Mathematical study proposes hesitant graph energy for business-location ranking

A mathematical study developed a degree-based graph framework for handling preferences that experts may regard as true, false or uncertain, and demonstrated it in an illustrative business-location case. The researchers extended Zagreb indices to hesitancy fuzzy graphs, introduced a corresponding Zagreb matrix and energy measure, and established upper and lower limits for the new quantities. They then used the framework to combine preference relations from four assessors evaluating five candidate business sites. The study is a theoretical and methodological contribution supported by constructed examples, not a trial using observed business performance, and the authors call for testing with real-world datasets.

The central idea is to represent uncertainty explicitly in graph-based analysis. A fuzzy graph assigns a membership value between zero and one to each vertex and edge, indicating the extent to which that element participates in the graph. An intuitionistic fuzzy graph adds separate membership and non-membership degrees, from which a residual hesitation value can be derived. A hesitancy fuzzy graph, by contrast, gives vertices and edges truthfulness, falsehood and abstention components. This distinction is important for decisions in which information is incomplete or an expert is reluctant to reduce a judgment to a simple yes-or-no choice. The paper treats this kind of information as part of the data structure rather than applying an ordinary crisp graph calculation.

Graph-theoretic topological indices provide compact numerical descriptions of network structure, and the Zagreb indices are among the degree-based measures examined in the study. The paper describes the first Zagreb index as a degree-based topological index and the second as another quantity derived from the graph’s degree structure, although the supplied article presentation does not display their explicit formulas. Earlier work had considered fuzzy graph energy, Laplacian energy, fuzzy Zagreb matrices and fuzzy Zagreb energy, including related limits in intuitionistic fuzzy settings. The authors identify a specific gap: the Zagreb indices and their associated energy had not previously been defined and investigated for the hesitancy fuzzy graph framework used in their analysis.

To address that gap, the researchers define first and second Zagreb indices for hesitancy fuzzy graphs and represent each with truthfulness, falsehood and abstention components. They also define a hesitancy fuzzy Zagreb matrix and examine the eigenvalue sets associated with its component matrices. The matrix is constructed to reflect the neighborhood-based connectivity structure of the hesitancy fuzzy graph. From the relevant eigenvalues, the study defines hesitancy fuzzy Zagreb energy as a new spectral measure. The approach therefore combines the graph’s degree information with the three-part representation of hesitant information, but it does not assume that a single membership score is sufficient to describe every decision.

The theoretical portion establishes lower and upper limits for the hesitancy fuzzy Zagreb matrix and energy quantities. These results are derived for a hesitancy fuzzy graph with a specified number of vertices and edges and use mathematical tools including matrix-trace properties, the Cauchy–Schwarz inequality and the comparison between arithmetic and geometric means. The paper illustrates the bounds with a small example graph and calculates the relevant values so that they can be compared with the stated constraints. The examples demonstrate internal consistency between the definitions, calculations and proposed limits. They do not measure predictive accuracy, reliability in a commercial setting or the performance of actual firms, because no such outcomes are evaluated in the study.

The business-location application examines five potential sites, denoted as alternatives, and asks four experts with backgrounds in business, finance, real estate and law to assess them. The criteria discussed include closeness to the intended market, the locations of competitors, utility taxes and other levies, laws and regulations, accessibility and infrastructure, security and parking availability. Each expert supplies a hesitancy fuzzy preference relation for the alternatives. In these relations, every comparison carries a truthfulness, falsehood and abstention triple, allowing the assessors’ different judgments to be represented within one mathematical model. The example therefore concerns expert preference aggregation rather than a measured comparison of business outcomes at the five locations.

The authors convert each expert’s preference relation into a hesitancy fuzzy graph, form the corresponding hesitancy fuzzy Zagreb matrix and calculate its energy. These four expert-specific energy values are used to obtain weights for combining the individual preference relations. The aggregation stage uses an ordered weighted averaging operator, which produces a collective hesitancy fuzzy preference relation. From that collective structure, the researchers determine the degrees of the candidate vertices. The final step ranks the alternatives using the degree of truthfulness associated with each vertex and nominates the option with the highest truthfulness degree as the preferred site. The study illustrates a complete computational route from uncertain preferences to a ranked decision within the designed case.

A comparison with a previously proposed approach based on spherical fuzzy Zagreb matrices shows where the new procedure differs. The earlier method aggregates expert findings with an average operator and evaluates alternatives using a truthfulness threshold above 0.5, treating options above that threshold as acceptable for further consideration. The new method instead combines truthfulness information with vertex-degree characteristics derived from the hesitancy fuzzy Zagreb matrices. The reported alternative-degree values therefore differ between the two strategies, although the authors state that the resulting ranking is consistent in the example. This consistency supports the internal coherence of the proposed method within the constructed case; it is not independent evidence that the method predicts profitability more accurately or leads to better business decisions.

Several limitations constrain what the study establishes. Its business application is a worked multi-attribute decision-analysis example based on supplied expert preferences, not an observational analysis of business openings, revenues, costs, customer traffic or subsequent site performance. The mathematical bounds describe permitted values under the model’s assumptions, while the comparison with the spherical fuzzy method checks only the supplied ranking. The authors also identify interval-valued and multivalued hesitancy fuzzy graphs as possible extensions of the framework. They further propose future investigation in resource allocation, infrastructure design and traffic-network analysis, but those applications are proposed directions rather than results reported in the paper. Until empirical testing is available, the contribution is best understood as a formal set of definitions, spectral quantities, bounds and a worked decision-support procedure for representing explicit hesitation in graph-based preference analysis.

Subject of Research: Mathematical development and illustrative application of hesitancy fuzzy graph energy for business-location decision analysis

Article Title: Hesitancy fuzzy Zagreb energy for hesitancy fuzzy graphs with application to optimal business location selection

Article References: Abinaya, A., & Deepa, G. (2026). Hesitancy fuzzy Zagreb energy for hesitancy fuzzy graphs with application to optimal business location selection. Frontiers in Applied Mathematics and Statistics, 12, Article 1989314. https://doi.org/10.3389/fams.2026.1989314

Image Credits: AI Generated

DOI: 10.3389/fams.2026.1989314

Keywords: hesitancy fuzzy graphs, Zagreb indices, Zagreb energy, graph matrices, spectral analysis, uncertain preferences, decision analysis, business location, fuzzy mathematics, multi-criteria decisions, Hesitancy, fuzzy