Reciprocal Property Research Articles

Algebraic analysis of pairwise comparisons with non-reciprocal property

The law of comparative judgement usually implies the reciprocal property of pairwise comparisons. But time error and position preference could yield some judgements that contradict this identity assumption. This paper proposes an algebraic approach to dealing with the challenge of non-reciprocal property in pairwise comparisons. The concept of non-reciprocal pairwise comparison matrix (NrPCM) is used to characterize the situation of non-reciprocal property. Algebraic analysis of NrPCMs is carried out, where a linear space is constructed by proposing a set of basis. Then the linear space is decomposed into the direct sum of some subspaces. The properties of the space are analyzed such as inner product, decomposition and coordinate. Finally, we apply the algebraic method to investigate the Condorcet paradox under the impact of indifferent voters. A novel transformation formula of non-reciprocal pairwise comparisons is proposed, where the non-reciprocal property holds. It is revealed that the occurrence of voting paradox can be understood clearly according to the transitivity property of NrPCMs.

On non-bipartite graphs with strong reciprocal eigenvalue property

Let G be a simple connected graph and A(G) be the adjacency matrix of G. A diagonal matrix with diagonal entries ±1 is called a signature matrix. If A(G) is nonsingular and X=SA(G)−1S−1 is entrywise nonnegative for some signature matrix S, then X can be viewed as the adjacency matrix of a unique weighted graph. It is called the inverse of G, denoted by G+. A graph G is said to have the reciprocal eigenvalue property (property(R)) if A(G) is nonsingular, and 1λ is an eigenvalue of A(G) whenever λ is an eigenvalue of A(G). Further, if λ and 1λ have the same multiplicity for each eigenvalue λ, then G is said to have the strong reciprocal eigenvalue property (property (SR)). It is known that for a tree T, the following conditions are equivalent: a) T+ is isomorphic to T, b) T has property (R), c) T has property (SR) and d) T is a corona tree (it is a tree which is obtained from another tree by adding a new pendant at each vertex).Studies on the inverses, property (R) and property (SR) of bipartite graphs are available in the literature. However, their studies for the non-bipartite graphs are rarely done. In this article, we study the inverse and property (SR) for non-bipartite graphs. We first introduce an operation, which helps us to study the inverses of non-bipartite graphs. As a consequence, we supply a class of non-bipartite graphs for which the inverse graph G+ exists and G+ is isomorphic to G. It follows that each graph G in this class has property (SR).

Reciprocal eigenvalue properties using the zeta and Möbius functions

In this paper, we develop a new approach to study the spectral properties of Boolean graphs using the zeta and Möbius functions on the Boolean algebra Bn of order 2n. This approach yields new proofs of the previously known results about the reciprocal eigenvalue property of Boolean graphs. Further, this approach allows us to extend the results to a more general setting of the zero-divisor graphs Γ(P) of complement-closed and convex subposets P of Bn. To do this, we consider the left linear representation of the incidence algebra of a poset P on the vector space of all real-valued functions V(P) on P. We then write down the adjacency operator A of the graph Γ(P) as the composition of two linear operators on V(P), namely, the operator that multiplies elements of V(P) on the left by the zeta function ζ of P and the complementation operator. This allows us to obtain the determinant of A and the inverse of A in terms of the Möbius function μ of the complement-closed posets P. Additionally, if we impose convexity on the poset P, then we obtain the strong reciprocal or strong anti-reciprocal eigenvalue property of Γ(P) and also obtain the absolute palindromicity of the characteristic polynomial of A. This produces a large family of examples of graphs having the strong reciprocal or strong anti-reciprocal eigenvalue property.

Singular graphs and the reciprocal eigenvalue property

Let G be a simple connected graph and A(G) be its adjacency matrix. The terms singularity, eigenvalues, and characteristic polynomial of G mean those of A(G). A nonsingular graph G is said to have the reciprocal eigenvalue property if the reciprocal of each eigenvalue of G is also an eigenvalue. A graph G (possibly singular) is said to have the weak reciprocal eigenvalue property if the reciprocal of each nonzero eigenvalue of it is also an eigenvalue. In Barik et al. (2022) [3], the authors proved that there is no nontrivial tree with the weak reciprocal eigenvalue property and posed the following question: “Does there exist a nontrivial graph with the weak reciprocal eigenvalue property?” Suppose that G is singular and the characteristic polynomial of G is xn−k(xk+a1xk−1+⋯+ak). Assume that A(G) has rank k, so that ak≠0. Can we ever have |ak|=1? The answer turns out to be negative. As an application, we settle the question posed in Barik et al. (2022) [3]. Another similar application is also mentioned. It is natural to wonder, “Does there exist a nontrivial, simple, connected weighted graph with the weak reciprocal eigenvalue property?” We provide a class of such graphs. Furthermore, we extend our results to weighted graphs.

Signed graphs with strong anti-reciprocal eigenvalue property

The ordered pair , where G is an underlying graph and is a signature function, is called a signed graph. A nonsingular signed graph is said to satisfy strong reciprocal (or strong anti-reciprocal) eigenvalue property if for each eigenvalue there exists (or ) in the spectrum of having same multiplicities, if we remove this multiplicity constraint then the signed graph is said to satisfy reciprocal (respectively anti-reciprocal) eigenvalue property. In this article, we investigate strong anti-reciprocal eigenvalue property in some families of signed graphs.

Reciprocal properties of random fields on undirected graphs

AbstractWe clarify and refine the definition of a reciprocal random field on an undirected graph, with the reciprocal chain as a special case, by introducing four new properties: the factorizing, global, local, and pairwise reciprocal properties, in decreasing order of strength, with respect to a set of nodes $\delta$ . They reduce to the better-known Markov properties if $\delta$ is the empty set, or, with the exception of the local property, if $\delta$ is a complete set. Conditions for each reciprocal property to imply the next stronger property are derived, and it is shown that, conditionally on the values at a set of nodes $\delta_0$ , all four properties are preserved for the subgraph induced by the remaining nodes, with respect to the node set $\delta\setminus\delta_0$ . We note that many of the above results are new even for reciprocal chains.

The future of education related to SDG 16 (peace, ‎justice and strong institutions): A position paper

The spreading aspects of the seventeen sustainable development goals have been of concern to ‎researchers in education worldwide. These goals have shed the light and grasped the attention to ‎very crucial issues. The significance of these issues (the cores of each SDG) not only lie in their ‎importance but also in their connectedness and mutual influence among each other. Education, ‎being the highest peak of achievement in every society, and hence in the whole (small) world, is ‎directly affected by each SDG. That was the reason for considering the future of Education in sight ‎of one of the most effective sustainable goals, namely, peace, justice and strong institutions. The ‎current paper is a position paper that discusses the float of the sixteenth sustainable goal along the ‎future of Education. The researchers propose two statements: firstly, SDG 4 (Quality Education) ‎and SDG 16 possess a reciprocal interaction property (i.e. they have mutual impact ‎on each other). ‎Secondly, this cyclic process has to start by first accomplishing SDG 16‎‏.

Note on “The breaking of additively reciprocal property of fuzzy preference relations and its implication to decision making under uncertainty”

A recent article by Liu et al. [8] devised measurement models to calculate additive consistency indices and weak transitivity indices for fuzzy preference relations (FPRs) with additive reciprocity breaking (ARB), and developed some decision models, such as the equivalency between FPRs with ARB and inconsistent FPRs with additive reciprocity, the optimization based weak transitivity improving model and priority vector derivation models. This paper utilizes six numerical illustrations to show that the measurement models are technically incorrect and sometimes the decision models are invalid or create contradictory results. New frameworks are proposed to define and measure additive consistency and weak transitivity for FPRs with ARB. Computational formulas are brought forward to acquire interval-valued or intuitionistic fuzzy priority vectors from additively consistent FPRs with ARB, and a linear programming model is built to ascertain interval-valued or intuitionistic fuzzy priority vectors from inconsistent FPRs with ARB.

On weighted noncorona graphs with property R and −SR

Let Gw be a simple weighted graph with adjacency matrix A(Gw). The set of all eigenvalues of A(Gw) is called the spectrum of weighted graph Gw denoted by σ(Gw). The reciprocal eigenvalue property (or property R) for a connected weighted nonsingular graph Gw is defined as, if η ∈ σ(Gw) then 1 η ∈ σ(Gw). Further, if η and 1 η have the same multiplicities for each η ∈ σ(Gw) then this graph is said to have strong reciprocal eigenvalue property (or property SR). Similarly, a connected weighted nonsingular graph Gw is said to have anti-reciprocal eigenvalue property (or property −R) if η ∈ σ(Gw) then −1 η ∈ σ(Gw). Furthermore, if η and −1 η have the same multiplicities for each η ∈ σ(Gw) then strong anti-reciprocal eigenvalue property (or property −SR) holds for the weighted graph Gw. In this article, classes of weighted noncorona graphs satisfying property R and property −SR are studied.

A prioritization approach of non-reciprocal fuzzy preference relations and its extension

In decision making problems, preference relations are the widely used tools to simulate the opinions of decision makers (DMs). It is important to elicit priorities from a preference relation in order to reach an optimal solution. In this paper, we propose an optimization-based method to derive the priority vector from a fuzzy preference relation (FPR) without additively reciprocal property (ARP). First, the concept of non-reciprocal fuzzy preference relations (NrFPRs) is used to characterize the case of FPRs without ARP. An interval additive reciprocal matrix (IARM) is decomposed into two NrFPRs. Second, the variance similarity measure is proposed to quantify the similarity degree of two column/row vectors of NrFPRs. A double variance consistency index is constructed to measure the inconsistency degree of NrFPRs. An optimization-based model is proposed to elicit the priority vector from NrFPRs. Some interesting properties are addressed to show the effectiveness of the proposed approach. Third, the proposed prioritization method is extended to derive priorities from IARMs. Some comparisons with the existing prioritization methods of IARMs are reported using numerical examples. The observations reveal that the uncertainty exhibiting in IARMs can be captured by NrFPRs. Some decision-making models with IARMs could be revisited using the proposed method for NrFPRs.

A consensus building model in group decision making with non-reciprocal fuzzy preference relations

Group decision making (GDM) is a wisdom extracting process where a group of decision makers (DMs) could reach a consensus on the optimal solution to the choice problem with a finite set of alternatives. This paper reports a consensus model in GDM, where the opinions of experts are expressed as fuzzy preference relations (FPRs) without additively reciprocal property to cope with the existing uncertainty. The concept of non-reciprocal fuzzy preference relations (NrFPRs) is proposed to capture the considered situation. A novel additive consistency index is constructed to quantify the inconsistency degree of NrFPRs using the relationship of two column/row vectors. An optimization model is constructed, where a new fitness function is proposed by considering the consistency degrees of NrFPRs and the consensus level of a group of experts. A novel concept of acceptable consensus standard is proposed to characterize the acceptance of the consensus process. The particle swarm optimization (PSO) algorithm is utilized to solve the constructed optimization problem. As compared to the existing models, numerical results show that the proposed model can be used to effectively reach an optimal solution to a GDM problem with NrFPRs.

Trees with the reciprocal eigenvalue property

It was shown in 2006 that among the nonsingular trees T (whose adjacency matrix A(T) is nonsingular), the corona trees (trees that are obtained by taking any tree T and then adding a new pendant vertex at each vertex of T) are the only ones which satisfy the reciprocal eigenvalue property (λ is an eigenvalue of A(T) if and only if is an eigenvalue of A(T), where their multiplicities are allowed to be different). A general question remained open. Can there be a tree which has at least one zero eigenvalue and whose nonzero eigenvalues satisfy the reciprocal eigenvalue property? In this note, we show that there are no such trees with at least two vertices. The proof is a beautiful application of the product of graphs.

The breaking of additively reciprocal property of fuzzy preference relations and its implication to decision making under uncertainty

The existing works usually assume that fuzzy preference relations (FPRs) exhibit additively reciprocal property. In this study, we investigate the situation that FPRs have no additively reciprocal property and address its implication to decision making under uncertainty. First, the considered situation is captured by proposing the novel concept of additively reciprocal property breaking (ARPB) for FPRs. It is proved that FPRs with ARPB, interval additive reciprocal preference relations (IARPRs) and intuitionistic fuzzy preference relations (IFPRs) are transformed each other. Second, the equivalence between FPRs with ARPR and additive reciprocal matrices (ARMs) is defined by keeping the inconsistency level unchanged. An optimization model is proposed to adjust an FPR without weak transitivity to an ARM with weak transitivity. Third, a novel decision making model is developed by considering weak transitivity of FPRs as the minimum requirement of rational choices. A new algorithm is elaborated on where decision information could be expressed as FPRs with ARPR, IARPRs and IFPRs, respectively. Finally, numerical results are reported to show the advantages of the developed model by comparing with the existing ones. The observations reveal that the concept of ARPB offers a novel understanding for uncertainty in decision information.

A Deoxyribozyme-Initiated Self-Catalytic DNA Machine for Amplified Live-Cell Imaging of MicroRNA.

Functional DNA nanostructures have been widely used in various bioassay fields. Yet, the programmable assembly of functional DNA nanostructures in living cells still represents a challenging goal for guaranteeing the sensitive and specific biosensing utility. In this work, we report a self-catalytic DNA assembly (SDA) machine by using a feedback deoxyribozyme (DNAzyme)-amplified branched DNA assembly. This SDA system consists of catalytic self-assembly (CSA) and DNAzyme amplification modules for recognizing and amplifying the target analyte. The analyte initiates the CSA reaction, leading to the formation of Y-shaped DNA that carries two RNA-cleaving DNAzymes. One DNAzyme can then successively cleave the corresponding substrate and generate numerous additional inputs to activate new CSA reactions, thus realizing a self-catalytic amplification reaction. Simultaneously, the other DNAzyme is assembled as a versatile signal transducer for cleaving the fluorophore/quencher-modified substrate, leading to the generation of an amplified fluorescence readout. By incorporating a flexible auxiliary sensing module, the SDA system can be converted into a universal sensing platform for detecting cancerous biomarkers, e.g., a well-known oncogene microRNA-21 (miR-21). Moreover, the SDA system realized the precise intracellular miR-21 imaging in living cells, which is attributed to the reciprocal amplification property between CSA reactions and DNAzyme biocatalysis. This compact SDA amplifier machine provides a universal and facile toolbox for the highly efficient identification of cancerous biomarkers and thus holds great potential for early cancer diagnosis.

On the inverse of unicyclic 3-coloured digraphs

This article provides a combinatorial description of the inverse of the adjacency matrix of a non-singular 3-coloured digraph. The class of unicyclic 3-coloured digraphs with the cycle weight and with a unique perfect matching, denoted by , is considered in this article. We characterize the 3-coloured digraphs in whose inverses are again 3-coloured digraphs. Furthermore, the 3-coloured digraphs in whose inverses are bipartite are also characterized. It is proved that the inverses of the 3-coloured digraphs in are always Laplacian non-singular. Characterizations of unicyclic 3-coloured digraphs in possessing unicyclic inverses are also supplied in this article. As an application, we can obtain the class of unicyclic 3-coloured digraphs with the cycle weight satisfying the strong reciprocal eigenvalue property.

An uncertainty-induced axiomatic foundation of the analytic hierarchy process and its implication

Uncertainty is always encountered by decision makers (DMs) when addressing a complex decision-making problem. The existing axiomatic foundation of the analytic hierarchy process (AHP) cannot reflect the uncertainty experienced by DMs. In this paper, it is first to find that the uncertainty experienced by DMs can be characterized by the non-reciprocal property of pairwise comparisons. The novel concept of reciprocal symmetry breaking (RSB) is proposed to capture the situation without reciprocal property. Then an uncertainty-induced axiomatic foundation of the AHP model is reported, where the RSB is one of the proposed axioms. Some interesting results are derived from the new axioms involving the concept of approximate consistency and the method of eliciting priorities. An index is constructed to measure approximate consistency degree of pairwise comparison matrices (PCMs). Some comparisons with the typical AHP model are offered by carrying out some numerical examples. The obtained results reveal that the proposed axiomatic foundation and the derived facts form a novel operational basis of the AHP model under uncertainty.

Group decision support model based on sequential additive complementary pairwise comparisons

In group decision support systems, it is important on how to process and manage individual decision information. In the paper, a sequential model is proposed to manage individual judgements with additively reciprocal property over paired alternatives. The process of realizing additive complementary pairwise comparisons (ACPCs) is captured. A real-time feedback mechanism is constructed to address the irrational behavior of individuals. An optimization model is established and solved by using the particle swarm optimization (PSO) algorithm, such that the consistency of individual judgements can be improved fast yet effectively. For the aggregation of individual decision information in group decision making (GDM), the weighted averaging operator is used. It is found that when all individual judgements are acceptably additively consistent, the collective matrix is with acceptable additive consistency. Under the control of individual consistency degrees, the approach of reaching consensus in GDM is further proposed. By comparing with some existing models, the observations reveal that the sequential model of originating additive complementary pairwise comparisons possesses the ability to rationally manage individual decision information.

Classes of nonbipartite graphs with reciprocal eigenvalue property

Let G be a simple connected graph and A(G) be the adjacency matrix of G. The graph G is said to have the reciprocal eigenvalue property (R) if A(G) is nonsingular and 1λ is an eigenvalue of A(G) whenever λ is an eigenvalue of A(G). Further, if λ and 1λ have the same multiplicity, for each eigenvalue λ, then G is said to have the strong reciprocal eigenvalue property (SR). Till date, all the classes of bipartite graphs that are found to have property (R), are found to have property (SR) and it is not known whether these two properties are equivalent even for the bipartite graphs with a unique perfect matching. Among nonbipartite graphs, there is only one known graph class for which these two properties are not equivalent. In this article, we construct some more classes of nonbipartite graphs with property (R) but not (SR).

Group Decision Making Based on Flexibility Degree of Fuzzy Numbers Under a Confidence Level

There is always vagueness experienced by experts in practical decision making processes. Fuzzy sets are introduced for their ability to model objects and phenomena with a flexible degree. Then, it is more advisable to utilize fuzzy numbers to express the opinions of decision makers (DMs) for the purpose of reflecting the flexibility of DMs in a decision making process. Of much importance is how to quantify the flexibility degree of fuzzy numbers. In this article, a novel method for computing the flexibility degree of fuzzy numbers is generally presented by using the concept of α-cut sets. In particular, some novel formulas are proposed to quantify the flexibility degrees of triangular and trapezoidal fuzzy numbers by equipping a bounded universe, respectively. The flexibility degree of a preference relation with triangular fuzzy numbers is computed by considering the effects of the applied scale and the reciprocal property. Furthermore, a new group decision making (GDM) model is formed when triangular fuzzy additive reciprocal preference relations (TFARPRs) are used to evaluate the judgments of DMs. A flexibility degree induced ordered weighted averaging operator is constructed to aggregate individual TFARPRs by offering more importance to that with less flexibility. Finally, some numerical results are reported to illustrate the new definitions and the proposed model. The sensitivity of confidence levels to the final decision reached by a group of experts is analyzed. The observations reveal that the flexibility degrees of DMs under various confidence levels are worth to be considered in the GDM problem with a dominant position, and the existing shortcomings are overcome.

Retrodirective metasurfaces from non-reciprocal to reciprocal using impedance modulation for high-super-cell-periodicity designs

The growing popularity of metasurfaces and their exceptional physical phenomena has gained remarkable interest in terms of propagation and electromagnetic wave control. Particularly, for the case of retroreflection, retrodirective gradient metasurfaces are well known for their reciprocal responses in terms of retroreflection specially for subwavelength super-cell periodicity designs. Recently, we have observed that for metasurfaces with high super-cell periodicities greater than $$2\lambda $$ , the retrodirective metasurface loses its reciprocal response in terms of retrodirectivity. The main challenge in this case is to retain the reciprocal response, as well as achieving efficient responses for retroreflection at the higher orders of diffraction that exist at these periodicities. Thus, in this paper we demonstrate the design of a retrodirective metasurface from non-reciprocal to reciprocal following the generalized phase law of reflection and introducing the impedance modulation technique, for a metasurface operating at 14.7 GHz with a super-cell periodicity of $$2.88 \lambda $$ . On the other hand, with a well-engineered surface impedance of a super-cell at these periodicities, efficient responses of retroreflection can be achieved at the higher orders of diffraction, giving rise to a retrodirective metasurface design operating simultaneously for multiple incident angles. Retroreflection has been achieved at eleven angles simultaneously, and the reciprocal property was retained at the desired angle of incidence $$10^{\circ }$$ .