In this paper, we mainly define a semidirect product version of the Schützenberger product and also a new two-sided semidirect product construction for arbitrary two monoids. Then, as main results, we present a generating and a relator set for these two products. Additionally, to explain why these products have been defined, we investigate the regularity for the semidirect product version of Schützenberger products and the subgroup separability for this new two-sided semidirect product.

We consider the connected graphs with a unique vertex of maximum degree \(3\). Two subfamilies of such graphs are characterized and ordered completely by their indices. Moreover, a conjecture about the complete ordering of all graphs in this set is proposed.

Let \(G = (V(G), E(G))\) be a simple graph and \(T(G)\) be the set of vertices and edges of \(G\). Let \(C\) be a \(k\)-color set. A (proper) total \(k\)-coloring \(f\) of \(G\) is a function \(f: T(G) \rightarrow C\) such that no adjacent or incident elements of \(T(G)\) receive the same color. For any \(u \in V(G)\), denote \(C(u) = \{f(u)\} \cup \{f(uv) | uv \in E(G)\}\). The total \(k\)-coloring \(f\) of \(G\) is called the adjacent vertex-distinguishing if \(C(u) \neq C(v)\) for any edge \(uv \in E(G)\). And the smallest number of colors is called the adjacent vertex-distinguishing total chromatic number \(\chi_{at}(G)\) of \(G\). Let \(G\) be a connected graph. If there exists a vertex \(v \in V(G)\) such that \(G – v\) is a tree, then \(G\) is a \(1\)-tree. In this paper, we will determine the adjacent vertex-distinguishing total chromatic number of \(1\)-trees.

In this paper, we extend the study on packing and covering of complete directed graph \(D_t\) with Mendelsohn triples \([6]\). Mainly, the maximum packing of \(D_t-P\) and \(D_t\cup{P}\) with Mendelsohn triples are obtained respectively, where \(P\) is a vertex-disjoint union of directed cycles in \(D_t\).

In the theory of orthogonal arrays, an orthogonal array is called schematic if its rows form an association scheme with respect to Hamming distances. Which orthogonal arrays are schematic orthogonal arrays and how to classify them is an open problem proposed by Hedayat et al. \([12]\). In this paper, we study the Hamming distances of the rows in orthogonal arrays and construct association schemes according to the distances. The paper gives the partial solution of the problem by Hedayat et al. for symmetric and some asymmetric orthogonal arrays of strength two.

The Padmakar-Ivan \((PI)\) index is a Wiener-Szeged-like topological index which reflects certain structural features of organic molecules. In this paper, we study the PI index of gated amalgam.

The nullity of a graph is the multiplicity of the eigenvalue zero in its spectrum. In this paper, we give formulae to calculate the nullity of \(n\)-vertex bicyclic graphs by means of the maximum matching number.

This note calculates the essential norm of a recently introduced integral-type operator from the Hilbert-Bergman weighted space \(A^2_\alpha(\mathbb{B}), \alpha \geq -1\) to a Bloch-type space on the unit ball \(\mathbb{B} \subset \mathbb{C}^n\).

Let \(G\) be a graph and let \(\sigma_k(G)\) be the minimum degree sum of an independent set of \(k\) vertices. For \(S \subseteq V(G)\) with \(|S| \geq k\), let \(\Delta_k(S)\) denote the maximum value among the degree sums of the subset of \(k\) vertices in \(S\). A cycle \(C\) of a graph \(G\) is said to be a dominating cycle if \(V(G \setminus C)\) is an independent set. In \([2]\), Bondy showed that if \(G\) is a \(2\)-connected graph with \(\sigma_3(G) \geq |V(G)| + 2\), then any longest cycle of \(G\) is a dominating cycle. In this paper, we improve it as follows: if \(G\) is a 2-connected graph with \(\Delta_3(S) \geq |V(G)| + 2\) for every independent set \(S\) of order \(\kappa(G) + 1\), then any longest cycle of \(G\) is a dominating cycle.

Let \(B\) be an \(m \times n\) array in which each symbol appears at most \(k\) times. We show that if \(k \leq \frac{n(n-1)}{8(m+n-2)} + 1\) then \(B\) has a transversal.