arXiv subjects

Valisoa R. M. Rakotonarivo

Publications and source records attributed to Valisoa R. M. Rakotonarivo.

4 recordsLinked to original sources

arXiv2026

Ratio of the number of $\mathbf{1}$-nearly independent vertex subsets and the Merrifield-Simmons index

The number $σ_k(G)$ of induced subgraphs with size $k$ of a graph $G$ was introduced recently as the number of $k$-nearly independent vertex subsets of $G$. Results highlighting similarity and difference in the behaviours of $σ_1$ and $σ_0$, have been reported. In this paper, we provide more comparison tools, by studying the ratio $\frac{σ_1(G)}{σ_{0}(G)}$. We establish sharp lower and upper bounds for this ratio over various classes of graphs, including connected graphs, trees, and forests.

math.CO↗

arXiv2026

Characterization of Trees with Maximum Security

The rank (also known as protection number or leaf-height) of a vertex in a rooted tree is the minimum distance between the vertex and any of its leaf descendants. We consider the sum of ranks over all vertices (known as the security) in binary trees, and produce a classification of families of binary trees for which the security is maximized. In addition, extremal results relating to maximum rank among all vertices in families of trees is discussed.

math.CO↗

arXiv2024

On minimizing the Wiener index of unicyclic graphs with fixed girth and given degree sequence

The Wiener index of a graph is the sum of all the distances between any pair of vertices. We aim to describe graphs which minimize the Wiener index among all unicyclic graphs with fixed girth and given degree sequence. Depending on where the centroid of the graph is, we will present three candidates for the minimization, namely the greedy unicyclic graph, the cycle-centered graph and the out-greedy unicyclic graph.

math.CO↗

arXiv2024

On the Sombor index of trees with degree restrictions

We study the Sombor index of trees with various degree restrictions. In addition to rediscovering that among all trees with a given degree sequence, the greedy tree minimises the Sombor index and the alternating greedy tree maximises it, we also provide a full characterisation of all trees that have those maximum or minimum values. Moreover, we compare trees with different degree sequences and deduce a few corollaries.

math.CO↗