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.

In this note, we characterize graphs with a given small matching number. Specifically, we characterize graphs with minimum degree at least \(2\) and matching number at most \(3\). The characterization when the matching number is at most \(2\) strengthens the result of Lai and Yan’s that characterized the non-supereulerian \(2\)-edge connected graphs with matching at most \(2\). Furthermore, the characterization of graphs with matching number at most \(3\) addresses a conjecture of Lai and Yan in [SuperEulerian graphs and matchings, Applied Mathematics Letters 24 (2011) 1867-1869].

Let \(D(G)\) be the distance matrix of a connected graph \(G\). The distance spectral radius of \(G\) is the largest eigenvalue of \(D(G)\) and has been proposed as a molecular structure descriptor. In this paper, we study the distance spectral radius of graphs with a given independence number. Special attention is paid to graphs with a given independence number and maximal distance spectral radius.