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.
A three-valued function $f\: V\rightarrow \{-1,0,1\}$ defined on the vertices of a graph $G=(V,E)$ is a minus total dominating function (MTDF) if the sum of its function values over any open neighborhood is at least one. That is, for every $v\in V$, $f(N(v))\ge 1$, where $N(v)$ consists of every vertex adjacent to $v$. The weight of an MTDF is $f(V)=\sum f(v)$, over all vertices $v\in V$. The minus total domination number of a graph $G$, denoted $\gamma _t^{-}(G)$, equals the minimum weight of an MTDF of $G$. In this paper, we discuss some properties of minus total domination on a graph $G$ and obtain a few lower bounds for $\gamma _t^{-}(G)$.