Let \(D\) be a finite and simple digraph with vertex set \(V(D)\), and let \(f: V(D) \to \{-1, 1\}\) be a two-valued function. If \(\sum_{x \in N_D^-[v]} f(x) \geq 1\) for each \(v \in V(D)\), where \(N_D^-[v]\) consists of \(v\) and all vertices of \(D\) from which arcs go into \(v\), then \(f\) is a signed dominating function on \(D\). The sum \(\sum_{v \in V(D)} f(v)\) is called the weight of \(f\). The signed domination number, denoted by \(\gamma_S(D)\), of \(D\) is the minimum weight of a signed dominating function on \(D\). In this work, we present different lower bounds on \(\gamma_S(D)\) for general digraphs, show that these bounds are sharp, and give an improvement of a known lower bound obtained by Karami in 2009 [H. Karami, S.M. Sheikholeslami, A. Khodkar, Lower bounds on the signed domination numbers of directed graphs, Discrete Math. 309 (2009), 2567-2570]. Some of our results are extensions of well-known properties of the signed domination number of graphs.

Let \(G\) be a graph of order at least \(2k\) and \(s_1, s_2, \ldots, s_k, t_1, t_2, \ldots, t_k\) be any \(2k\) distinct vertices of \(G\). If there exist \(k\) disjoint paths \(P_1, P_2, \ldots, P_k\) such that \(P_i\) is an \(s_i – t_i\) path for \(1 \leq i \leq k\), we call \(G\) \(k\)-linked. K. Kawarabayashi et al. showed that if \(n \geq 4k – 1\) (\(k \geq 2\)) with \(\sigma_2(G) \geq n + 2k – 3\), then \(G\) is \(k\)-linked. Li et al. showed that if \(G\) is a graph of order \(n \geq 232k\) with \(\sigma_2^*(G) \geq n + 2k – 3\), then \(G\) is \(k\)-linked. For sufficiently large \(n\), it implied the result of K. Kawarabayashi et al. The main purpose of this paper is to lower the bound of \(n\) in the result of Li et al. We show that if \(G\) is a graph of order \(n \geq 111k + 9\) with \(\sigma_2^*(G) \geq n + 2k – 3\), then \(G\) is \(k\)-linked. Thus, we improve the order bound to \(111k + 9\), and when \(n \geq 111k + 9\), it implies the result of \(K\). Kawarabayashi \(et al\).

The classification of all dihedral f-tilings of the Riemannian sphere \(S^2\) ,whose prototiles are two right triangles with at least one isosceles, is given.The combinatorial structure and the symmetry group of each tiling is also achieved.

In [4], the author introduced a new metric on the space \(\text{Mat}_{m \times s}(\mathbb{Z}_q)\), which is the module space of all \(m \times s\) matrices with entries from the finite ring \(\mathbb{Z}_q\) (\(q \geq 2\)), generalizing the classical Lee metric [5] and the array RT-metric [8], and named this metric as GLRTP-metric, which is further renamed as LRTJ-metric (Lee-Rosenbloom-Tsfasman-Jain Metric) in [1]. In this paper, we introduce a complete weight enumerator for codes over \(\text{Mat}_{m \times s}(\mathbb{Z}_q)\) endowed with the LRTJ-metric and obtain a MacWilliams-type identity with respect to this new metric for the complete weight enumerator.

The Zagreb indices and the modified Zagreb indices are important topological indices in mathematical chemistry. In this paper we study the relationship between the modified Zagreb indices and the reformulated modified Zagreb indices with respect to trees.

In this work, we first prove that every prime number \(p \equiv 1 \pmod{4}\) can be written in the form \(p = P^2 – 4Q\)with two positive integers \(P\) and \(Q\). Then, we define the sequence \(B_n(P, Q)\) to be \(B_0 = 2\), \(B_1 = P\), and \(B_n = PB_{n-1} – QB_{n-2}\) for \(n \geq 2\), and derive some algebraic identities on it. Also, we formulate the limit of the cross-ratio for four consecutive numbers \(B_n\), \(B_{n+1}\), \(B_{n+2}\), and \(B_{n+3}\).

An edge set \(F\)is called a restricted edge-cut if \(G – F\) is disconnected and contains no isolated vertices. The minimum cardinality over all restricted edge-cuts is called the restricted edge-connectivity of \(G\), and denoted by \(\lambda'(G)\). A graph \(G\) is called \(\lambda’\)-optimal if \(\lambda'(G) = \xi(G)\), where \(\xi(G) = \min\{d_G(u) + d_G(v) – 2: uv \in E(G)\}\). In this note, we obtain a sufficient condition for a \(k( \geq 3)\)-regular connected graph with two orbits to be \(\lambda’\)-optimal.

Let \(G = (V, E)\) be a connected graph. An edge set \(S \subset E\) is a \(k\)-restricted edge cut if \(G – S\) is disconnected and every component of \(G – S\) has at least \(k\) vertices. The \(k\)-restricted edge connectivity \(\lambda_k(G)\) of \(G\) is the cardinality of a minimum \(k\)-restricted edge cut of \(G\). A graph \(G\) is \(\lambda_k\)-connected if \(k\)-restricted edge cuts exist. A graph \(G\) is called \(\lambda_k\)-optimal if \(\lambda_k(G) = \xi_k(G)\), where \[\xi_k(G) = \min\{|[X, Y]|: X \subseteq V, |X| = k \text{ and } G[X] \text{ is connected}\};\] Here, \(G[X]\) is the subgraph of \(G$\) induced by the vertex subset \(X \subseteq V\), and \(Y = V \setminus X\) is the complement of \(X\); \([X, Y]\) is the set of edges with one end in \(X\) and the other in \(Y\). \(G\) is said to be super-\(\lambda_k\) if each minimum \(k\)-restricted edge cut isolates a connected subgraph of order \(k\). In this paper, we give some sufficient conditions for triangle-free graphs to be super-\(\lambda_3\).

The \(k\)-path-connectivity \(\pi_k(G)\) of a graph \(G\) was introduced by Hager in \(1986\). Recently, Mao investigated the \(3\)-path-connectivity of lexicographic product graphs. Denote by \(G \circ H\) the lexicographic product of two graphs \(G\) and \(H\). In this paper, we prove that \(\pi_4(G \circ H) \geq \lfloor\frac{|V(H)|-2}{3}\rfloor\) for any two connected graphs \(G\) and \(H\). Moreover, the bound is sharp. We also derive an upper bound of \(\pi_4(G \circ H)\), that is, \(\pi_4(G \circ H) \leq 2\pi_4(G)|V(H)|\).

This paper devotes to the investigation of \(3\)-designs admitting the special projective linear group \(\mathrm{PSL}(2, 2^n)\) as an automorphism, and we determine all the possible values of \(n\) in the simple \(3-(2^n + 1, 7, \lambda)\) designs admitting \(\mathrm{PSL}(2,2^n)\) as an automorphism group.