymptotic analysis of unlabelled k-trees using singularity analysis.…”

“…The substraction operation in the resulting functional equations of generating series severely complicates any follow-up probabilistic analyses, since the corresponding Boltzmann sampling procedures have to employ a costly rejection process. In order to tackle this, we combine the front-colouring method by Gainer-Dewar and Gessel [14] with the framework of unlabelled R-enriched trees by the second author [27] and the cycle-pointing method by Bodirsky, Fusy, Kang and Vigerske [5]. The latter approach is based on the idea of considering graphs marked at a cyclic permutation of some of the vertices, such that the cycle appears in at least one automorphism.…”

The Laplacian energy and Laplacian eigenvalues of the K-trees

“…Recently there is a lot of work on the Laplacian eigenvalues, especially the Laplacian index of a graph. This paper is concerned with the Laplacian energy of k-trees [6]. The class of k − trees may be defined recursively: a k-tree is either a complete graph on k vertices or a graph obtained from a smaller k-tree by adjoining a new vertex together with k edges connecting it to a k-clique.…”