Next to the shortest path distance, the second most popular distance function between vertices in a graph is the commute distance (resistance distance). For two vertices u and v, the hitting time Huv is the expected time it takes a random walk to travel from u to v. The commute time is its symmetrized version Cuv = Huv + Hvu. In our paper we study the behavior of hitting times and commute dista...

Let $G$ be a finite group and $cd^*(G)$ be the set of nonlinear irreducible character degrees of  $G$. Suppose that $rho(G)$ denotes the set of primes dividing some element of $cd^*(G)$. The bipartite divisor graph for the set of character degrees which is denoted by $B(G)$, is a bipartite graph whose vertices are the disjoint union of $rho(G)$ and $cd^*(G)$, and a vertex $p in rho(G)$ is conne...

Let G be a connected simple graph, and let f be a mapping from V (G) to the set of integers. This paper is concerned with the existence of a spanning tree in which each vertex v has degree at least f(v). We show that if ∣∣ΓG(S)∣∣− f(S)+ |S| ≥ 1 for any nonempty subset S ⊆ L, then a connected graph G has a spanning tree such that dT (x) ≥ f(x) for all x ∈ V (G), where ΓG(S) is the set of neighbo...