The existence question for a \(3\)-\((16,7,5)\) design is open, In this paper, we examine possible automorphisms of this design. We consider a minimum subset of basic permutations consisting of cycles of prime length \(p\) and prove that if a \(3\)-\((16,7,5)\) design exists, then it is either rigid or admits basic automorphisms with cycles of length \(2\) or \(3\).

We define a product summation of ordered partition \(f_j(n,m,r) = \sum{c_1^r c_2^r \ldots c_j^rc_{j+1} \ldots c_m}\), where the sum is over all positive integers \(c_1, c_2, \ldots, c_m\) with \(c_1 + c_2 + \cdots + c_m = n\) and \(0 \leq j \leq m\). We concentrate on \(f_m(n,m,r)\) in this paper. The main results are as follows:

(1) The generating function for \(f_m(n,m,r)\) and the explicit formula for \(f_m(n,m,2) , f_m(n,m,3)\) and \(f_m(n,m, 4)\) are obtained.

(2) The relationship between \(f_j(n,m,r)\) for \(r = 2,3\) and the Fibonacci and Lucas numbers is found.

It is shown that for \(2 \leq t \leq n-3\), a strict \(t\)-SB\((n,n-1)\) design does not exist, but for \(n \geq 3\), a non-strict \(2\)-SB\((n,n-1)\) design exists. The concept of large sets for Steiner triple systems is extended to SB designs and examples of large sets for SB designs are given.

It is shown that every well-defined solution to the second-order difference equation in the title, when \((A_n)_{n \in 0}\) is a two-periodic sequence such that \(\max\{A_0, A_1\} \geq 0\), is eventually periodic with period two. In the case \(\max\{A_0, A_1\} \leq 0\), it is shown the existence of unbounded solutions, by describing all solutions in terms of \(A_0\), \(A_1\), \(x_{-1}\), and \(x_0\).

This paper considers the folded hypercube \(FQ_n\) as an enhancement on the hypercube, and obtains some algebraic properties of \(FQ_n\). Using these properties, the authors show that for any two vertices \(x\) and \(y\) in \(FQ_n\), with distance \(d\) and any integers \(h \in \{d, n+1- d\}\) and \(l\) with \(h \leq l \leq 2^n – 1\), \(FQ_n\) contains an \(xy\)-path of length \(l\) and no \(xy\)-path of other length, provided that \(l\) and \(h\) have the same parity.

Let \(G\) be a \(2\)-tough graph on at least five vertices and let \(e_1, e_2\) be a pair of arbitrarily given edges of \(G\). Then
(a) There exists a \(2\)-factor in G containing \(e_1, e_2\).
(b) There exists a \(2\)-factor in G avoiding \(e_1, e_2\).
(c) There exists a \(2\)-factor in G containing \(e_1\) and avoiding \(e_2\).

In this paper, a sufficient condition is obtained for the global asymptotic stability of the following system of difference equations
\[x_{n+1} = \frac{x_ny_{n-1}}{x_ny_{n-1}+1} ,y_{n+1}=\frac{y_n x_{n-1}}{y_nx_{n-1} + 1} , \quad n = 0, 1, 2, \ldots,\]
where the initial values \((x_k, y_k) \in (0, \infty) (\text{for} k=-1,0)\).

Erdős and Sós conjectured in \(1962\) that if the average degree of a graph \(G\) exceeds \(k – 2\), then \(G\) contains every tree on \(k\) vertices. Results from Sauer and Spencer (and independent results from Zhou) prove the special case where \(G\) has \(k\) vertices. Results from Slater, Teo, and Yap prove the case where \(G\) has \(k + 1\) vertices. In \(1996\), Woźniak proved the case where \(G\) has \(k + 2\) vertices. We prove the conjecture for the case where \(G\) has \(k + 3\) vertices.

A semi-double graph is a connected multi-graph such that each multi-edge consists of two edges. If there is at most one loop at each vertex of a semi-double graph, then this graph is called a single-petal graph. Via the degree-sum of nonadjacent vertices, the up-embeddability of semi-double graphs and single-petal graphs are discussed in this paper. And the results obtained in this paper can be extended to determine the up-embeddability of multi-graphs and pseudographs.

The commuting graph of an arbitrary ring \(R\), denoted by \(\Gamma(R)\), is the graph whose vertices are all non-central elements of \(R\), and two distinct vertices \(a\) and \(b\) are adjacent if and only if \(ab = ba\). In this paper, we investigate the connectivity, the diameter, the maximum degree and the minimum degree of the commuting graph of the quaternion algebra \(\mathbb{Z}_n[i, j, k]\).