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In this paper, the group homomorphism graph is introduced and investigated. The group homomorphism graph, denoted by H I (G , G ′) , is an undirected graph in which the vertex set contains all homomorphisms excluding the monomorphisms and the zero homomorphism from the group G to the group G ′ , and two distinct vertices are adjacent if and only if the intersection of their kernels is non-trivial. We investigate the interplay between the graph-theoretic properties of this graph with algebraic properties of the groups. In this work, connectedness, diameter, clique number, chromatic number, domination number, independence number, etc. are found. [ABSTRACT FROM AUTHOR]

A regular matroid M on a finite set E is represented by a totally unimodular matrix. The set of vectors from Z E orthogonal to rows of the matrix form a regular chain group N. Assume that ψ is a homomorphism from N into a finite additive Abelian group A and let A ψ [ N ] be the set of vectors g from (A − 0) E , such that ∑ e ∈ E g (e) · f (e) = ψ (f) for each f ∈ N (where · is a scalar multiplication). We show that | A ψ [ N ] | can be evaluated by a polynomial function of | A | . In particular, if ψ (f) = 0 for each f ∈ N , then the corresponding assigning polynomial is the classical characteristic polynomial of M. [ABSTRACT FROM AUTHOR]

This research aims to introduce and explore the theory of neutrosophic soft hyperalgebras (N S H A s), focusing on their core principles and potential applications in decision-making under uncertainty. By defining key operations such as intersection and union, we clarify the foundational characteristics of N S H A s and their relationship to soft hyperalgebras. The concepts of ξ β -identity N S H A and ξ -absolute N S H A are also examined to better understand their properties. The practical relevance of N S H A is demonstrated through applications in various fields, highlighting its adaptability in addressing complex decision-making scenarios. This approach offers a novel, more precise method for navigating uncertainty in areas such as project methodology selection, sensitivity analysis, and AI chatbot selection. [ABSTRACT FROM AUTHOR]

The minimum spanning tree problem in graphs under budget-type degree constraints (DCMST) is a well-known NP-hard problem. Spanning trees do not always exist, and the optimum can not be approximated within a constant factor. Recently, solutions have been proposed to solve degree-constrained spanning problems in the case of limited momentary capacities of the nodes. For a given node, the constraint represents a limited degree of the node for each visit. Finding the solution with minimum cost is NP-hard and the related algorithms are not trivial. This paper focuses on this new spanning problem with heterogeneous capacity-like degree bounds. The minimum cost solution corresponds to a graph-related structure, i.e., a hierarchy. We study the conditions of its existence, and we propose its exact computation, a heuristic algorithm, and its approximation. [ABSTRACT FROM AUTHOR]

This article presents a new idea for an extension of the fuzzy INK algebra called bipolar fuzzy INK subalgebra. The objective of this study is to define the features that distinguish bipolar fuzzy INK-subalgebras of INK-algebras. The algebraic operations on these sub-algebras are also studied. The thorough examination allows us to prove a number of theorems that shed light on the connections between the higher and lower-level sets related to these ideas. In addition, several related topics are thoroughly examined, and the idea of homomorphism for bipolar fuzzy INK sub-algebras is introduced. [ABSTRACT FROM AUTHOR]

The middle graph M(G) of a graph G is an intersection graph on the vertex set V (G) of any graph G. Let E(G) be an edge set of G and F = V 0(G) [ E(G), where V 0(G) indicates the family of all one vertex subsets of the set V (G). This concept was introduced by T. Hamada and I. Yoshimura [4]. M. Chandramouleeswaran et al., studied isomorphism and automorphism groups for semiring valued graph (S-valued graph). In this paper, we study the morphisms and its properties of middle graph of S-valued graphs. [ABSTRACT FROM AUTHOR]

The diamond product of a graph G(V, E) with a graph H(V',E') denoted by G ◊ H is a graph whose a vertex set V(G ◊ H) is a Hom(G,H) and an edge set E(G ◊ H) = {{f,g}\f,g ∈ Hom(G,H) and {f (x),g(x)} ∈ E', for all x ∈ V(G)}. A round robin tournament problem involves creating a schedule where each participant plays against every other participant exactly once. This research represents the application of the diamond product of path graph and complete graph to 2n-participants round robin tournament problem. Moreover, the research also represents an algorithm to find a solution of 2n-participants round robin tournament problem. [ABSTRACT FROM AUTHOR]

We say that a signed graph is k $k$‐critical if it is not k $k$‐colorable but every one of its proper subgraphs is k $k$‐colorable. Using the definition of colorability due to Naserasr, Wang, and Zhu that extends the notion of circular colorability, we prove that every 3‐critical signed graph on n $n$ vertices has at least 3n−12 $\frac{3n-1}{2}$ edges, and that this bound is asymptotically tight. It follows that every signed planar or projective‐planar graph of girth at least 6 is (circular) 3‐colorable, and for the projective‐planar case, this girth condition is best possible. To prove our main result, we reformulate it in terms of the existence of a homomorphism to the signed graph C3* ${C}_{3}^{* }$, which is the positive triangle augmented with a negative loop on each vertex. [ABSTRACT FROM AUTHOR]

Understanding the concept of group theory and to apply it in other field of sciences has been a problem among undergraduate students. Although there are some attempt by Dubinsky et al, [1], where some finite groups were discussed among students, the paper was limited to theoretical aspect of the topic. This research is therefore designed to explore clearly the procedure for constructing finite groups for better understanding of the subject area in the given domain. The research is channeled towards the use of group theory in Network Dynamics, which serves as concrete application of finite groups to the students. Some minor open problems with regards to algebraic graph theory was discussed which lead to network dynamics and groupoid formalism. [ABSTRACT FROM AUTHOR]

This article presents a new idea for an extension of the fuzzy INK algebra called bipolar fuzzy INK subalgebra. The objective of this study is to define the features that distinguish bipolar fuzzy INK-subalgebras of INK-algebras. The algebraic operations on these sub-algebras are also studied. The thorough examination allows us to prove a number of theorems that shed light on the connections between the higher and lower-level sets related to these ideas. In addition, several related topics are thoroughly examined, and the idea of homomorphism for bipolar fuzzy INK sub-algebras is introduced.

An essential and quite different class of functions is the triangular norm and its related operators (uninorms, nullnorms, and associative copulas). They are used widely in many disciplines, including fuzzy set theory, probability and statistics, decision sciences, and others. This paper proposes the notion of hybrid Ξ -norm and defines the concepts of Ξ -hybrid ideals, Ξ -hybrid h -ideals in a hemiring . Some equivalent conditions are obtained for a hybrid structure to be a Ξ -hybrid left h -ideal, and it is proved that every imaginable Ξ -hybrid left h -ideal of hemiring is a hybrid left h -ideal, but not conversely by an example. [ABSTRACT FROM AUTHOR]

The notion of complex interval-valued q-neutrosophic subbisemiring (CIVqNSBS) is developed and examined. Additionally, we examine the homomorphic features and significant attributes of CIVqNSBS. We suggest the CIVqNSBS level sets for bisemirings. Consider a complex neutrosophic subset of bisemiring Δ, denoted as-→ℵ, if and only if every non-empty level set--→Z(∂,♭) is a subbisemiring, where ∂, ♭ ∈ D[0, 1], then-→Z = (-→Z⊺ℵ,-→Z גℵ,-→ZℲℵ) is a CIVqNSBS of Δ. Let-→ℵ be the strongest complex neutrosophic relation of bisemiring Δ, and let -→Ψ be a CIVqNSBS of bisemiring Δ, if and only if -→Ψ is a CIVqNSBS of Δ × Δ, then -→ℵ is a CIVqNSBS of bisemiring Δ. We show that homomorphic images of all CIVqNSBSs are CIVqNSBSs, and homomorphic pre-images of all CIVqNSBSs are CIVqNSBSs. There are examples given to illustrate our results. [ABSTRACT FROM AUTHOR]

Fuzzy graphs find numerous applications in real life. One of the extensions of fuzzy graphs is Fermatean fuzzy graphs. Here, we introduce the concepts of Fermatean fuzzy set to the domain of graph theory and obtain a novel type of graph, referred to as Fermatean fuzzy graph (FFG ). The article establishes fundamental terms such as strong FFG , complete FFG , regular FFG , path, degree, total degree, homomorphism, and isomorphism of FFG , as well as the complement of FFG . Alongside the introduction of these concepts, their important properties, theorems, and illustrative examples are defined and discussed. Finally, an application has surfaced suggesting the utilization of FFG s to scrutinize the key factors affecting the productivity of Tata Motors, paving the way for a comprehensive analysis of the company's operational efficiency using score function. [ABSTRACT FROM AUTHOR]

A graph H is common if the limit as n → ∞ of the minimum density of monochro-matic labelled copies of H in an edge colouring of Kn with red and blue is attained by a sequence of quasirandom colourings. We apply an information-theoretic approach to show that certain graphs obtained from odd cycles and paths via gluing operations are common. In fact, for every pair (H1, H2) of such graphs, there exists p ∈ (0, 1) such that an appropriate linear combination of red copies of H1 and blue copies of H2 is minimized by a quasirandom colouring in which ... edges are red; such a pair (H1, H2) is said to be (p, 1 - p)-common. Our approach exploits a strengthening of the common graph property for odd cycles that was recently proved using Schur convexity. We also exhibit a (p, 1 - p)-common pair (H1, H2) such that H2 is uncommon. [ABSTRACT FROM AUTHOR]

This paper describes several cases of adjunction in the homomorphism preorder of relational structures. We say that two functors Λ and Γ between thin categories of relational structures are adjoint if for all structures A and B, we have that Λ⁢(A) maps homomorphically to B if and only if A maps homomorphically to Γ⁢(𝐁). If this is the case, Λ is called the left adjoint to Γ and Γ the right adjoint to Λ. In 2015, Foniok and Tardif described some functors on the category of digraphs that allow both left and right adjoints. The main contribution of Foniok and Tardif is a construction of right adjoints to some of the functors identified as right adjoints by Pultr in 1970. We generalise results of Foniok and Tardif to arbitrary relational structures, and coincidently, we also provide more right adjoints on digraphs, and since these constructions are connected to finite duality, we also provide a new construction of duals to trees. Our results are inspired by an application in promise constraint satisfaction -- it has been shown that such functors can be used as efficient reductions between these problems. [ABSTRACT FROM AUTHOR]

Subgraph densities have been defined, and served as basic tools, both in the case of graphons (limits of dense graph sequences) and graphings (limits of bounded-degree graph sequences). While limit objects have been described for the "middle ranges", the notion of subgraph densities in these limit objects remains elusive. We define subgraph densities in the orthogonality graphs on the unit spheres in dimension d , under an appropriate sparsity condition on the subgraphs. These orthogonality graphs exhibit the main difficulties of defining subgraphs in the "middle" range, and so we expect their study to serve as a key example to defining subgraph densities in more general Markov spaces. Interest in studying homomorphisms of a finite graph G into orthogonality graphs is supported by the fact that such homomorphisms are just the orthonormal representations of the complementary graph. [ABSTRACT FROM AUTHOR]

In this paper, the concept of circular r $r$‐flow in a mono‐directed signed graph (G,σ) $(G,\sigma)$ is introduced. That is a pair (D,f) $(D,f)$, where D $D$ is an orientation on G $G$ and f:E(G)→(−r,r) $f:E(G)\to (-r,r)$ satisfies that ∣f(e)∣∈[1,r−1] $| f(e)| \in [1,r-1]$ for each positive edge e $e$ and ∣f(e)∣∈[0,r2−1]∪[r2+1,r) $| f(e)| \in [0,\frac{r}{2}-1]\cup [\frac{r}{2}+1,r)$ for each negative edge e $e$, and the total in‐flow equals the total out‐flow at each vertex. This is the dual notion of circular colorings of signed graphs and is distinct from the concept of circular flows in bi‐directed graphs associated with signed graphs studied in the literature. We first explore the connection between circular 2kk−1 $\frac{2k}{k-1}$‐flows and modulo k $k$‐orientations in signed graphs. Then we focus on the upper bounds for the circular flow indices of signed graphs in terms of the edge‐connectivity, where the circular flow index of a signed graph is the minimum value r $r$ such that it admits a circular r $r$‐flow. We prove that every 3‐edge‐connected signed graph admits a circular 6‐flow and every 4‐edge‐connected signed graph admits a circular 4‐flow. More generally, for k≥2 $k\ge 2$, we show that every (3k−1) $(3k-1)$‐edge‐connected signed graph admits a circular 2kk−1 $\frac{2k}{k-1}$‐flow, every 3k $3k$‐edge‐connected signed graph has a circular r $r$‐flow with r<2kk−1 $r\lt \frac{2k}{k-1}$, and every (3k+1) $(3k+1)$‐edge‐connected signed graph admits a circular 4k+22k−1 $\frac{4k+2}{2k-1}$‐flow. Moreover, the (6k−2) $(6k-2)$‐edge‐connectivity condition is shown to be sufficient for a signed Eulerian graph to admit a circular 4k2k−1 $\frac{4k}{2k-1}$‐flow, and applying this result to planar graphs, we conclude that every signed bipartite planar graph of negative girth at least 6k−2 $6k-2$ admits a homomorphism to the negative even cycles C−2k ${C}_{-2k}$. [ABSTRACT FROM AUTHOR]

We introduce the notion of q-neutrosophic cubic vague subbisemiring (q-NSCVSBS) and level set of q-NSCVSBS of a bisemiring. The q-NSCVSBS is a new concept of subbisemirings of bisemirings. Let Ξ be a neutrosophic vague subset of Λ. Then 0 = ([T-Ξ, T+Ξ ], [I-Ξ, I+Ξ ], [F-Ξ, F+Ξ ]) is a q-NSCVSBS of Λ if and only if all non empty level set 0(ϱ1,ϱ2,s) is also a SBS of Λ for every ϱ1, ϱ2, s ∈ [0, 1]. Let Ξ be the q-NSCVSBS of Λ and Υ be the strongest cubic q-neutrosophic vague relation of Λ. Then Ξ is a q-NSCVSBS of Λ × Λ. Let Ξ be the q-NSCVSBS of Λ, show that pseudo cubic q-neutrosophic vague coset (ςΞ)p is also a q-NSCVSBS of Λ, for all ς ∈ Λ. Let Ξ1,Ξ2, ...,Ξn be the any family of q - NSCV SBSs of Λ1,Λ2, ...,Λn respectively, then Ξ1 × Ξ2 × ... × Ξn is also a q-NSCVSBS of Λ1 × Λ2 × ... × Λn. The homomorphic image of every q-NSCVSBS is also a q-NSCVSBS. The homomorphic pre-image of every q-NSCVSBS is also a q-NSCVSBS. [ABSTRACT FROM AUTHOR]

Let A,B be two rings and n ⩾ 2 be an integer. An additive map h: A → B is called an n-Jordan homomorphism if h(xn) = h(x)n for all x ∈ A; h is called an n-homomorphism or an anti-n-homomorphism if h(Πni=1 xi) = Πn i=1 h(xi) or h(Πni=1 xi) = Πn i=0 h(xn-i ∈ A. We give the following variation of a theorem on n-Jordan homomorphisms due to I.N. Herstein: Let n ≥ 2 be an integer and h be an n-Jordan homomorphism from a ring A into a ring B of characteristic greater than n. Suppose further that A has a unit e, then h = h(e)τ, where h(e) is in the centralizer of h(A) and τ is a Jordan homomorphism. By using this variation, we deduce the following result of G. An: Let A and B be two rings, where A has a unit and B is of characteristic greater than an integer n ≥ 2. If every Jordan homomorphism from A into B is a homomorphism (anti-homomorphism), then every n-Jordan homomorphism from A into B is an n-homomorphism (anti-n-homomorphism). As a consequence of an appropriate lemma, we also obtain the following result of E. Gselmann: Let A,B be two commutative rings and B is of characteristic greater than an integer n ≥ 2. Then every n-Jordan homomorphism from A into B is an n-homomorphism. [ABSTRACT FROM AUTHOR]1, ..., xn ∈ A. We give the following variation of a theorem on n-Jordan homomorphisms due to I.N. Herstein: Let n ≥ 2 be an integer and h be an n-Jordan homomorphism from a ring A into a ring B of characteristic greater than n. Suppose further that A has a unit e, then h = h(e)τ, where h(e) is in the centralizer of h(A) and τ is a Jordan homomorphism. By using this variation, we deduce the following result of G. An: Let A and B be two rings, where A has a unit and B is of characteristic greater than an integer n ≥ 2. If every Jordan homomorphism from A into B is a homomorphism (anti-homomorphism), then every n-Jordan homomorphism from A into B is an n-homomorphism (anti-n-homomorphism). As a consequence of an appropriate lemma, we also obtain the following result of E. Gselmann: Let A,B be two commutative rings and B is of characteristic greater than an integer n ≥ 2. Then every n-Jordan homomorphism from A into B is an n-homomorphism. [ABSTRACT FROM AUTHOR]

The generic homomorphism problem, which asks whether an input graph \(G\) admits a homomorphism into a fixed target graph \(H\) , has been widely studied in the literature. In this article, we provide a fine-grained complexity classification of the running time of the homomorphism problem with respect to the clique-width of \(G\) (denoted \({\operatorname{cw}}\)) for virtually all choices of \(H\) under the Strong Exponential Time Hypothesis. In particular, we identify a property of \(H\) called the signature number \(s(H)\) and show that for each \(H\) , the homomorphism problem can be solved in time \(\mathcal{O^{*}}(s(H)^{{\operatorname{cw}}})\). Crucially, we then show that this algorithm can be used to obtain essentially tight upper bounds. Specifically, we provide a reduction that yields matching lower bounds for each \(H\) that is either a projective core or a graph admitting a factorization with additional properties—allowing us to cover all possible target graphs under long-standing conjectures. [ABSTRACT FROM AUTHOR]

In this paper, we give another definition of Ruston elements and almost Ruston elements, which is equivalent to the definitions given by Mouton and Raubenheimer in the case that the homomorphism has a closed range and Riesz property. For two homomorphisms, we consider the preserver problems of Fredholm theory and Fredholm spectrum theory. In addition, we study the spectral mapping theorems of Fredholm (Weyl, Browder, Ruston, and almost Ruston) elements relative to a homomorphism. Last but not least, the dependence of Fredholm theory on three homomorphisms is considered, and meanwhile, the transitivity of Fredholm theory relative to three homomorphisms is illustrated. Furthermore, we consider the Fredholm theory relative to more homomorphisms. [ABSTRACT FROM AUTHOR]

In this research paper, our aim is to introduce the new concept of neutrosophic doubt fuzzy bi-ideal of BS-algebras as an extension of doubt fuzzy bi-ideal of BS-algebras and investigated its algebraic nature. Neutrosophic doubt fuzzy bi-ideal of BS-algebras is also applied in Cartesian product. Finally, we also provide the homomorphic behaviour of Neutrosophic doubt fuzzy bi-ideal of BS-algebras.

This article has two main parts. In the first part we show that some of the basic theory of generalized polynomials on commutative semi-groups can be extended to all semigroups. In the second part we show that if a sub-semigroup S of a group G generates G in the sense that G = S · S−1, then a generalized polynomial on S with values in an Abelian group H can be extended to a generalized polynomial on G into H. Finally there is a short discussion of the extendability of exponential functions and generalized exponential polynomials.

In this paper, we discuss an important and nontrivial theorem on evaluation homomorphisms. We state this theorem as a canonical duality between the family of all smooth mappings f ∈ Hom(M,M′) of a smooth real finite-dimensional manifold M into a similar manifold M′ and the family of homomorphisms φ of the algebra C∞ (M′) of smooth scalar-valued functions on M′ into the analogous algebra C∞ (M) on M, φ ∈ Hom (C∞ (M′), C∞ (M)). This formulation possesses the maximum natural generality and, at the same time, allows it to be used in applications in the standard canonical form. [ABSTRACT FROM AUTHOR]

In this research paper, our aim is to introduce the new concept of neutrosophic doubt fuzzy bi-ideal of BSalgebras as an extension of doubt fuzzy bi-ideal of BS-algebras and investigated its algebraic nature. Neutrosophic doubt fuzzy bi-ideal of BS-algebras is also applied in Cartesian product. Finally, we also provide the homomorphic behaviour of Neutrosophic doubt fuzzy bi-ideal of BS-algebras. [ABSTRACT FROM AUTHOR]

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We show that for every integer m≥2$$ m\ge 2 $$ and large n$$ n $$, every properly edge‐colored graph on n$$ n $$ vertices with at least n(logn)53$$ n{\left(\log n\right)}^{53} $$ edges contains a rainbow subdivision of Km$$ {K}_m $$. This is sharp up to a polylogarithmic factor. Our proof method exploits the connection between the mixing time of random walks and expansion in graphs. [ABSTRACT FROM AUTHOR]

An attribute-based authority system helps in providing file/system access to only authorized users. However, they face specific challenges in securing the dataset from attacks. Hence, a novel hybrid Chimp Optimization -based Elliptical Curve Cryptosystem Model (CbECCM) was designed in this article. The main objective of this model is to overcome these demerits of the existing system and to provide file access to the authorized user. The chimp fitness in the user authorization module improves the system's confidentiality. The presented work was validated with a crop-recommendation dataset, and the outcomes are estimated. Moreover, the homomorphism algorithm increases the data integrity and reduces the time complexity. Finally, the robustness of the developed model was checked by launching the Denial of Service (DoS) attack on the cloud servers. The performance of the designed model is estimated and validated with a comparative analysis. The performance analysis shows that the developed model attained a high confidential rate of 100%. The comparative analysis proves that the developed model achieved better outcomes than others. [ABSTRACT FROM AUTHOR]

The present study investigates the relation between derivations and hyperideals on ordered hyperrings with no zero divisors. Also, we identify some results for the ordered hyperrings induced by the homomorphism of the ordered hyperrings by derivations. The present work explores some aspects of derivations in ordered hyperrings. Also, we establish some results in connection with homomorphisms and hyperideals. Furthermore, we describe prime hyperideals associated to a derivation d on an ordered hyperring T and derive several results about homomorphisms and derivations on ordered hyperrings. [ABSTRACT FROM AUTHOR]

We introduce the concept of an interval-valued neutrosophic cubic vague subbisemiring (IVNCVSBS) and level set of IVNCVSBS of a bisemiring. An IVNCVSBS is the new extension of neutrosophic subbisemirings and SBS over bisemirings. Let ℵ be a neutrosophic vague subset in Ξ, we show that i = ([k - ℵ, k + ℵ ג], [ - ℵ ג, + ℵ ], [' - ℵ, ' + ℵ ]) is a IVNCVSBS of Ξ if and only if all non empty level set i ('1,'2,s) is a SBS of Ξ for all '1, '2, s ∈ [0, 1]. Let ℵ be the IVNCVSBS of Ξ and Υ be the strongest cubic neutrosophic vague relation of Ξ. To prove that ℵ is a IVNCVSBS of Ξ n Ξ. Let ℵ be any IVNCVSBS of Ξ, prove that pseudo cubic neutrosophic vague coset (ςℵ) p is a IVNCVSBS of Ξ, for all ς ∈ Ξ. Let ℵ1, ℵ2, ..., ℵn be the family of IV NCV SBSs of Ξ1, Ξ2, ..., Ξn respectively. To prove that ℵ1 nℵ2 n...nℵn is a IVNCVSBS of Ξ1 nΞ2 n...nΞn. The homomorphic image of every IVNCVSBS is a IVNCVSBS. The homomorphic pre-image of every IVNCVSBS is a IVNCVSBS. Examples are provided to strengthen our results. [ABSTRACT FROM AUTHOR]

The objective of this paper is to investigate the innovative concept of complex neutrosophic subbisemiring. The novelty of the complex neutrosophic subbisemiring lies in its wide range of truth, indeterminacy, and false function values. It goes beyond the range of [0; 1] in the complex plane in contrast to the traditional range [0; 1]. Therefore, these three functions can be described mathematically using a complex number in the complex neutrosophic subbisemiring. We develop and analyze the concept of complex interval-valued neutrosophic subbisemiring (CIVNSBS). Moreover, we study homomorphic characteristics and important properties of CIVNSBS. We propose the level sets of CIVNSBS and complex interval valued neutrosophic normal subbisemiring (CIVNNSBS) of bisemirings. Moreover, we introduce ... CIVNSBS of bisemiring S if and only if bV is a CIVNSBS of S x S. We illustrate that homomorphic images of every CIVNSBS is a CIVNSBS and homomorphic pre-images of every CIVNSBS is a CIVNSBS. Examples are provided to illustrate our results. [ABSTRACT FROM AUTHOR]

In this paper, we introduce a c-prime fuzzy graph of a nearring with respect to a level ideal of a fuzzy ideal. We find a relation between properties of the fuzzy ideal and properties of the fuzzy graph. We introduce ideal symmetry of the fuzzy graph and obtain conditions under which the graph is ideal symmetric. We find a relation between nearring homomorphisms and graph homomorphisms. We investigate conditions required for the homomorphic image of a c-prime fuzzy ideal to be a c-prime fuzzy ideal. [ABSTRACT FROM AUTHOR]

Word-representable graphs, which are the same as semi-transitively orientable graphs, generalize several fundamental classes of graphs. In this paper we propose a novel approach to study word-representability of graphs using a technique of homomorphisms. As a proof of concept, we apply our method to show word-representability of the simplified graph of overlapping permutations that we introduce in this paper. For another application, we obtain results on word-representability of certain subgraphs of simplified de Bruijn graphs that were introduced recently by Petyuk and studied in the context of word-representability. [ABSTRACT FROM AUTHOR]

The aim of this sequence of work is to investigate polynomial equations satisfied by additive functions. As a result of this, new characterization theorems for homomorphisms and derivations can be given. More exactly, in this paper the following type of equation is considered ∑ i = 1 n f i (x p i) g i (x q i) = 0 x ∈ F , where n is a positive integer, F ⊂ C is a field, f i , g i : F → C are additive functions and p i , q i are positive integers for all i = 1 , ... , n . [ABSTRACT FROM AUTHOR]

Neutosophic vague subbisemirings (NSVSBS) are discussed here, as well as their level sets. Subbisemirings are a generalization of bisemirings, and NSVSBSs are a generalization of sub-bisemirings. A number of illustrative examples are provided to illustrate the theory for (ξ, τ )-NSVSBS over bisemiring theory. Following is an outline of the preliminary definitions and results presented in Section 2. The concept of a NSVSBS is introduced in Section 3.

Our aim is to explore the idea of neutrosophic N-ideals in near-subtraction semigroups in this article and obtain some outcomes that are equivalent to them. We also illustrate the notion of a neutrosophic κ-intersection. Additionally, in a near-subtraction semigroup, we examine the term homomorphism of a neutrosophic κ-structure and establish some conclusions based on a homomorphic neutrosophic κ-structure preimage of a neutrosophic κ-left (respectively, right) ideal.

A regular matroid M on a finite set E is represented by a totally unimodular matrix. The set of vectors from ZE orthogonal to rows of the matrix form a regular chain group N. Assume that ψ is a homomorphism from N into a finite additive Abelian group A and let Aψ[N] be the set of vectors g from (A−0)E, such that ∑e∈Eg(e)·f(e)=ψ(f) for each f∈N (where · is a scalar multiplication). We show that |Aψ[N]| can be evaluated by a polynomial function of |A|. In particular, if ψ(f)=0 for each f∈N, then the corresponding assigning polynomial is the classical characteristic polynomial of M.

This research aims to introduce and explore the theory of neutrosophic soft hyperalgebras (NSHAs), focusing on their core principles and potential applications in decision-making under uncertainty. By defining key operations such as intersection and union, we clarify the foundational characteristics of NSHAs and their relationship to soft hyperalgebras. The concepts of ξβ-identity NSHA and ξ-absolute NSHA are also examined to better understand their properties. The practical relevance of NSHA is demonstrated through applications in various fields, highlighting its adaptability in addressing complex decision-making scenarios. This approach offers a novel, more precise method for navigating uncertainty in areas such as project methodology selection, sensitivity analysis, and AI chatbot selection.

The minimum spanning tree problem in graphs under budget-type degree constraints (DCMST) is a well-known NP-hard problem. Spanning trees do not always exist, and the optimum can not be approximated within a constant factor. Recently, solutions have been proposed to solve degree-constrained spanning problems in the case of limited momentary capacities of the nodes. For a given node, the constraint represents a limited degree of the node for each visit. Finding the solution with minimum cost is NP-hard and the related algorithms are not trivial. This paper focuses on this new spanning problem with heterogeneous capacity-like degree bounds. The minimum cost solution corresponds to a graph-related structure, i.e., a hierarchy. We study the conditions of its existence, and we propose its exact computation, a heuristic algorithm, and its approximation.

Isomorphism and homomorphism are fundamental topics in abstract algebra, but how mathematicians characterise the utility of these concepts remains understudied. Based on interviews with nine research-active mathematicians, we confirmed use of previously noted conceptual metaphors and noted new metaphors and metaphor clusters. We expand on prior work by highlighting five thematic contrasts between metaphors for these concepts: context dependent sameness, classification of objects versus computing maps, indistinguishable versus distinct objects, structure-preservation versus structure loss, and information gain versus structure loss. Implications include the centrality of sameness to thinking about isomorphism and homomorphism but the need for care in how that sameness is framed; the value of being able to attend to the object and map; the importance of being able to determine which structures are preserved and which are lost in context; and the existence of a conceptual purpose for homomorphism as a tool for changing perspectives. [ABSTRACT FROM AUTHOR]

The foundational concepts of subalgebra, ideal, and homomorphism within the domain of BF-algebra were originally introduced by Andrzej Walendziak [A. Walendziak, On BF-Algebras, Math. Slovaca, 57(2) (2007), 119-128]. In this paper, we introduce innovative concepts related to Neutrosophic BF-Subalgebras and Neutrosophic BFIdeals derived from the application of Subalgebra, Ideal, and Homomorphism principles to Neutrosophic sets. We explore the outcomes concerning a Neutrosophic BF-Ideal with respect to principle of homomorphism, homomorphic image of a Neutrosophic BF-Ideal satisfying the sup-inf property, and homomorphic pre-images within the context of a Neutrosophic set embedded in BF-algebra. The outcomes of the above study are equally applicable to Neutrosophic BF-Subalgebra Lastly, we delve into the conceptual understanding of a level set of a Neutrosophic BF-Ideal within a BF-algebra. In the future, the above study can be extended to address various types of implicative ideals and filters within BF-algebra. Moreover, these Neutrosophic BF-Subalgebras and Neutrosophic BF-Ideals can be applied to neutrosophic soft sets within the context of BF-algebra, which are used for various decision making techniques. [ABSTRACT FROM AUTHOR]

This paper introduces the idea of a neutrosophic vague subbisemiring (NSVSBS), level sets of NSVSBS, and (p; ƍ)-neutrosophic vague subbisemiring ((p;ƍ)-NSVSBS) of a bisemiring. NSVSBSs are generalizations of neutrosophic subbisemirings and SBS based on bisemirings. Let - be a neutrosophic vague subset in B, we show that V=([Tʌ-, T+ ]ʌ [Iʌʌ ʌ I+ ]ʌ [F&#652ʌ ʌ ;F+ ʌ ]) is a NSVSBS of B if and only if all non empty level set V(t1,t2,s) is a SBS of B for t1, t2, ϵ 2[0,1]. In the case that ʌ is a NSVSBS of a bisemiring B and V is the strongest neutrosophic vague relation of B, we prove that λ is a NSVSBS of B × B. Let Λ be any NSVSBS of B, prove that pseudo neutrosophic vague coset (τΛ)p is a NSVSBS of B, for every τεB. Let τ1;τ2,...,τn be the family of NSV SBSs of B1; B2,...Bn respectively. We show that τ1 × τ2 ×....× τn is a NSVSBS of B1 × B2 &#215,...× Bn. The homomorphic image of every NSVSBS is a NSVSBS. The homomorphic pre-image of every NSVSBS is a NSVSBS. Examples are provided to strengthen our results. [ABSTRACT FROM AUTHOR]

We show that any two Hadamard graphs on the same number of vertices are quantum isomorphic. This follows from a more general recipe for showing quantum isomorphism of graphs arising from certain association schemes. The main result is built from three tools. A remarkable recent result [20] of Mančinska and Roberson shows that graphs G and H are quantum isomorphic if and only if, for any planar graph F , the number of graph homomorphisms from F to G is equal to the number of graph homomorphisms from F to H. A generalization of partition functions called "scaffolds" [23] affords some basic reduction rules such as series-parallel reduction and can be applied to counting homomorphisms. The final tool is the classical theorem of Epifanov showing that any plane graph can be reduced to a single vertex and no edges by extended series-parallel reductions and Delta-Wye transformations. This last sort of transformation is available to us in the case of exactly triply regular association schemes. The paper includes open problems and directions for future research. [ABSTRACT FROM AUTHOR]

This article presents the concepts of neutrosophic INK-subalgebras and INK-ideals of INK-algebras. We also studied neutrosophic INK-subalgebras and INK-ideals that depend on homomorphisms and anti-homomorphisms. [ABSTRACT FROM AUTHOR]

Concepts of generalized commutative intuitionistic fuzzy finite state machines and generalized switching intuitionistic fuzzy finite state machines are proposed in this paper, and properties of these two kinds of intuitionistic fuzzy finite state machines are investigated. By using the conditions associated with generalized commutativity, a kind of congruences of intuitionistic fuzzy finite state machines is defined. Homomorphic properties of generalized commutative intuitionistic fuzzy finite state machine are also discussed. Moreover, products of generalized commutative intuitionistic fuzzy finite state machines and products of generalized switching intuitionistic fuzzy finite state machines are studied, and some related properties are proved. [ABSTRACT FROM AUTHOR]

In this paper we consider generalized monomial functions f , g : F → C (of possibly different degree) that also fulfill f (P (x)) = Q (g (x)) x ∈ F , where P ∈ F [ x ] and Q ∈ C [ x ] are given (classical) polynomials. [ABSTRACT FROM AUTHOR]