Let $G$ be a graph with $n$ vertices, $m$ edges and a vertex degree sequence $(d_1,d_2,\cdots ,d_n)$, where $d_1\ge d_2\ge \cdots \ge d_n$. The spectral radius and the largest Laplacian eigenvalue are denoted by $\rho \left(G\right)$ and $\mu \left(G\right)$, respectively. We determine the graphs with $$\rho \left(G\right)=\frac{d_n-1}{2}+\sqrt{2m-n{d}_n+\frac{{(d_n+1)}^2}{4}}$$ and the graphs with $d_n\ge 1$ and $$\mu \left(G\right)=d_n+\frac{1}{2}+\sqrt{{\sum }_{i=1}^n{d}_i(d_i-d_n)+{\left(d_n-\frac{1}{2}\right)}^2}.$$ We also present some sharp lower bounds for the Laplacian eigenvalues of a connected graph.

A three-valued function $fV\to \{-1,0,1\}$ 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 $v\in V$, $f\left(N\right(v\left)\right)\ge 1$, where $N\left(v\right)$ consists of every vertex adjacent to $v$. The weight of an MTDF is $f\left(V\right)=\sum f\left(v\right)$, over all vertices $v\in V$. The minus total domination number of a graph $G$, denoted ${\gamma }_t^{-}\left(G\right)$, equals the minimum weight of an MTDF of $G$. In this paper, we discuss some properties of minus total domination on a graph...

A set D of vertices in a graph G = (V,E) is a dominating set of G if every vertex in V-D is adjacent to some vertex in D. The domination number γ(G) of G is the minimum cardinality of a dominating set. We define the cobondage number $b_c\left(G\right)$ of G to be the minimum cardinality among the sets of edges X ⊆ P₂(V) - E, where P₂(V) = X ⊆ V:|X| = 2 such that γ(G+X) < γ(G). In this paper, the exact values of bc(G) for some standard graphs are found and some bounds are obtained. Also, a Nordhaus-Gaddum...

For an ordered set $W=\{w_1,w_2,\cdots ,w_k\}$ of vertices in a connected graph $G$ and a vertex $v$ of $G$, the code of $v$ with respect to $W$ is the $k$-vector $$c_W\left(v\right)=(d(v,w_1),d(v,w_2),\cdots ,d(v,w_k)).$$ The set $W$ is an independent resolving set for $G$ if (1) $W$ is independent in $G$ and (2) distinct vertices have distinct codes with respect to $W$. The cardinality of a minimum independent resolving set in $G$ is the independent resolving number $\mathrm{i}r\left(G\right)$. We study the existence of independent resolving sets in graphs, characterize all nontrivial connected graphs $G$ of order...