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The multiplication of cells leads to consider the cell division model with mutation. Large cell counts therefore appear, implying that the distribution of the final number of mutant cells is a heavy tailed distribution. This distribution can be interpreted as a compound Poisson distribution, which depends on two parameters: the average number of mutations and the fitness parameter (heavy tail index). In this work, we specify some conditions that ensure a stability property of heavy-tailed distributions, namely, the distribution of random sum of random variables is a heavy-tailed distribution. We apply the obtained result to the compound Poisson distribution. To estimate the tail index, the Hill estimator and the generating function method are used. A comparative study is performed using these two estimators. [ABSTRACT FROM AUTHOR]

In this paper, we introduce the modified q-Genocchi polynomials, investigate their properties, and give their generating function. We obtain a number of new relations and properties for q-Genocchi polynomials, such as addition formula, explicit formula, derivative formula, and multiplication formula. As an application, a new q-analogue of Genocchi zeta function is presented by applying the Mellin transform to the generating function of the modified q-Genocchi polynomials. Finally, we define the q-Genocchi zeta-type functions and then prove their interpolation by the modified q-Genocchi polynomials at negative integers. [ABSTRACT FROM AUTHOR]

This paper is about the positive part U q + of the q -deformed enveloping algebra U q ( sl ˆ 2). The algebra U q + admits an embedding, due to Rosso, into a q -shuffle algebra ▪. The underlying vector space of ▪ is the free algebra on two generators x , y. Therefore, the algebra ▪ has a basis consisting of the words in x , y. Let U denote the image of U q + under the Rosso embedding. In our first main result, we find all the words in x , y that are contained in U. One type of solution is called alternating. The alternating words have been studied by Terwilliger. There is another type of solution, which we call doubly alternating. In our second main result, we display many commutator relations involving the doubly alternating words. In our third main result, we describe how the doubly alternating words are related to the alternating words. [ABSTRACT FROM AUTHOR]

The initial segmentation phase is crucial in image processing to simplify the image representation and extract some desired features. Different methods and techniques have been proposed for image multi-level thresholding, but they are still stuck in local optima and need improvement. Recently, a metaheuristic optimization algorithm called Golden Jackal Optimizer (GJO) has been proposed as an alternative solution. The GJO has been adopted as a good solution for many optimization problems. However, the GJO attempted to solve the convergence problem to a local minimum during execution, often leading to unsatisfactory results. Most variants of GJO are based on chaotic systems due to their easy implementation and remarkable capacity to avoid being trapped in local optima. This paper proposes a Polynomial Chebychev Symmetric Chaotic-based GJO (PCSCGJO) algorithm by combining a recently developed chaotic generating function to achieve better segmentation results. This variant improves the GJO by introducing the chaotic generating function of the Chebyshev polynomials as an update process while searching for the optimal solution. Simulation results prove the effectiveness of the PCSCGJO method and its ability to deal with different medical color images. The quality of the segmented images obtained by the proposed method was compared to well-known metaheuristic algorithms using performance metrics such as PSNR, SSIM, FSIM, and MSE. Consequently, the metrics values show that the suggested technique outperforms the other methods regarding quality and accuracy. [ABSTRACT FROM AUTHOR]

Integer compositions of n are viewed as bargraphs with n circular nodes or square cells in which the ith part of the composition x i is given by the ith column of the bargraph with x i nodes or cells. The sun is at infinity in the north west of our two dimensional model and each node/cell may or may not be lit depending on whether it stands in the shadow cast by another node/cell to its left. We study the number of lit nodes in an integer composition of n and later we modify this to yield the number of lit square cells. We then count the number of columns being lit which leads naturally to those cases where only the first column is lit. We prove the theorem that the generating function for the latter is the same as the generating function for compositions in which the first part is strictly smallest. This theorem has interesting q-series identities as corollaries which allow us to deduce in a simple way the asymptotics for both the number of lit nodes and columns as n → ∞ . [ABSTRACT FROM AUTHOR]

Four classes of multiple symmetric sums over cyclic products of binomial coefficients are examined. By incorporating the generating function approach and recursive construction method, they are expressed analytically as coefficients of rational functions. Several recurrence relations and generating functions are explicitly determined when the dimension of the multiple sums does not exceed five. [ABSTRACT FROM AUTHOR]

Based on the finite-difference method, the considered Riesz space-fractional diffusion equations result in a series of linear systems, whose coefficient matrices are composed of the identity matrix and the product of diagonal matrix and Toeplitz matrix. With the aid of the Toeplitz structure contained in the discrete linear system, a class of banded preconditioner with shift compensation (BSC preconditioner) is designed to improve the convergence rates of the Krylov subspace iteration methods. The BSC preconditioner can be viewed as a modification of the banded preconditioner with diagonal compensation (BDC preconditioner) proposed by F.-R. Lin et al. (F.-R. Lin, S.-W. Yang and X.-Q. Jin, J. Comput. Phys. 256, 109–117 2014), but unlike the BDC preconditioner, it retains the same Toeplitz-like structure as the coefficient matrix of the discrete linear system. Although the theoretical result on the eigenvalues of the BDC-preconditioned matrix has not been given so far, the eigenvalue distributions of the BSC-preconditioned matrix and the BDC-preconditioned matrix can be demonstrated simultaneously by using the structure of the BSC preconditioner. In addition, theoretical analysis shows that the eigenvalues of the BSC-preconditioned matrix are real and located in a positive interval [ 1 , ν) with ν weakly dependent on h, while the eigenvalues of the BDC-preconditioned matrix are weakly clustered around those of the BSC-preconditioned matrix, which indicates that the BSC preconditioner is almost as effective as the BDC preconditioner. Numerical experiments reveal that both the BSC preconditioner and BDC preconditioner can significantly accelerate the convergence rates of the Krylov subspace iteration methods, and can show h-independent convergence behavior when the order of the fractional derivative approaches 2. [ABSTRACT FROM AUTHOR]

We propose a systematic approach to construct a new family of stochastic symplectic schemes for the strong approximation of the solution of stochastic Hamiltonian systems. Our approach is based both on B-series and generating functions. The proposed schemes are a generalization of the implicit midpoint rule, they require derivatives of the Hamiltonian functions of at most order two, and are constructed by defining a generating function. We construct some schemes with strong convergence order one and a half, and we illustrate numerically their long term performance. [ABSTRACT FROM AUTHOR]

In this research, new representations of basic functions are proposed based on the new types of fuzzy partition and a subnormal generating function. The generalized uniform fuzzy partitions in subnormal case, i.e. in case a generating function K is not normal (generalized normal case), and simpler form of fuzzy transform (FzT) components based on these new representations of the generalized uniform fuzzy partitions are indicated. The main properties of a new uniform fuzzy partition are suggested. New theorems and lemmas are proved. [ABSTRACT FROM AUTHOR]

In this paper, we define double almost-Riordan arrays and find that the set of all double almost-Riordan arrays forms a group, called the double almost-Riordan group. We also obtain the sequence characteristics of double almost-Riordan arrays and give the production matrices of two types for double almost-Riordan arrays. In addition, we discuss the algebraic properties of the double almost-Riordan group, and finally give the compression of double almost-Riordan arrays and their sequence characteristics. [ABSTRACT FROM AUTHOR]

The Riordan group of Riordan arrays was first described in 1991, and since then, it has provided useful tools for the study of areas such as combinatorial identities, polynomial sequences (including families of orthogonal polynomials), lattice path enumeration, and linear recurrences. Useful extensions of the idea of a Riordan array have included almost Riordan arrays, double Riordan arrays, and their generalizations. After giving a brief overview of the Riordan group, we define two further extensions of the notion of Riordan arrays, and we give a number of applications for these extensions. The relevance of these applications indicates that these new extensions are worthy of study. The first extension is that of the reverse symmetrization of a Riordan array, for which we give two applications. The first application of this symmetrization is to the study of a family of Riordan arrays whose symmetrizations lead to the famous Robbins numbers as well as to numbers associated with the 20 vertex model of mathematical physics. We provide closed-form expressions for the elements of these arrays, and we also give a canonical Catalan factorization for them. We also describe an alternative family of Riordan arrays whose symmetrizations lead to the same integer sequences. The second application of this symmetrization process is to the area of the enumeration of lattice paths. We remain with the applications to lattice paths for the second extension of Riordan arrays that we introduce, which is the interleaved Riordan array. The methods used include generating functions, linear algebra, weighted compositions, and linear recurrences. In the case of the symmetrization process applied to Riordan arrays, we focus on the principal minor sequences of the resulting square matrices in the context of integrable lattice models. [ABSTRACT FROM AUTHOR]

Let F be the finite field of order q and M (n , r , F) be the set of n × n matrices of rank r over the field F. For α ∈ F and A ∈ M (n , F) , let Z A , r α = { X ∈ M (n , r , F) | Tr (A X) = α }. In this article, we solve the problem of determining the cardinality of Z A , r α. We also solve the generalization of the problem to rectangular matrices. [ABSTRACT FROM AUTHOR]

In this paper, we introduce a new family of sequences related to Horadam-type sequences. Specifically, we consider the repunit sequence { r n } n ≥ 0 , which is defined by the initial terms r 0 = 0 and r 1 = 1 and follows the Horadam recurrence relation given by r n = 11 r n − 1 − 10 r n − 2 for n ≥ 2 . Many studies have explored generalizations of integer sequences in different directions: some by preserving the initial terms, some by preserving the recurrence relation, and some by considering different numerical sets beyond positive integers. In this article, we take the third approach. Specifically, we study these sequences in the context of the tricomplex ring T. We define the Tricomplex Repunit sequence { t r n } n ≥ 0 , with initial terms t r 0 = (0 , 1 , 11) and t r 1 = (1 , 11 , 111) , and governed by the recurrence relation t r n = 11 t r n − 1 − 10 t r n − 2 , for n ≥ 2 . This sequence is also a Horadam-type sequence but defined in the tricomplex ring T. In this paper, we establish the properties of the Tricomplex Repunit sequence and establish several new as well as well-known identities associated with it, including Binet's formula, Tagiuri–Vajda's identity, d'Ocagne's identity, and Catalan's identity. We also derive the generating function for this sequence. Furthermore, we study various additional properties of these generalized sequences and establish results concerning the summation of terms related to the Tricomplex Repunit sequence, and one of our main goals is to determine analogous or symmetrical properties for the Tricomplex Repunit sequence to those we know for the ordinary repunit sequence. [ABSTRACT FROM AUTHOR]

In this study, we establish some properties of Bronze Fibonacci and Bronze Lucas sequences. Then we find the relationships between the roots of the characteristic equation of these sequences with these sequences. What is interesting here is that even though the roots change, equality is still maintained. Also, we derive the special relations between the terms of these sequences. We give the important relations among these sequences, positive and negative index terms, with the sum of the squares of two consecutive terms being related to these sequences. In addition, we present the application of generalized Bronze Fibonacci sequences to hyperbolic quaternions. For these hyperbolic quaternions, we give the summation formulas, generating functions, etc. Moreover, we obtain the Binet formulas in two different ways. The first is in the known classical way and the second is with the help of the sequence's generating functions. In addition, we calculate the special identities of these hyperbolic quaternions. Furthermore, we examine the relationships between the hyperbolic Bronze Fibonacci and Bronze Lucas quaternions. Finally, the terms of the generalized Bronze Fibonacci sequences are associated with their hyperbolic quaternion values. [ABSTRACT FROM AUTHOR]

In the paper, the authors present the positivity and decreasing property of the normalized tails of the series expansion of the generating function of the classical Bernoulli numbers and prove the increasing property of the ratio between two normalized tails of the series expansion of the generating function of the classical Bernoulli numbers by showing the increasing property of the ratio between two Bernoulli polynomials. [ABSTRACT FROM AUTHOR]

We explain a discontinuous drop in the exponential growth rate for certain multivariate generating functions at a critical parameter value in even dimensions d ≤ 4. This result depends on computations in the homology of the algebraic variety where the generating function has a pole. These computations are similar to, and inspired by, a thread of research in applications of complex algebraic geometry to hyperbolic PDEs, going back to Leray, Petrowski, Atiyah, Bott and Gårding. As a consequence, we give a topological explanation for certain asymptotic phenomena appearing in the combinatorics and number theory literature. Furthermore, we show how to combine topological methods with symbolic algebraic computation to determine explicitly the dominant asymptotics for such multivariate generating functions, giving a significant new tool to attack the so-called connection problem for asymptotics of P-recursive sequences. This in turn enables the rigorous determination of integer coefficients in the Morse-Smale complex, which are difficult to determine using direct geometric methods. [ABSTRACT FROM AUTHOR]

Generalizing previous results, we introduce and study a new statistic on words, that we call rectangle capacity. For two fixed positive integers r and s , this statistic counts the number of occurrences of a rectangle of size r × s in the bargraph representation of a word. We find the bivariate generating function for the distribution on words of the number of r × s rectangles and the generating function for their total number over all words. We also obtain the analog results for Catalan words. [ABSTRACT FROM AUTHOR]

Interacting in open networks carries certain risks. To ensure the information security of network interaction participants, cryptographic protocols (CrP) are used. High levels of security can be achieved through their formal verification. A common formal method for verifying CrP is model checking. In this work, we propose using the TLA+/TLC toolset to check models of CrP. This toolset is widely applied in various practical fields. The protocol model is defined in the TLA+ specification language, as well as the required security properties in the form of invariants. The model of a protocol describes its behavior as a transition system containing all possible states of the protocol model and transitions between them. The TLC model checker is employed to automatically verify that the model meets the required properties. The task of verifying CrP has its specifics. This study proposes three modeling techniques that take into account the specifics of both the task and the TLA+/TLC toolset being used. The first technique involves replacing a system consisting of an arbitrary number of agents with a three-agent system. This simplifies the model and reduces its state space. The second technique is related to representing transmitted messages as a hierarchical structure, allowing encrypted messages to be nested within others. The third technique consists of optimizing the model to improve the performance of the TLC model checker by defining a function that generates only those elements leading to transitions between states in the model. These techniques simplify the model and reduce verification time. We demonstrate the application of these results on a simple protocol example — the Needham-Schroeder public key authentication protocol. After detecting a known vulnerability in the original protocol by using TLC, we model and verify an improved version. Verification results show that the new version of the protocol does not have this vulnerability.

The production matrix plays an important role in characterizing a Riordan array. Recently, Barry explored the notion of the n -th production matrix and characterized the Riordan arrays corresponding to the second and third production matrices respectively. This paper is devoted to study the n -th production matrix and its corresponding Riordan arrays systematically. Our work is threefold. First, we show that every n -th production matrix can be factorized into a product of n matrices associated with the ordinary production matrix. Second, we prove a characterization of the Riordan array corresponding to the n -th production matrix, which was conjectured by Barry. Third, we claim that if the ordinary production matrix of a Riordan array is totally positive, so are the n -th production matrix and its corresponding Riordan arrays. Our results are illustrated by the generalized Catalan array which includes many well-known Riordan arrays as special cases. [ABSTRACT FROM AUTHOR]

This paper investigates the power-saving mechanisms of Discontinuous Reception (DRX), a technique used in wireless communication networks to reduce energy consumption. By employing a discrete-time Geo/Geo/1 queueing model with differentiated vacations and system disasters, we aim to more accurately capture the intermittent nature of data arrivals, often overlooked in continuous-time models. Our research addresses the existing gap in the literature by providing a more realistic representation of DRX behaviour. Understanding the performance and characteristics of DRX is crucial for optimizing energy efficiency and improving the overall performance of wireless networks. This paper contributes to this understanding by deriving steady-state probabilities, calculating key performance metrics, and visualizing the system behaviour through graphical analysis. [ABSTRACT FROM AUTHOR]

Persistence modules and barcodes are used in symplectic topology to define various invariants of Hamiltonian diffeomorphisms, however numerical methods for computing these barcodes are not yet well developed. In this paper we define one such invariant called the generating function barcode of compactly supported Hamiltonian diffeomorphisms of R 2 n by applying Morse theory to generating functions quadratic at infinity associated to such Hamiltonian diffeomorphisms and provide an algorithm (i.e a finite sequence of explicit calculation steps) that approximates it. [ABSTRACT FROM AUTHOR]

Quantum N -toroidal algebras are generalization of the quantum toroidal algebras. In this paper, we construct two types of level-one vertex representations of the quantum N -toroidal algebras in type B , which also realize the quantum toroidal algebras of type B as special cases. [ABSTRACT FROM AUTHOR]

In this study, we explore the impact of stochastic resetting on the dynamics of random walks on a T-fractal network. By employing the generating function technique, we establish a recursive relation between the generating function of the first passage time (FPT) and derive the relationship between the mean first passage time (MFPT) with resetting and the generating function of the FPT without resetting. Our analysis covers various scenarios for a random walker reaching a target site from the starting position; for each case, we determine the optimal resetting probability γ * that minimizes the MFPT. We compare the results with the MFPT without resetting and find that the inclusion of resetting significantly enhances the search efficiency, particularly as the size of the network increases. Our findings highlight the potential of stochastic resetting as an effective strategy for the optimization of search processes in complex networks, offering valuable insights for applications in various fields in which efficient search strategies are crucial. [ABSTRACT FROM AUTHOR]

In this paper, we define a novel family of arithmetic sequences associated with the Fibonacci numbers. Consider the ordinary Fibonacci sequence { f n } n ∈ N 0 having initial terms f 0 = 0 , and f 1 = 1 and recurrence relation f n = f n − 1 + f n − 2 (n ≥ 2) . In many studies, authors worked on the generalizations of integer sequences in different ways, some by preserving the initial terms, others by preserving the recurrence relation, and some for numeric sets other than positive integers. Here, we will follow the third path. So, in this article, we study a new extension t f n ∗ , with initial terms t f 0 ∗ = (f 0 ∗ , f 1 ∗ , f 2 ∗) and t f 1 ∗ = (f 1 ∗ , f 2 ∗ , f 3 ∗) , which is generated by the recurrence relation t f n ∗ = t f n − 1 ∗ + t f n − 2 ∗ (n ≥ 2) , the Fibonacci-type sequence. The aim of this paper is to define Tricomplex Fibonacci numbers as an extension of the Fibonacci sequence and to examine some of their properties such as the recurrence relation, summation formula and generating function, and some classical identities. [ABSTRACT FROM AUTHOR]

Given an integer n ≥ 4, how many inequivalent quadrilaterals with ordered integer sides and perimeter n are there? Denoting such number by Q(n), the answer is given by the following closed formula: ... . [ABSTRACT FROM AUTHOR]

In this paper, we define quadra Fibona-Pell hybrid numbers by using the definition of hybrid numbers. After that, we introduce quadra Fibona-Pell hybrinomials and investigate some properties of quadra Fibona-Pell hybrid numbers and hybrinomials. Finally, we show the matrix representation of quadra Fibona-Pell hybrinomials. [ABSTRACT FROM AUTHOR]

Özdemir [1] introduced Hybrid numbers as a generalization of complex, hyperbolic and dual numbers. Hybrinomial sequence is the combination of hybrid numbers and polynomial sequence. In this paper we consider special kind of hybrinomial sequence, namely the Van Der Laan hybrinomial sequence. Binet-Like Formula, generating function and exponential generating function of this sequence are shown in this paper. Properties and some summation identities for Van Der Laan polynomial sequence and hybrinomial sequence are represented in this paper. In addition, some interesting summation identities of Van Der Laan hybrid numbers are obtained. [ABSTRACT FROM AUTHOR]

In this paper, we derive some nonlinear differential equations from generating function of generalized harmonic numbers and give some identities involving generalized harmonic numbers and special numbers by using these differential equations. For example, for any positive integers N, n, r, a and any integer m ≥2, ... where S1 (n, k) is Stirling number of the first kind. [ABSTRACT FROM AUTHOR]

This paper presents a generalization of the sequence defined by the third-order recurrence relation Vn (aj, Pj) = ³Σj=1PjVn-j, n ≥ 4, P3 ≠ 0 with initial terms Vj = aj, where ajand pj j = 1,2,3, are any non-zero real numbers. The generating function and Binet's formula are derived for this generalized tribonacci sequence. Classical second-order generalized Fibonacci sequences and other existing sequences based on second-order recurrence relations are implicitly included in this analysis. These derived sequences are discussed as special cases of the generalization. A pictorial representation is provided, illustrating the growth and variation of tribonacci numbers for different initial terms aj and coefficients pj. Additionally, the tribonacci constant is examined and visually represented. It is observed that the constant is influenced solely by the coefficients pj of the recurrence relation and is unaffected by the initial terms aj. [ABSTRACT FROM AUTHOR]

A three-parameter probability distribution is derived from a composed cumulative distribution function, which is itself a family of bounded support transformations. The transformed model called the doubly bounded exponential distribution, which exhibits decreasing shaped density while the hazard rate has increasing shape. Some statistical properties are obtained in closed form, such as the moments and various entropy functions. The parameter estimation is carried out by the methods of maximum likelihood estimate, least squares estimate, weighted least squares estimate, Anderson-Darling estimate, and Cramér-von Mises estimate. The performance of these estimators is assessed through a Monte Carlo simulation study. The identifiability of the DB-Exp model's parameters is also investigated. The proposed distribution can produce a higher performance than several well-known bounded distributions in the literature, as shown by an application to rainfall data. [ABSTRACT FROM AUTHOR]

It is known that the following five counting problems lead to the same integer sequence f t (n) : the number of nonequivalent compact Huffman codes of length n over an alphabet of t letters, the number of "nonequivalent" complete rooted t-ary trees (level-greedy trees) with n leaves, the number of "proper" words (in the sense of Even and Lempel), the number of bounded degree sequences (in the sense of Komlós, Moser, and Nemetz), and the number of ways of writing 1 = 1 t x 1 + ⋯ + 1 t x n with integers 0 ≤ x 1 ≤ x 2 ≤ ⋯ ≤ x n . In this work, we show that one can compute this sequence for all n < N with essentially one power series division. In total we need at most N 1 + ε additions and multiplications of integers of cN bits (for a positive constant c < 1 depending on t only) or N 2 + ε bit operations, respectively, for any ε > 0 . This improves an earlier bound by Even and Lempel who needed O (N 3) operations in the integer ring or O (N 4) bit operations, respectively. [ABSTRACT FROM AUTHOR]

In this study, the k -Oresme and k -Oresme-Lucas sequences are defined, and some terms of these sequence are given. Then, the relations between the terms of the k -Oresme and k -Oresme-Lucas sequences are presented. In addition, we give these sequences the Binet formulas, generating functions, Cassini identity, Catalan identity etc. Moreover, the k -Oresme and k -Oresme-Lucas sequences are associated with Fibonacci, Pell numbers and Lucas, and Pell- Lucas numbers, respectively. Finally, the Catalan transforms of these sequences are given and Hankel transforms are applied to these Catalan sequences and associated with the terms of the sequence. [ABSTRACT FROM AUTHOR]

In this paper we consider a forced oscillator with a discontinuous restoring force. By the Aubry–Mather theory we prove that there exist infinitely many periodic and quasiperiodic solutions. The proof relies on analysing the generating function of the system. The approach is applicable to studying the dynamics of more general forced nonsmooth oscillators of Hamiltonian type. [ABSTRACT FROM AUTHOR]

There is a well-known bijection between finite binary sequences and integer partitions. Sequences of length r correspond to partitions of perimeter r + 1 . Motivated by work on rational numbers in the Calkin–Wilf tree, we classify partitions whose corresponding binary sequence is a palindrome. We give a generating function that counts these partitions, and describe how to efficiently generate all of them. Atypically for partition generating functions, we find an unusual significance to prime degrees. Specifically, we prove there are nontrivial palindrome partitions of n except when n = 3 or n + 1 is prime. We find an interesting new "branching diagram" for partitions, similar to Young's lattice, with an action of the Klein four group corresponding to natural operations on the binary sequences. [ABSTRACT FROM AUTHOR]

AbstractThis paper characterises the shape of the Young diagram associated with integer partitions in terms of two newly invented statistics which are defined in the text: namely interior vertices and horizontal edges. Generating functions and associated asymptotic results are developed for both interior vertices and horizontal edges. [ABSTRACT FROM AUTHOR]

AbstractInversion sequences of length n are positive integer sequences e1 e2 …en such that 1 ≤ ei ≤ i for all 1 ≤ i ≤ n. These sequences are in bijection with the permutations of [n]. This paper focuses on the polyomino or barpgraph representation of the inversion sequences. More precisely, we study the distribution of lattice points on these polyominoes. We find the generating functions respect to the length, the number of interior vertices, corners, and vertices of a given degree. We also give simple explicit formulas for the total values of these parameters over all polyominoes of inversion sequences of length n. Throughout this work, we use symbolic computer algebra to facilitate the computations. [ABSTRACT FROM AUTHOR]

In this paper, we obtain a combinatorial expression for the Perron root and eigenvectors of a non-negative integer matrix using techniques from symbolic dynamics. We associate such a matrix with a multigraph and consider the edge shift corresponding to it. This gives rise to a collection of forbidden words F which correspond to the non-existence of an edge between two vertices, and a collection of repeated words R with multiplicities which correspond to multiple edges between two vertices. In general, for given collections F of forbidden words and R of repeated words with pre-assigned multiplicities, we construct a generalized language as a multiset. A combinatorial expression that enumerates the number of words of fixed length in this generalized language gives the Perron root and eigenvectors of the adjacency matrix. We also obtain conditions under which such a generalized language is a language of an edge shift. [ABSTRACT FROM AUTHOR]