The degrees, number of edges, spectral radius and weakly Hamilton-connectedness of bipartite graphs

TL;DR

This paper investigates conditions based on degrees, edges, and spectral radius that ensure a balanced bipartite graph is weakly Hamilton-connected, extending classical Hamiltonian concepts to bipartite graphs.

Contribution

It introduces new spectral and degree-based criteria for weakly Hamilton-connectedness in balanced bipartite graphs, a less explored area.

Findings

Spectral radius bounds imply weakly Hamilton-connectedness.

Degree and edge count conditions guarantee weakly Hamilton-connectedness.

Provides theoretical conditions for bipartite graph connectivity.

Abstract

A path of a graph $G$ is called a Hamilton path if it passes through all the vertices of $G$. A graph is Hamilton-connected if any two vertices are connected by a Hamilton path. Note that any bipartite graph is not Hamilton-connected. We consider the weak version of Hamilton-connected property among bipartite graphs. A weakly Hamilton-connected graph is a balanced bipartite graph $G=(X,Y,E)$ that there is a Hamilton path between any vertex $x\in X$ and $y\in Y$. In this paper, we present some degrees, number of edges, and spectral radius conditions for a simple balanced bipartite graph to be weakly Hamilton-connected.