We consider a variation of a classical Turán-type extremal problem due to Bollobás \([2,p. 398, no. 13]\) as follows: determine the smallest even integer \(\sigma(C^k,n)\) such that every graphic sequence \(\pi = (d_1,d_2,\ldots,d_n)\) with term sum \(\sigma(\pi) = d_1 + d_2 + \cdots + d_n \geq \sigma(C^k,n)\) has a realization \(G\) containing a cycle with \(k\) chords incident to a vertex on the cycle. Moreover, we also consider a variation of a classical Turán-type extremal result due to Faudree and Schelp \([7]\) as follows: determine the smallest even integer \(\sigma(P_\ell,n)\) such that every graphic sequence \(\pi = (d_1,d_2,\ldots,d_n)\) with \(\sigma(\pi) \geq \sigma(P_\ell,n)\) has a realization \(G\) containing \(P_\ell\) as a subgraph, where \(P_\ell\) is the path of length 2. In this paper, we determine the values of \(\sigma(P_\ell,n)\) for \(n \geq \ell+1\) and the values of \(\sigma(C^k,n)\) for \(n \geq (k+3)(2k+5)\).

The hyperbolic Fibonacci function, which is the continuous extension of Binet’s formula for the Fibonacci number, transforms the Fibonacci number theory into a “continuous” theory because every identity for the hyperbolic Fibonacci function has its discrete analogy in the framework of the Fibonacci number. In this new paper, we define three important generalizations of the \(k\)-Fibonacci sine, cosine, and quasi-sine hyperbolic functions and then carry over many concepts and techniques that we learned in a standard setting for the \(k\)-Fibonacci sine, cosine, and quasi-sine hyperbolic functions to the generalizations of these functions.

A new construction of authentication codes with arbitration from pseudo-symplectic geometry over finite fields is given. The parameters and the probabilities of deceptions of the codes are also computed.

It was conjectured in a recently published paper that for any integer \(k \geq 8\) and any even integer \(n\) with \(2k+3 < n < 2k+\lfloor\frac{k}{2}\rfloor+3\), the \(k\)th power \(C_n^k\) of the \(n\)-cycle is not a divisor graph. In this paper, we prove this conjecture, hence obtaining a complete characterization of those powers of cycles which are divisor graphs.

Inspired by a recent paper by Giulietti, Korchmàros and Torres \([3]\), we provide equations for some quotient curves of the Deligne-Lusztig curve associated to the Suzuki group \(S_z(q)\).

In this paper, we study the global behavior of the nonnegative equilibrium points of the difference equation

\[x_{n+1} = \frac{ax_{n-2l}}{b+c\prod\limits_{i=0}^{k+1}x_{n-2i}}, \quad n=0,1,\ldots,\]

where \(a\), \(b\), and \(c\) are nonnegative parameters, initial conditions are nonnegative real numbers, and \(k\) and \(l\) are nonnegative integers, with \(l \leq k+1\).

The chromatic polynomial of a graph \(\Gamma\), \(C(\Gamma; \lambda)\), is the polynomial in \(\lambda\) which counts the number of distinct proper vertex \(\lambda\)-colorings of \(\Gamma\), given \(\lambda\) colors. By applying the addition-contraction method, chromatic polynomials of some sequences of \(2\)-connected graphs satisfy a number of recursive relations. We will show that by knowing the chromatic polynomial of a few small graphs, the chromatic polynomial of each of these sequences can be computed by utilizing either matrices or generating functions.

An \(f\)-coloring of a graph \(G\) is an edge-coloring of \(G\) such that each color appears at each vertex \(v \in V(G)\) at most \(f(v)\) times. The minimum number of colors needed to \(f\)-color \(G\) is called the \(f\)-chromatic index of \(G\). A simple graph \(G\) is of \(f\)-class 1 if the \(f\)-chromatic index of \(G\) equals \(\Delta_f(G)\), where \(\Delta_f(G) = \max_{v\in V(G)}\{\left\lceil\frac{d(v)}{f(v)}\right\rceil\}\). In this article, we find a new sufficient condition for a simple graph to be of \(f\)-class 1, which is strictly better than a condition presented by Zhang and Liu in 2008 and is sharp. Combining the previous conclusions with this new condition, we improve a result of Zhang and Liu in 2007.

We provide the specifics of how affine planes of orders three, four, and five can be used to partition the full design comprising all triples on \(9, 16\), and \(25\) elements, respectively. Key results of the approach for order five are generalized to reveal when there is potential for using suitable affine planes of order \(n\) to partition the complete sets of \(n^2\) triples into sets of mutually disjoint triples covering either all \(n^2\), or else precisely \(n^2 – 1\), elements.

In this paper, the notion of left-right and right-left \(f\)-derivation of a BCC-algebra is introduced, and some related properties are investigated. Also, we consider regular \(f\)-derivation and \(d\)-invariant on \(f\)-ideals in BCC-algebras.