In this paper, we recall Konhauser polynomials. Approximation properties of these operators are obtained with the help of the Korovkin theorem. The order of convergence of these operators is computed by means of modulus of continuity, Peetre’s K-functional, and the elements of the Lipschitz class. Also, we introduce the \(r\)-th order generalization of these operators and we evaluate this generalization by the operators defined in this paper. Finally, we give an application to differential equations.

A graph \(X\) is said to be End-regular (resp., End-orthodox, End-inverse) if its endomorphism monoid \(\mathrm{End}(X)\) is a regular (resp., orthodox, inverse) semigroup. In this paper, End-regular (resp., End-orthodox, End-inverse) graphs which are the join of split graphs \(X\) and \(Y\) are characterized. It is also proved that \(X + Y\) is never End-inverse for any split graphs \(X\) and \(Y\).

Some new families of complete caps in Galois affine spaces \({AG}(N,q)\) of dimension \(N \equiv 0 \pmod{4}\) and odd order \(q \leq 127\) are constructed. No smaller complete caps appear to be known.

We give two Frankl-like results on set systems with restrictions on set difference sizes and set symmetric difference sizes modulo prime powers. Based on a similar method, we also give a bound on codes satisfying the properties of Hamming distance modulo prime powers.

In this note, a resolvable \((K_4 – e)\)-design of order \(296\) is constructed. Combining the results of \([2, 3, 4]\), the existence spectrum of resolvable \((K_4 – e)\)-designs of order \(v\) is the set \(\{v : v \equiv 16 \pmod{20}, v \geq 16\}\).

We study permutations of the set \([n] = \{1, 2, \ldots, n\}\) written in cycle notation, for which each cycle forms an increasing or decreasing interval of positive integers. More generally, permutations whose cycle elements form arithmetic progressions are considered. We also investigate the class of generalized interval permutations, where each cycle can be rearranged in increasing order to form an interval of consecutive positive integers.

In this paper, we study the symmetry for the generalized twisted Genocchi polynomials and numbers. We give some interesting identities of the power sums and the generalized twisted Genocchi polynomials using the symmetric properties for the \(p\)-adic invariant \(q\)-integral on \(\mathbb{Z}_p\).

In this paper, we use a simple method to derive different recurrence relations on the recursive sequence order-\(k\) and their sums, which are more general than that given in literature [J.Feng, More Identities on the Tribonacci Numbers, Ars Combinatoria, \(100(2011), 73-78]\). By using the generating matrices, we get more identities on the recursive sequence order-\(k\) and their sums, which are more general than that given in literature [E.Kihg, Tribonacci Sequences with Certain Indices and Their Sums, Ars Combinatoria, \(86(2008), 13-22]\) .

By applying discharging methods and properties of critical graphs, we proved that every simple planar graph \(G\) with \(\Delta(G) \geq 5\) is of class 1, if any 4-cycle is not adjacent to a 5-cycle in \(G\).

A graph \(G\) is pancyclic if it contains a cycle of every length from 3 to \(|V(G)|\) inclusive. A graph \(G\) is panconnected if there exists a path of length \(l\) joining any two different vertices \(x\) and \(y\) with \(d_G(x,y) \leq l \leq |V(G)| – 1\), where \(d_G(x,y)\) denotes the distance between \(x\) and \(y\) in \(G\). A hamiltonian graph \(G\) is panpositionable if for any two different vertices \(x\) and \(y\) of \(G\) and any integer \(k\) with \(d_G(x,y) \leq k \leq |V(G)|/2\), there exists a hamiltonian cycle \(C\) of \(G\) with \(d_C(x,y) = k\), where \(d_C(x,y)\) denotes the distance between \(x\) and \(y\) in a hamiltonian cycle \(C\) of \(G\). It is obvious that panconnected graphs are pancyclic, and panpositionable graphs are pancyclic.

The above properties can be studied in bipartite graphs after some modification. A graph \(H = (V_0 \cup V_1, E)\) is bipartite if \(V(H) = V_0 \cup V_1\) and \(E(H)\) is a subset of \(\{(u,v) | u \in V_0 \text{ and } v \in V_1\}\). A graph is bipancyclic if it contains a cycle of every even length from 4 to \(2\lfloor |V(H)|/2 \rfloor\) inclusive. A graph \(H\) is bipanconnected if there exists a path of length \(l\) joining any two different vertices \(x\) and \(y\) with \(d_H(x,y) \leq l \leq |V(H)| – 1\), where \(d_H(x,y)\) denotes the distance between \(x\) and \(y\) in \(H\) and \(l – d_H(x,y)\) is even. A hamiltonian graph \(H\) is bipanpositionable if for any two different vertices \(x\) and \(y\) of \(H\) and for any integer \(k\) with \(d_H(x,y) \leq k \leq |V(H)|/2\), there exists a hamiltonian cycle \(C\) of \(H\) with \(d_C(x,y) = k\), where \(d_C(x,y)\) denotes the distance between \(x\) and \(y\) in a hamiltonian cycle \(C\) of \(H\) and \(k – d_H(x,y)\) is even. It can be shown that bipanconnected graphs are bipancyclic, and bipanpositionable graphs are bipancyclic.

In this paper, we present some examples of pancyclic graphs that are neither panconnected nor panpositionable, some examples of panconnected graphs that are not panpositionable, and some examples of graphs that are panconnected and panpositionable, for nonbipartite graphs. Corresponding examples for bipartite graphs are discussed. The existence of panpositionable (or bipanpositionable, resp.) graphs that are not panconnected (or bipanconnected, resp.) is still an open problem.