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.

The \({corona}\) of two graphs \(G\) and \(H\), written as \(G \odot H\), is defined as the graph obtained by taking one copy of \(G\) and \(|V(G)|\) copies of \(H\), and joining by an edge the \(i\)th vertex of \(G\) to every vertex in the \(i\)th copy of \(H\). In this paper, we present the explicit formulae of the (modified) Schultz and Zagreb indices in the corona of two graphs.

A geodetic (resp. monophonic) dominating set in a connected graph \(G \) is any set of vertices of \(G\) which is both a geodetic (resp.monophonic) set and a dominating set in \(G\). This paper establishes some relationships between geodetic domination and monophonic domination in a graph. It also investigates the geodetic domination and monophonic domination in the join, corona and composition of
connected graphs.