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.
In this paper we consider the following problem: Over the class of all simple connected unicyclic graphs on $n$ vertices with girth $g$ ($n$, $g$ being fixed), which graph minimizes the Laplacian spectral radius? Let $U_{n,g}$ be the lollipop graph obtained by appending a pendent vertex of a path on $n-g$ $(n> g)$ vertices to a vertex of a cycle on $g\geq 3$ vertices. We prove that the graph $U_{n,g}$ uniquely minimizes the Laplacian spectral radius for $n\geq 2g-1$ when $g$ is even and for $n\geq 3g-1$ when $g$ is odd.
Let µ_{n-1}(G) be the algebraic connectivity, and let µ_{1}(G) be the Laplacian spectral radius of a k-connected graph G with n vertices and m edges. In this paper, we prove that {\mu _{n - 1}}(G) \geqslant \frac{{2n{k^2}}}{{(n(n - 1) - 2m)(n + k - 2) + 2{k^2}}} , with equality if and only if G is the complete graph Kn or Kn − e. Moreover, if G is non-regular, then {\mu _1}(G) < 2\Delta - \frac{{2(n\Delta - 2m){k^2}}}{{2(n\Delta - 2m)({n^2} - 2n + 2k) + n{k^2}}} , where ▵ stands for the maximum degree of G. Remark that in some cases, these two inequalities improve some previously known results., Xiaodan Chen, Yaoping Hou., and Obsahuje seznam literatury
The Laplacian spectral radius of a graph is the largest eigenvalue of the associated Laplacian matrix. In this paper, we improve Shi's upper bound for the Laplacian spectral radius of irregular graphs and present some new bounds for the Laplacian spectral radius of some classes of graphs.