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| Format: | Preprint |
| Published: |
2026
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| Online Access: | https://arxiv.org/abs/2604.20900 |
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| _version_ | 1866918462905909248 |
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| author | Goswami, Angshuman R. |
| author_facet | Goswami, Angshuman R. |
| contents | The primary objective of this paper is to investigate the notions of geometric and sequential convexity within a graph-theoretic framework, with the aim of examining various structural properties and exploring the connection between these two branches of mathematics.
A simple connected vertex-weighted graph $G(V,E)$ with a non-empty set of leaf vertices is said to be star-convex if there exists at least one node $u\in V(G)$ such that, for every chosen leaf vertex $v$, there is a monotone path (either increasing or decreasing) connecting $v$ to $u$. One of the main results states that a graph $G$ is star-convex if and only if there exists a tree $T\subseteq G$ that contains all leaf vertices and is itself star-convex.
On the other hand, a sequence $\big(u_n\big)_{n=0}^{\infty}$ is said to be convex if it satisfies the following inequality $$ 2u_{i}\leq u_{i-1}+u_{i+1}\qquad \mbox{for all}\quad i\in \mathbb{N}. $$ We demonstrate that, under minimal assumptions, a class of convex sequences can be embedded into a spider graph so as to make it star-convex. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2604_20900 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | On Weighted Star--Convex Graphs Goswami, Angshuman R. General Mathematics The primary objective of this paper is to investigate the notions of geometric and sequential convexity within a graph-theoretic framework, with the aim of examining various structural properties and exploring the connection between these two branches of mathematics. A simple connected vertex-weighted graph $G(V,E)$ with a non-empty set of leaf vertices is said to be star-convex if there exists at least one node $u\in V(G)$ such that, for every chosen leaf vertex $v$, there is a monotone path (either increasing or decreasing) connecting $v$ to $u$. One of the main results states that a graph $G$ is star-convex if and only if there exists a tree $T\subseteq G$ that contains all leaf vertices and is itself star-convex. On the other hand, a sequence $\big(u_n\big)_{n=0}^{\infty}$ is said to be convex if it satisfies the following inequality $$ 2u_{i}\leq u_{i-1}+u_{i+1}\qquad \mbox{for all}\quad i\in \mathbb{N}. $$ We demonstrate that, under minimal assumptions, a class of convex sequences can be embedded into a spider graph so as to make it star-convex. |
| title | On Weighted Star--Convex Graphs |
| topic | General Mathematics |
| url | https://arxiv.org/abs/2604.20900 |