Every set of natural numbers determines a generating function convergent for \( q \in (-1, 1) \) whose behavior as \( q \to 1^- \) determines a germ. These germs admit a natural partial ordering that can be used to compare sets of natural numbers in a manner that generalizes both cardinality of finite sets and density of infinite sets. For any finite set \( D \) of positive integers, call a set \( S \) “\( D \)-avoiding” if no two elements of \( S \) differ by an element of \( D \). We study the problem of determining, for fixed \( D \), all \( D \)- avoiding sets that are maximal in the germ order. In many cases, we can show that there is exactly one such set. We apply this to the study of one-dimensional packing problems.

We prove some combinatorial identities by an analytic method. We use the property that singular integrals of particular functions include binomial coefficients. In this paper, we prove combinatorial identities from the fact that two results of the particular function calculated as two ways using the residue theorem in the complex function theory are the same. These combinatorial identities are the generalization of a combinatorial identity that has been already obtained

Bargraphs are column convex polyominoes, where the lower edge lies on a horizontal axis. We consider the inner site-perimeter, which is the total number of cells inside the bargraph that have at least one edge in common with an outside cell and obtain the generating function that counts this statistic. From this we find the average inner perimeter and the asymptotic expression for this average as the semi-perimeter tends to infinity. We finally consider those bargraphs where the inner site-perimeter is exactly equal to the area of the bargraph.

Let \( G \) be a graph with \( n \) vertices, then the \( c \)-dominating matrix of \( G \) is the square matrix of order \( n \) whose \( (i, j) \)-entry is equal to 1 if the \( i \)-th and \( j \)-th vertex of \( G \) are adjacent, is also equal to 1 if \( i = j \), \( v_i \in D_c \), and zero otherwise.

The \( c \)-dominating energy of a graph \( G \), is defined as the sum of the absolute values of all eigenvalues of the \( c \)-dominating matrix.

The main purposes of this paper are to introduce the \( c \)-dominating Estrada index of a graph. Moreover, to obtain upper and lower bounds for the \( c \)-dominating Estrada index and investigate the relations between the \( c \)-dominating Estrada in

Let \( G(V,E) \) be a simple connected graph with vertex set \( V \) and edge set \( E \). The Wiener index in the graph is \(W(G) = \sum_{\{u,v\} \subseteq V} d(u,v),\) where \( d(u,v) \) is the distance between \( u \) and \( v \), and the Hosoya polynomial of \( G \) is \(H(G, x) = \sum_{\{u,v\} \subseteq V} x^{d(u,v)}.\) The hyper-Wiener index of \( G \) is \(WW(G) = \frac{1}{2} \left( W(G) + \sum_{\{u,v\} \subseteq V} d^2(u,v) \right).\) In this paper, we compute the Wiener index, Hosoya polynomial, and hyper-Wiener index of Jahangir graph \( J_{8,m} \) for \( m \geq 3 \).

Hybrid numbers are generalization of complex, hyperbolic and dual numbers. In this paper we introduce and study Fibonacci-Pell hybrinomials, i.e. polynomials, which are a generalization of hybrid numbers of the Fibonacci type.

The hub-integrity of a graph is given by the minimum of \( |S| + m(G – S) \), where \( S \) is a hub set and \( m(G – S) \) is the maximum order of the components of \( G – S \). In this paper, the concept of hub edge-integrity is introduced as a new measure of the stability of a graph \( G \), and it is defined as \(HEI(G) = \min\{|S| + m(G – S)\},\) where \( S \) is an edge hub set and \( m(G – S) \) is the order of a maximum component of \( G – S \). Furthermore, an \( HEI \)-set of \( G \) is any set \( S \) for which this minimum is attained. Several properties and bounds on the \( HEI \) are presented, and the relationship between \( HEI \) and other parameters is investigated. The \( HEI \) of some classes of graphs is also computed.

A graph \( G(R) \) is said to be a zero divisor graph of a commutative ring \( R \) with identity if \( x_1, x_2 \in V(G(R)) \) and \( (x_1, x_2) \in E(G(R)) \) if and only if \( x_1 \cdot x_2 = 0 \). The vertex-degree-based eccentric topological indices of zero divisor graphs of commutative rings \( \mathbb{Z}_{p^2} \times \mathbb{Z}_{q^2} \) are studied in this paper, with \( p \) and \( q \) being primes.

The sums \( S(x, t) \) of the centered remainders \( kt – \lfloor kt \rfloor – 1/2 \) over \( k \leq x \) and corresponding Dirichlet series were studied by A. Ostrowski, E. Hecke, H. Behnke, and S. Lang for fixed real irrational numbers \( t \). Their work was originally inspired by Weyl’s equidistribution results modulo 1 for sequences in number theory.

In a series of former papers, we obtained limit functions which describe scaling properties of the Farey sequence of order \( n \) for \( n \to \infty \) in the vicinity of any fixed fraction \( a/b \) and which are independent of \( a/b \). We extend this theory on the sums \( S(x, t) \) and also obtain a scaling behavior with a new limit function. This method leads to a refinement of results given by Ostrowski and Lang and establishes a new proof for the analytic continuation of related Dirichlet series. We will also present explicit relations to the theory of Farey sequences.

Bulgarian solitaire is played on \( n \) cards divided into several piles; a move consists of picking one card from each pile to form a new pile. This can be seen as a process on the set of integer partitions of \( n \): if sorted configurations are represented by Young diagrams, a move in the solitaire consists of picking all cards in the bottom layer of the diagram and inserting the picked cards as a new column. Here we consider a generalization, \( L \)-solitaire, wherein a fixed set of layers \( L \) (that includes the bottom layer) are picked to form a new column.

\( L \)-solitaire has the property that if a stable configuration of \( n \) cards exists it is unique. Moreover, the Young diagram of a configuration is convex if and only if it is a stable (fixpoint) configuration of some \( L \)-solitaire. If the Young diagrams representing card configurations are scaled down to have unit area, the stable configurations corresponding to an infinite sequence of pick-layer sets \( (L_1, L_2, \ldots) \) may tend to a limit shape \( \phi \). We show that every convex \( \phi \) with certain properties can arise as the limit shape of some sequence of \( L_n \). We conjecture that recurrent configurations have the same limit shapes as stable configurations.

For the special case \( L_n = \{1, 1 + \lfloor 1/q_n \rfloor, 1 + \lfloor 2/q_n \rfloor, \ldots\} \), where the pick layers are approximately equidistant with average distance \( 1/q_n \) for some \( q_n \in (0,1] \), these limit shapes are linear (in case \( nq_n^2 \to 0 \)), exponential (in case \( nq_n^2 \to \infty \)), or interpolating between these shapes (in case \( nq_n^2 \to C > 0 \)).