It has been known for at least \(2500\) years that mathematics and music are directly related. This article explains and extends ideas originating with Euler involving labeling parts of graphs with notes in such a way that other parts of the graphs correspond in a natural way to chords. The principal focus of this research is the notion of diatonic labelings of cubic graphs, that is, labeling the edges with pitch classes in such a way that vertices are incident with edges labeled with the pitch classes of a triad in a given diatonic scale. The pitch classes are represented in a natural way with elements of \(\mathbb{Z}_{12}\), the integers modulo twelve.

Several classes of cubic graphs are investigated and shown to be diatonic. Among the graphs considered are Platonic Solids, cylinders, and Generalized Petersen Graphs. It is shown that there are diatonic cubic graphs on \(n\) vertices for even \(n \geq 14\). Also, it is shown that there are cubic graphs on \(n\) vertices that do not have diatonic labelings for all even \(n \geq 4\). The question of forbidden subgraphs is investigated, and a forbidden subgraph for diatonic graphs, or “clash”, is demonstrated.

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 \(G\).