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 \( nq_n^2 \to C > 0 \)).

We introduce a polygonal cylinder \( C_{m,n} \), using the Cartesian product of paths \( P_m \) and \( P_n \) and using topological identification of vertices and edges of two opposite sides of \( P_m \times P_n \), and give its Hosoya polynomial, which, depending on odd and even \( m \), is covered in seven separate cases.

In this paper we find recurrence relations for the asymptotic probability a vertex is \(k\) protected in all Motzkin trees. We use a similar technique to calculate the probabilities for balanced vertices of rank \(k\). From this we calculate upper and lower bounds for the probability a vertex is balanced and upper and lower bounds for the expected rank of balanced vertices.