An antipodal labeling is a function \(f\ \)from the vertices of \(G\) to the set of natural numbers such that it satisfies the condition \(d(u,v) + \left| f(u) – f(v) \right| \geq d\), where d is the diameter of \(G\ \)and \(d(u,v)\) is the shortest distance between every pair of distinct vertices \(u\) and \(v\) of \(G.\) The span of an antipodal labeling \(f\ \)is \(sp(f) = \max\{|f(u) – \ f\ (v)|:u,\ v\, \in \, V(G)\}.\) The antipodal number of~G, denoted by~an(G), is the minimum span of all antipodal labeling of~G. In this paper, we determine the antipodal number of Mongolian tent and Torus grid.
A module \(M\) over a commutative ring is termed an \(SCDF\)-module if every Dedekind finite object in \(\sigma[M]\) is finitely cogenerated. Utilizing this concept, we explore several properties and characterize various types of \(SCDF\)-modules. These include local \(SCDF\)-modules, finitely generated $SCDF$-modules, and hollow \(SCDF\)-modules with \(Rad(M) = 0 \neq M\). Additionally, we examine \(QF\) SCDF-modules in the context of duo-ri
Let \(G=(V(G),E(G))\) be a graph with \(p\) vertices and \(q\) edges. A graph \(G\) of size \(q\) is said to be odd graceful if there exists an injection \(\lambda: V(G) \to {0,1,2,\ldots,2q-1}\) such that assigning each edge \(xy\) the label or weight \(|\lambda(x) – \lambda(y)|\) results in the set of edge labels being \({1,3,5,\ldots,2q-1}\). This concept was introduced in 1991 by Gananajothi. In this paper, we examine the odd graceful labeling of the \(W\)-tree, denoted as \(WT(n,k)\).
This paper introduces a novel type of convex function known as the refined modified \((h,m)\)-convex function, which is a generalization of the traditional \((h,m)\)-convex function. We establish Hadamard-type inequalities for this new definition by utilizing the Caputo \(k\)-fractional derivative. Specifically, we derive two integral identities that involve the nth order derivatives of given functions and use them to prove the estimation of Hadamard-type inequalities for the Caputo \(k\)-fractional derivatives of refined modified \((h,m)\)-convex functions. The results obtained in this research demonstrate the versatility of the refined modified \((h,m)\)-convex function and the usefulness of Caputo \(k\)-fractional derivatives in establishing important inequalities. Our work contributes to the existing body of knowledge on convex functions and offers insights into the applications of fractional calculus in mathematical analysis. The research findings have the potential to pave the way for future studies in the area of convex functions and fractional calculus, as well as in other areas of mathematical research.
In this paper, we utilize the \(\sigma\) category to introduce \(EKFN\)-modules, which extend the concept of the \(EKFN\)-ring. After presenting some properties, we demonstrate, under certain hypotheses, that if \(M\) is an \(EKFN\)-module, then the following equivalences hold: the class of uniserial modules coincides with the class of \(cu\)-uniserial modules; \(EKFN\)-modules correspond to the class of locally noetherian modules; and the class of \(CD\)-modules is a subset of the \(EKFN\)-modules.
Let \(G\) be a finite solvable group and \(\Delta\) be the subset of \(\Upsilon \times \Upsilon\), where \(\Upsilon\) is the set of all pairs of size two commuting elements in \(G\). If \(G\) operates on a transitive \(G\) – space by the action \((\upsilon_{1},\upsilon_{2})^{g}=(\upsilon_{1}^{g},\upsilon_{2}^{g})\); \(\upsilon_{1},\upsilon_{2} \in \Upsilon\) and \(g \in G\), then orbits of \(G\) are called orbitals. The subset \(\Delta_{o}=\{(\upsilon,\upsilon);\upsilon \in \Upsilon, (\upsilon,\upsilon) \in \Upsilon \times \Upsilon\}\) represents \(G’s\) diagonal orbital.
The orbital regular graph is a graph on which \(G\) acts regularly on the vertices and the edge set. In this paper, we obtain the orbital regular graphs for some finite solvable groups using a regular action. Furthermore, the number of edges for each of a group’s orbitals is obtained.
By combining the telescoping method with Cassini–like formulae, we evaluate, in closed forms, four classes of sums about products of two arctangent functions with their argument involving Pell and Pell–Lucas polynomials. Several infinite series identities for Fibonacci and Lucas numbers are deduced as consequences.
Integer partitions of \( n \) are viewed as bargraphs (i.e., Ferrers diagrams rotated anticlockwise by 90 degrees) in which the \( i \)-th part of the partition \( x_i \) is given by the \( i \)-th column of the bargraph with \( x_i \) cells. The sun is at infinity in the northwest of our two-dimensional model, and each partition casts a shadow in accordance with the rules of physics. The number of unit squares in this shadow but not being part of the partition is found through a bivariate generating function in \( q \) tracking partition size and \( u \) tracking shadow. To do this, we define triangular \( q \)-binomial coefficients which are analogous to standard \( q \)-binomial coefficients, and we obtain a formula for these. This is used to obtain a generating function for the total number of shaded cells in (weakly decreasing)
partitions of \( n \).
We consider a scalar-valued implicit function of many variables, and provide two closed formulae for all of its partial derivatives. One formula is based on products of partial derivatives of the defining function, the other one involves fewer products of building blocks of multinomial type, and we study the combinatorics of the coefficients showing up in both formulae.
Tensors, or multi-linear forms, are important objects in a variety of areas from analytics, to combinatorics, to computational complexity theory. Notions of tensor rank aim to quantify the “complexity” of these forms, and are thus also important. While there is one single definition of rank that completely captures the complexity of matrices (and thus linear transformations), there is no definitive analog for tensors. Rather, many notions of tensor rank have been defined over the years, each with their own set of uses.
In this paper we survey the popular notions of tensor rank. We give a brief history of their introduction, motivating their existence, and discuss some of their applications in computer science. We also give proof sketches of recent results by Lovett, and Cohen and Moshkovitz, which prove asymptotic equivalence between three key notions of tensor rank over finite fields with at least three elements.
We prove two conjectures due to Sun concerning binomial-harmonic sums. First, we introduce a proof of a formula for Catalan’s constant that had been conjectured by Sun in 2014. Then, using a similar approach as in our first proof, we solve an open problem due to Sun involving the sequence of alternating odd harmonic numbers. Our methods, more broadly, allow us to reduce difficult binomial-harmonic sums to finite combinations of dilogarithms that are evaluable using previously known algorithms.
The aim of this work is to establish congruences \( \pmod{p^2} \) involving the trinomial coefficients \( \binom{np-1}{p-1}_2 \) and \( \binom{np-1}{(p-1)/2}_2 \) arising from the expansion of the powers of the polynomial \( 1 + x + x^2 \). In main results, we extend some known congruences involving the binomial coefficients \( \binom{np-1}{p-1} \) and \( \binom{np-1}{(p-1)/2} \), and establish congruences linking binomial coefficients and harmonic numbers.
In analogy with the semi-Fibonacci partitions studied recently by Andrews, we define semi-\( m \)-Pell compositions. We find that these are in bijection with certain weakly unimodal \( m \)-ary compositions. We give generating functions, bijective proofs, and a number of unexpected congruences for these objects. In the special case of \( m = 2 \), we have a new combinatorial interpretation of the semi-Pell sequence and connections to other objects.
The aim of this paper is to introduce and study a new class of analytic functions which generalize the classes of \(\lambda\)-Spirallike Janowski functions. In particular, we gave the representation theorem, the right side of the covering theorem, starlikeness estimates and some properties related to the functions in the class \( S_\lambda ( T, H, F ) \).
criteria to verify log-convexity of sequences is presented. Iterating this criteria produces infinitely log-convex sequences. As an application, several classical examples of sequences arising in Combinatorics and Special Functions are presented. The paper concludes with a conjecture regarding coefficients of chromatic polynomials.
We discuss the VC-dimension of a class of multiples of integers and primes (equivalently indicator functions) and demonstrate connections to prime counting functions. Additionally, we prove limit theorems for the behavior of an empirical risk minimization rule as well as the weights assigned to the output hypothesis in AdaBoost for these “prime-identifying” indicator functions, when we sample \( mn \) i.i.d. points uniformly from the integers \(\{2, \ldots, n\}\).
Integer compositions and related counting problems are a rich and ubiquitous topic in enumerative combinatorics. In this paper we explore the definition of symmetric and asymmetric peaks and valleys over compositions. In particular, we compute an explicit formula for the generating function for the number of integer compositions according to the number of parts, symmetric, and asymmetric peaks and valleys.
In this paper we show some identities come from the \( q \)-identities of Euler, Jacobi, Gauss, and Rogers-Ramanujan. Some of these identities relate the function sum of divisors of a positive integer \( n \) and the number of integer partitions of \( n \). One of the most intriguing results found here is given by the next equation, for \( n > 0 \),
\[
\sum_{l=1}^n \frac{1}{l!} \sum_{w_1+w_2+\cdots+w_l \in C(n)} \frac{\sigma_1(w_1) \sigma_1(w_2) \cdots \sigma_1(w_l)}{w_1 w_2 \cdots w_l} = p_1(n),
\]
where \( \sigma_1(n) \) is the sum of all positive divisors of \( n \), \( p_1(n) \) is the number of integer partitions of \( n \), and \( C(n) \) is the set of integer compositions of \( n \). In the last section, we show seven applications, one of them is a series expansion for
\[
\frac{(q^{a_1};q^{b_1})_\infty (q^{a_2};q^{b_2})_\infty \cdots (q^{a_k};q^{b_k})_\infty}
{(q^{c_1};q^{d_1})_\infty (q^{c_2};q^{d_2})_\infty \cdots (q^{c_r};q^{d_r})_\infty},
\]
where \( a_1, \ldots, a_k, b_1, \ldots, b_k, c_1, \ldots, c_r, d_1, \ldots, d_r \) are positive integers, and \( |q| < 1 \).
Between Bernoulli/Euler polynomials and Pell/Lucas polynomials, convolution sums are evaluated in closed form via the generating function method. Several interesting identities involving Fibonacci and Lucas numbers are shown as consequences including those due to Byrd \( (1975) \) and Frontczak \( (2020) \).
The notion of length spectrum for natural numbers was introduced by Pong in \([5]\). In this article, we answer the question of how often one can recover a random number from its length spectrum. We also include a quick deduction of a result of LeVeque in \([4]\) on the average order of the size of length spectra.
This paper uses exponential sum methods to show that if \( E \subset M_2(\mathbb{Z}/p^r) \)
is a finite set of \( 2 \times 2 \) matrices with sufficiently large density and \( j \) is any unit in the finite ring \( \mathbb{Z}/p^r \), then there exist at least two elements of \( E \) whose difference has determinant \( j \).
In this paper, we introduce a generalized family of numbers and polynomials of one or more variables attached to the formal composition \( f \cdot (g \circ h) \) of generating functions \( f \), \( g \), and \( h \). We give explicit formulae and apply the obtained result to two special families of polynomials; the first concerns the generalization of some polynomials applied to the theory of hyperbolic differential equations recently introduced and studied by \( M. \, Mihoubi \) and \( M. \, Sahari \). The second concerns two-variable Laguerre-based generalized Hermite-Euler polynomials introduced and should be updated to studied recently by \( N. \, U. \, Khan \, \textit{et al.} \).
In this paper, we show that the generalized exponential polynomials and the generalized Fubini polynomials satisfy certain binomial identities and that these identities characterize the mentioned polynomials (up to an affine transformation of the variable) among the class of the normalized Sheffer sequences.
Let \( A \) be a subset of a finite field \( \mathbb{F} \). When \( \mathbb{F} \) has prime order, we show that there is an absolute constant \( c > 0 \) such that, if \( A \) is both sum-free and equal to the set of its multiplicative inverses, then \( |A| < (0.25 – c)|\mathbb{F}| + o(|\mathbb{F}|) \) as \( |\mathbb{F}| \to \infty \). We contrast this with the result that such sets exist with size at least \( 0.25|\mathbb{F}| – o(|\mathbb{F}|) \) when \( \mathbb{F} \) has characteristic 2.
In this paper, we will recover the generating functions of Tribonacci numbers and Chebychev polynomials of first and second kind. By making use of the operator defined in this paper, we give some new generating functions for the binary products of Tribonacci with some remarkable numbers and polynomials. The technique used here is based on the theory of the so-called symmetric functions.
It is shown that if \( V \subseteq \mathbb{F}_p^{n \times \cdots \times n} \) is a subspace of \( d \)-tensors with dimension at least \( tn^{d-1} \), then there is a subspace \( W \subseteq V \) of dimension at least \( t / (dr) – 1 \) whose nonzero elements all have analytic rank \( \Omega_{d, p}(r) \). As an application, we generalize a result of Altman on Szemerédi’s theorem with random differences.
Extensions of a set partition obtained by imposing bounds on the size of the parts and the coloring of some of the elements are examined. Combinatorial properties and the generating functions of some counting sequences associated with these partitions are established. Connections with Riordan arrays are presented.
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
In this Paper, we establish a new application of the Mittag-Lefier Function method that will enlarge the application to the non linear Riccati Differential equations with fractional order. This method provides results that converge promptly to the exact solution. The description of fractional derivatives is made in the Caputo sense. To emphasize the consistency of the approach, few illustrations are presented to support the outcomes. The outcomes declare that the procedure is very constructive and relavent for determining non linear Ricati differential equations of fractional order.
One of the important features of an interconnection network is its ability to efficiently simulate programs or parallel algorithms written for other architectures. Such a simulation problem can be mathematically formulated as a graph embedding problem. In this paper, we embed complete multipartite graphs into certain trees, such as \(k\)-rooted complete binary trees and \(k\)-rooted sibling trees.
In this paper we compute the \(P_3\)-forcing number of honeycomb network. A dynamic coloring of the vertices of a graph \(G\) starts with an initial subset \(S\) of colored vertices, with all remaining vertices being non-colored. At each discrete time interval, a colored vertex with exactly one non-colored neighbor forces this non-colored neighbor to be colored. The initial set \(S\) is called a forcing set of \(G\) if, by iteratively applying the forcing process, every vertex in G becomes colored. If the initial set \(S\) has the added property that it induces a subgraph of \(G\) whose components are all paths of length 3, then \(S\) is called a \(P_3\)-forcing set of \(G\). A Ps-forcing set of \(G\) of minimum cardinality is called the \(P_3\)-forcing number of G denoted by \(ZP_3(G)\).
In this paper, we introduced a new concept called nonsplit monophonic set and its relative parameter nonsplit monophonic number \(m_{ns}(G)\). Some certain properties of nonsplit monophonic sets are discussed. The nonsplit monophonic number of standard graphs are investigated. Some existence theorems on nonsplit monophonic number are established.
In graph theory and network analysis, centrality measures identify the most important vertices within a graph. In a connected graph, closeness centrality of a node is a measure of centrality, calculated as the reciprocal of the sum of the lengths of the shortest paths between the node and all other nodes in the graph. In this paper, we compute closeness centrality for a class of neural networks and the sibling trees, classified as a family of interconnection networks.
Let \(G = (V, E)\) be a graph. A total dominating set of \(G\) which intersects every minimum total dominating set in \(G\) is called a transversal total dominating set. The minimum cardinality of a transversal total dominating set is called the transversal total domination number of G, denoted by \(\gamma_{tt}(G)\). In this paper, we begin to study this parameter. We calculate \(\gamma_{tt}(G)\) for some families of graphs. Further some bounds and relations with other domination parameters are obtained for \(\gamma_{tt}(G)\).
Let \(G\) be any graph. The concept of paired domination was introduced having gaurd backup concept in mind. We introduce pendant domination concept, for which at least one guard is assigned a backup, A dominating set \(S\) in \(G\) is called a pendant dominating set if \((S)\) contains at least one pendant vertex. The least cardinality of a pendant dominating set is called the pendant domination number of G denoted by \(\gamma_{pe}(G)\). In this article, we initiate the study of this parameter. The exact value of \(\gamma_{pe}(G)\) for some families of standard graphs are obtained and some bounds are estimated. We also study the proprties of the parameter and interrelation with other invarients.
Transportation Problem (TP) is the exceptional case to obtain the minimum cost. A new hypothesis is discussed for getting minimal cost in transportation problem in this paper and also Vogel’s Approximation Method (VAM) and MODI method are analyzed with the proposed method. This approach is examined with various numerical illustrations.
This paper mainly surveys the literature on bulk queueing models and its applications. Distributed Different systems in the zone of queuing speculation merging mass queuing architecture. These mass queueing models are often related to confirm the clog issues. Through this diagram, associate degree challenge has been created to envision the paintings accomplished on mass strains, showing various wonders and also the goal is to present enough facts to inspectors, directors and enterprise those that are dependent on the usage of queueing hypothesis to counsel blockage troubles and need to find the needs of enthusiasm of applicable models near the appliance.
Frank Harary and Allen J. Schwenk have given a formula for counting the number of non-isomorphic caterpillars on \(n\) vertices with \(n ≥ 3\). Inspired by the formula of Frank Harary and Allen
J. Schwenk, in this paper, we give a formula for counting the number of non-isomorphic caterpillars with the same degree sequence.
The line graph \(L(G)\) of a connected graph G, has vertex set identical with the set of edges of \(G\), and two vertices of \(L(G)\) are adjacent if and only if the corresponding edges are adjacent in \(G\). Ivan Gutman et al examined the dependency of certain physio-chemical properties of alkanes in boiling point, molar volume, and molar refraction, heat of vapourization, critical temperature, critical pressure and surface tension on the Bertz indices of \(L'(G)\) Dobrynin and Melnikov conjectured that there exists no nontrivial tree \(T\) and \(i≥3\), such that \(W(L'(T)) = W(T)\). In this paper we study Wiener and Zagreb indices for line graphs of Complete graph, Complete bipartite graph and wheel graph.
A set S of vertices in a graph G is called a dominating set of G if every vertex in V(G)\S is adjacent to some vertex in S. A set S is said to be a power dominating set of G if every vertex in the system is monitored by the set S following a set of rules for power system monitoring. The power domination number of G is the minimum cardinality of a power dominating set of G. In this paper, we solve the power domination number for certain nanotori such as H-Naphtelanic, \(C_5C_6C_7[m,n]\) nanotori and \(C_4C_6C_8[m,n]\) nanotori.
Let \(G_k, (k ≥ 0)\) be the family of graphs that have exactly k cycles. For \(0 ≤ k ≤ 3\), we compute the Hadwiger number for graphs in \(G_k\) and further deduce that the Hadwiger Conjecture is true for such families of graphs.
Split domination number of a graph is the cardinality of a minimum dominating set whose removal disconnects the graph. In this paper, we define a special family of Halin graphs and determine the split domination number of those graphs. We show that the construction yield non-isomorphic families of Halin graphs but with same split domination numbers.
A graph \(G(v,E)\) with \(n\) vertices is said to have modular multiplicative divisor bijection \(f: V(G)→{1,2,.., n}\) and the induced function \(f*: E(G) → {0,1,2,…, n – 1}\) where \(f*(uv)=f(u)f(v)(mod\,\,n)\) for all \(uv \in E(G)\) such that \(n\) divides the sum of all edge labels of \(G\). This paper studies MMD labeling of an even arbitrary supersubdivision (EASS) of corona related graphs.
In this paper, the distance and degree based topological indices for double silicate chain graph are obtained.
In this paper, we introduce a new form of fuzzy number named as Icosikaitetragonal fuzzy number with its membership function. It includes some basic arithmetic operations like addition, subtraction, multiplication and scalar multiplication by means of \(\alpha\)-cut with numerical illustrations.
In this paper, we determine the wirelength of embedding complete bipartite graphs \(K_{2^{n-1}, 2^{n-1}} into 1-rooted sibling tree \(ST_n^1\), and Cartesian product of 1-rooted sibling trees and paths.
A dominator coloring is a proper vertex coloring of a graph \(G\) such that each vertex is adjacent to all the vertices of at least one color class or either alone in its color class. The minimum cardinality among all dominator coloring of \(G\) is a dominator chromatic number of \(G\), denoted by \(X_d(G)\). On removal of a vertex the dominator chromatic number may increase or decrease or remain unaltered. In this paper, we have characterized nontrivial trees for which dominator chromatic number is stable.
1970-2025 CP (Manitoba, Canada) unless otherwise stated.