The main objective of this paper is to develop Krein-Milman-type theorems and Ulam-type stability results for graphs. To establish these results, we introduce several meaningful definitions of vertex-weighted convex graphs inspired by the concept of sequential convexity. We also present a close relationship between the two discrete structures, namely sequential convexity and perfect binary trees. We show that if a graph satisfies a certain convexity property approximately, then this property can be made exact by minimally perturbing the weights assigned to its vertices. Furthermore, we study several structural characterisations, formulate convex minorants for weighted graphs, and derive sandwich-type results. Special emphasis is placed on trees, and an investigation of extremal value problems is also carried out
For a given geometric graph-convexity on a graph $G$ equipped with a weight function on the vertices with value in $\mathbb{Z}$, the Max Weight Convex Set problem consists in determining the convex set $S$ with maximum weight (sum of the weight of the vertices in $S$). Although the problem is NP-complete in general, it...
Fariza Aklouche, Pierre Bergé, M. Habib· 0 citations
In this paper, we establish formulas and sharp bounds for general Kulli-Basava indices and characterize graphs that attain these bounds. These indices have been shown to possess strong discriminating power for distinguishing nonisomorphic chemical structures. They are based on the edge neighborhood degrees of vertices...
A Neumaier graph is a non-complete edge-regular graph containing a regular clique; it is called strictly Neumaier if it is not strongly regular. In this paper we present a construction using finite rings that unifies several known results and yields three new families, each containing infinitely many strictly Neumaier...
A. Abiad, W. Castryck, M. De Boeck et al.· 0 citations
This paper introduces new characterizations, decomposition theorems, and stability results for convex sequences. We show that a sequence is convex precisely when its epigraph satisfies a midpoint convexity condition, thereby connecting discrete and geometric notions of convexity. A decomposition result proves that any...
In this work we establish several monotonicity and decomposition results in the framework of random regular graphs. Among other results, we show that, for a wide range of parameters d1≤d2, there exists a coupling of G(n,d1) and G(n,d2) satisfying that G(n,d1)⊆G(n,d2) with high probability, confirming a conjecture of Ga...
Lawrence Hollom, Lyuben Lichev, Adva Mond et al.· The Annals of Applied Probab...· 0 citations
