A total dominating set in a graph $G$ is a subset $X$ of $V(G)$ such that each vertex of $V(G)$ is adjacent to at least one vertex of $X$. The total domination number of $G$ is the minimum cardinality of a total dominating set. A function $f\colon V(G)\rightarrow \{-1,1\}$ is a signed dominating function (SDF) if the sum of its function values over any closed neighborhood is at least one. The weight of an SDF is the sum of its function values over all vertices. The signed domination number of $G$ is the minimum weight of an SDF on $G$. In this paper we present several upper bounds on the algebraic connectivity of a connected graph in terms of the total domination and signed domination numbers of the graph. Also, we give lower bounds on the Laplacian spectral radius of a connected graph in terms of the signed domination number of the graph.
We prove that a connected Riemannian manifold admitting a pair of nontrivial Einstein-Weyl structures (g, ±ω) with constant scalar curvature is either Einstein, or the dual field of ω is Killing. Next, let (Mn , g) be a complete and connected Riemannian manifold of dimension at least 3 admitting a pair of Einstein-Weyl structures (g,±ω). Then the Einstein-Weyl vector field E (dual to the 1-form ω) generates an infinitesimal harmonic transformation if and only if E is Killing.