Consider an n-set, say \(X_n = {1,2,…,n}\). An exponential generating function and recurrence relation for the number of subpermutations of \(X_n\), whose orbits are of size at most \(k \geq 0\) are obtained. Similar results for
the number of nilpotent subpermutations of nilpotency index at most \(k\), and exactly \k\) are also given, along with arithmetic and asypmtotic formulas for these numbers. \(1\) \(2\)

In this paper, we show that the crossing number of the complete tripartite graph \(K_{2,4,n}\) is \(6\left\lfloor\frac{n}{2}\right\rfloor \left\lfloor\frac{n-1}{2}\right\rfloor+2n\).

An \((n \times n)\) matrix \(A = (a_{ij})\) is called a Toeplitz matrix
if it has constant values along all diagonals parallel to the main diagonal.
A directed Toeplitz graph is a digraph with Toeplitz adjacency matrix.
In this paper, we discuss conditions for the existence of Hamiltonian cycles
in directed Toeplitz graphs.

For \(n \geq 2\) and a local field \(K\), let \(\Delta_n\) denote the affine building naturally associated to the symplectic group \(\mathrm{Sp}_{n}(K)\). We compute the spectral radius of the subgraph \(Y_n\) of \(\Delta_n\) induced by the special vertices in \(\Delta_n\), from which it follows that \(Y_n\) is an analogue of a family of expanders and is non-amenable.

The concept of \(t\)-(v, \(\lambda\)) trades of block designs has been studied in detail. See, for example, A.~S. Hedayat (1990) and Billington (2003). Latin trades have also been extensively studied under various names; see A.~D. Keedwell (2004) for a survey. Recently, Khanban, Mahdian, and Mahmoodian have extended the concept of Latin trades and introduced \(t\)-(\(v, k\)) Latin trades.In this paper, we study the spectrum of possible volumes of these trades, \(S(t, k)\). Firstly, similarly to trades of block designs, we consider \((t+2)\) numbers \(s_i = 2^{i+1}-2^{(t+1)-i} \), \(0 \leq i \leq t+1\), as critical points. Then, we show that \(s_i \in S(t,k)\) for any \(0 \leq i \leq t+1\), and if \(s \in (s_i, s_{i+1}, )\), \(0 \leq i \leq t\), then \(s \notin S(t, t+1)\). As an example, we precisely determine \(S(3, 4)\).

This paper investigates the relationship between the degree-sum of adjacent vertices, girth, and upper embeddability of graphs, combining it with edge-connectivity. The main result is:
Let \(G\) be a \(k\)-edge-connected simple graph with girth \(g\). If there exists an integer \(m\) (\(1 \leq m \leq g\)) such that for any \(m\) consecutively adjacent vertices \(x_i\) (\(i = 1, 2, \ldots, m\)) in any non-chord cycle \(C\) of \(G\), it holds that

\[\sum\limits_{i=1}^m d_G(x_i) > \frac{mn}{(k-1)^2+2} + \frac{km}{g}+(2-g)m,\]

where \(k = 1, 2, 3, n = |V(G)|\), then \(G\) is upper embeddable and the upper bound is best possible.

In this study, we define and investigate the Bivariate Gaussian Fibonacci and Bivariate Gaussian Lucas Polynomials. We derive generating functions, Binet formulas, explicit formulas, and partial derivatives of these polynomials. By defining these bivariate polynomials for special cases, we obtain:\(F_n(x, 1)\) as the Gaussian Fibonacci polynomials,\(L_n(x, 1)\) is the Gaussian Lucas polynomials,\( {F}_{n}(1, 1)\) as the Gaussian Fibonacci numbers, and \( {L}_{n}(1, 1)\) as the Gaussian Lucas numbers, as defined in \([19]\).

In this paper, we show that the set \(\{E_0(x), E_1(x), \ldots, E_n(x)\}\) of Euler polynomials is a basis for the space of polynomials of degree less than or equal to \(n\). From the properties of Euler basis polynomials, we derive some interesting identities on the product of two Bernoulli and Euler polynomials.

An \(n\)-colour even composition is defined as an \(n\)-colour composition with even parts. In this paper, we obtain generating functions, explicit formulas, and a recurrence formula for \(n\)-colour even compositions.