Let \(G\) be a graph with vertex set \(V\). A set \(D \subseteq V\) is a total restrained dominating set of \(G\) if every vertex in \(V\) has a neighbor in \(D\) and every vertex in \(V-D\) has a neighbor in \(V-D\). The minimum cardinality of a total restrained dominating set of \(G\) is called the total restrained domination number of \(G\), denoted by \(\gamma_{tr}(G)\). Cyman and Raczek \((2006)\) showed that if \(G\) is a connected graph of order \(n\) and minimum degree \(\delta\) such that \(2 \leq \delta \leq n-2\), then \(\gamma_{tr}(G) \leq n-\delta\). In this paper, we first introduce the concept of max-min total restrained domination number, denoted by \(\gamma_{tr}^M(G)\), of \(G\), and extend the above result by showing that \(\gamma_{tr}^M(G) \leq \gamma_{tr}(G) \leq n-\delta\). We then proceed to establish that \((1)\) \(\gamma_{tr}^M(G) \leq n-2\delta\) if \(n \geq 11\) and \(G\) contains a cut-vertex, and \((2)\) \(\gamma_{tr}(G) \leq n-4\) if \(n \geq 11\) and \(\delta \geq 2\).

In response surface analysis, it is generally assumed that the observations are independent and there is no effect of neighbouring units. But under the situation when the units are placed linearly with no gaps, the experimental units may experience neighbour or overlap effects from neighbouring units. Hence, for proper specification it is important to include the neighbour effects in the model. First order response surface mode! with neighbour effects from immediate left and right neighbouring units has been considered here and the conditions have been derived for the orthogonal estimation of coefficients of this model. The variance of estimated response has also been obtained and conditions for first order response surface model with neighbour effects to be rotatable have been obtained. A method of obtaining designs satisfying the derived conditions has been proposed. A first order rotatable design with neighbour effects using half replicate of \(2^3\) has also been given.

In [J. Guo, K. Wang, A construction of pooling designs with high degree of error correction, J. Combin. Theory Ser. A \(118(2011) 2056-2058]\), Guo and Wang proposed a new model for disjunct matrices. As a generalization of Guo-Wang’s designs, we obtain a
new family of pooling designs. Our designs and Guo-Wang’s designs have the same numbers of items and pools, but the error-tolerance property of our design is better than that of Guo-Wang’s designs under some conditions.

A \({vertex \;irregular\; total \;labeling}\) \(\sigma\) of a graph \(G\) is a labeling of vertices and edges of \(G\) with labels from the set \(\{1, 2, \ldots, k\}\) in such a way that for any two different vertices \(x\) and \(y\), their weights \(wt(x)\) and \(wt(y)\) are distinct. The \({weight}\) \(wt(x)\) of a vertex \(x\) in \(G\) is the sum of its label and the labels of all edges incident with \(x\). The minimum \(k\) for which the graph \(G\) has a vertex irregular total labeling is called the \({total \;vertex\; irregularity \;strength}\) of \(G\). In this paper, we study the total vertex irregularity strength for two families of graphs, namely Jahangir graphs and circulant graphs.

The Sum-Balaban index is defined as
\[SJ(G) = \frac{|E(G)|}{\mu+1} \sum\limits_{uv \in E(G)} \frac{1}{\sqrt{D_G(u)+D_G(v)}}\],
where \(\mu\) is the cyclomatic number of \(G\) and \(D_G(u)=\sum_{u\in V(G)}d_G(u,v)\). In this paper, we characterize the tree with the maximum Sum-Balaban index among all trees with \(n\) vertices and diameter \(d\). We also provide a new proof of the result that the star \(S_n\) is the graph which has the maximum Sum-Balaban index among all trees with \(n\) vertices. Furthermore, we propose a problem for further research.

A connected graph \(G = (V, E)\) is called a quasi-unicycle graph if there exists \(v_0 \in V\) such that \(G – v_0\) is a unicycle graph. Denote by \(\mathcal{G}(n, d_0)\) the set of quasi-unicycle graphs of order \(n\) with the vertex \(v_0\) of degree \(d_0\) such that \(G – v_0\) is a unicycle graph. In this paper, we determine the maximum spectral radii of quasi-unicycle graphs in \(\mathcal{G}(n, d_0)\).

Let \(Diag(G)\) and \(D(G)\) be the degree-diagonal matrix and distance matrix of \(G\), respectively. Define the multiplier \(Diag(G)D(G)\) as the degree distance matrix of \(G\). The degree distance of \(G\) is defined as \(D'(G) = \sum_{x \in V(G)} d_G(x) D(x)\), where \(d_G(u)\) is the degree of vertex \(x\), \(D_G(x)=\sum_{u\in V(G)}d_G(u,x)\) and \(d_G(u,x)\) is the distance between \(u\) and \(v\). Obviously, \(D'(G)\) is also the sum of elements of the degree distance matrix \(Diag(G)D(G)\) of \(G\). A connected graph \(G\) is a cactus if any two of its cycles have at most one common vertex. Let \(\mathcal{G}(n,r)\) be the set of cacti of order \(n\) and with \(r\) cycles. In this paper, we give the sharp lower bound of the degree distance of cacti among \(\mathcal{G}(n,r)\), and characterize the corresponding extremal cactus.

We introduce the concept of molds, which together with an appropriate weight function, gives all the information of a regular tournament. We use the molds to give a shorter proof of the characterization of domination graphs than the one given in \([4, 5]\), We also use the molds to give a lower and an upper bound of the dichromatic number for all regular tournaments with the same mold.

In this paper, we prove that every countable set of formulas of the propositional logic has at least one equivalent independent subset. We illustrate the situation by considering axioms for Boolean algebras; the proof of independence we give uses model forming.

In this paper, we introduce a new type of graph labeling known as \({super\; mean \;labeling}\). We investigate the super mean labeling for the Complete graph \(K_n\), the Star \(K_{1,n}\), the Cycle \(C_{2n+1}\), and the graph \(G_1 \cup G_2\), where \(G_1\) and \(G_2\) are super mean graphs, as well as some standard graphs.