On the number of independent subsets in trees with restricted degrees

Andriantiana, Eric Ould Dadah; Wagner, Stephan

doi:10.1016/j.mcm.2010.10.003

Cited by 6 publications

(2 citation statements)

References 19 publications

Supporting

Mentioning

Contrasting

Order By: Relevance

“…unconstrained) matchings; in particular, we show that on (not necessarily phylogenetic) trees with in total n nodes, maximum degree 3 and no degree-2 nodes adjacent to each other, there can be at most O(1.5895 n ) matchings. This supplements existing upper bounds of O(φ n ), valid for arbitrary trees, and O(1.5538 n ) for trees of maximum degree 3 where all interior nodes have degree 3 (see [2,Theorem 1 and Remark 5]; the precise constant is 1 + √ 2). In Section 2 we give preliminaries.…”

Section: Introductionsupporting

confidence: 71%

Convex characters, algorithms and matchings

Kelk¹,

Meuwese²,

Wagner³

2021

Preprint

Self Cite

View full text Add to dashboard Cite

Phylogenetic trees are used to model evolution: leaves are labelled to represent contemporary species ("taxa") and interior vertices represent extinct ancestors. Informally, convex characters are measurements on the contemporary species in which the subset of species (both contemporary and extinct) that share a given state, form a connected subtree. In [19] it was shown how to efficiently count, list and sample certain restricted subfamilies of convex characters, and algorithmic applications were given. We continue this work in a number of directions. First, we show how combining the enumeration of convex characters with existing parameterised algorithms can be used to speed up exponential-time algorithms for the maximum agreement forest problem in phylogenetics. Second, we re-visit the quantity g2(T ), defined as the number of convex characters on T in which each state appears on at least 2 taxa. We use this to give an algorithm with running time O(φ n • poly(n)), where φ ≈ 1.6181 is the golden ratio and n is the number of taxa in the input trees, for computation of maximum parsimony distance on two state characters. By further restricting the characters counted by g2(T ) we open an interesting bridge to the literature on enumeration of matchings. By crossing this bridge we improve the running time of the aforementioned parsimony distance algorithm to O(1.5895 n •poly(n)), and obtain a number of new results in themselves relevant to enumeration of matchings on at-most binary trees.

show abstract

Section: Introductionsupporting

confidence: 71%

Convex characters, algorithms and matchings

Kelk¹,

Meuwese²,

Wagner³

2021

Preprint

Self Cite

View full text Add to dashboard Cite

show abstract

“…Wang determined a class of unicyclic graphs and obtained the ordering of their Hosoya and Merrifield-Simmons indexes [139]. It turns out that graphs of minimal Hosoya index coincide with those of maximal Merrifield-Simmons index [125,140,141]. The absolute magnitudes of the coefficients of the HP and the matching polynomial of a caterpillar graph are identical to those of the sextet and resonance polynomials of a benzenoid system.…”

Section: Hosoya Polynomialmentioning

confidence: 99%

Counting Polynomials in Chemistry: Past, Present, and Perspectives

Joița,

Tomescu,

Jäntschi

2023

Symmetry

View full text Add to dashboard Cite

Counting polynomials find their way into chemical graph theory through quantum chemistry in two ways: as approximate solutions to the Schrödinger equation or by storing information in a mathematical form and trying to find a pattern in the roots of these expressions. Coefficients count how many times a property occurs, and exponents express the extent of the property. They help understand the origin of regularities in the chemistry of specific classes of compounds. Our objective is to accelerate the research of newcomers into chemical graph theory. One problem in understanding these concepts is in the different approaches and notations of each research study; some researchers provide online tools for computing these mathematical concepts, but these need to be maintained for functionality. We take advantage of similar mathematical aspects of 14 such polynomials that merge theoretical chemistry and pure mathematics; give examples, differences, and similarities; and relate them to recent research.

show abstract

The Fibonacci numbers of certain subgraphs of circulant graphs

Dosal-Trujillo

Galeana‐Sánchez

2015

AKCE International Journal of Graphs and Combinatorics

View full text Add to dashboard Cite

On the number of independent subsets in trees with restricted degrees

Cited by 6 publications

References 19 publications

Convex characters, algorithms and matchings

Convex characters, algorithms and matchings

Counting Polynomials in Chemistry: Past, Present, and Perspectives

The Fibonacci numbers of certain subgraphs of circulant graphs

Contact Info

Product

Resources

About