A study of the lexicographic-max product and its impact on vertex degree and total degree in fuzzy graphs

Introduction

Graph theory, one of the most elegant branches of discrete mathematics, traces its origins to the seminal work of Leonhard Euler in 1736 on the Konigsberg bridge problem. Since then, graph theory has evolved into a powerful tool for modeling relationships and solving problems across diverse scientific and engineering domains (Wilson, 1996) [43] . The classical graph model G = (V, E), consisting of a vertex set V and an edge set E, provides a binary framework where edges either exist or do not exist between pairs of vertices. While this crisp model has proven remarkably effective in many contexts, its inherent limitation lies in its inability to represent graded or uncertain relationships that characterize numerous real-world systems. The year 1965 marked a revolutionary development in the mathematical treatment of uncertainty with Lotfi A. Zadeh's introduction of fuzzy set theory. Zadeh's fundamental insight was to extend the classical notion of set membership from a binary property to a graded notion where elements can belong to a set with varying degrees of membership in the interval [0, 1] (Zadeh, 1965) [45] . This extension has had profound implications for mathematics, computer science, and engineering, fundamentally transforming our approach to modeling uncertainty and imprecision. The integration of fuzzy set theory with graph theory was initiated by Arnold Kaufmann in 1973, who introduced the basic notion of a fuzzy graph based on Zadeh's concept of fuzzy relations (Kaufmann, 1973) [18] . While Kaufmann established the foundational concepts, it was Azriel Rosenfeld's seminal 1975 paper that truly established fuzzy graph theory as a distinct mathematical discipline with rigorous foundations (Rosenfeld, 1975) [35] . Rosenfeld developed fuzzy analogues of fundamental graph-theoretic concepts including paths, cycles, connectedness, trees, bridges, and cut vertices, establishing the theoretical framework that continues to guide research in the field. A fuzzy graph G = (sigma, mu) consists of a nonempty set V together with a pair of functions sigma: V -> [0, 1] and mu: V x V -> [0, 1] such that mu(u, v) <= sigma(u) AND sigma(v) for all u, v in V. The value sigma(u) represents the membership degree of vertex u, while mu(u, v) represents the membership degree of the edge connecting u and v. This condition ensures that the edge membership does not exceed the membership of its endpoints, preserving the semantic interpretation of edge strength as a relationship between vertices (Rosenfeld, 1975; Mordeson and Nair, 2012) [35,24] . The degree of a vertex v in a fuzzy graph G = (sigma, mu), denoted deg(v), is defined as the sum of membership degrees of all edges incident with v. This definition extends the classical notion of vertex degree to the fuzzy setting by summing the membership degrees rather than counting edges. The total degree of a vertex v, denoted td(v), is defined as td(v) = deg(v) + sigma(v), incorporating both the connectivity strength and the vertex's own membership degree (Sunitha and Mathew, 2013) [22] . Graph product operations provide powerful tools for constructing complex graphs from simpler components, enabling the analysis of large-scale networks through decomposition. Among the fundamental graph products, the lexicographic product stands out for its hierarchical structure, making it particularly suitable for modeling multi-level systems. In crisp graph theory, the lexicographic product G1[G2] of two graphs has vertex set V1 x V2, with adjacency defined by either: (1) u1u2 in E1 and V1, V2 arbitrary in V2, or (2) u1 = u2 and V1V2 in E2. This structure naturally represents hierarchical networks where upper-level connectivity dominates (Imrich and Klavzar, 2000) [17] . The extension of lexicographic products to fuzzy graphs was pioneered by Radha and Arumugam (2015), who introduced both the lexicographic min-product and lexicographic maxproduct operations. In the lexicographic max-product G 1 [G 2 ]max, edge memberships are computed using the maximum (OR) operator rather than the minimum (AND) operator used in the min-product. Specifically, for edges arising from G 1 connections, mu((ui, vj), (uk, v l )) = mu1(ui, uk) OR sigma2(vj) OR sigma2(v l ), while for edges arising from G 2 connections (within the same G 1 layer), mu((ui, vj), (ui, v l )) = sigma1(ui) OR mu2(vj, v l ). This maximum-based aggregation gives preferential treatment to stronger connections, making the lexicographic max-product particularly appropriate for hierarchical network models where dominant relationships should propagate through the structure. Subsequent work by Syamala and Balasubramanian (2019) extended these concepts to intuitionistic fuzzy graphs, while Borzooei and Hoseini (2020) investigated lexicographic products of vague graphs. Recent studies have further explored the application of lexicographic products in q-rung orthopair fuzzy graphs and picture fuzzy graphs, demonstrating the continued relevance and extensibility of this operation (Akram et al. 2023; Shao et al. 2023) [2,37] . Despite these advances, several important questions regarding the lexicographic max-product remain insufficiently explored.

In particular, while formulas for vertex degree in the lexicographic max-product have been established under specific conditions on the factor graphs, a comprehensive analysis of total degree behavior and the conditions under which regularity properties are preserved requires further investigation. Moreover, the practical computational implications of these formulas and their validation through detailed numerical examples merit additional attention. The present study addresses these gaps by providing a systematic investigation of the lexicographic max-product and its impact on vertex degree and total degree in fuzzy graphs. The specific objectives of this paper are: (1) to establish explicit formulas for computing the degree and total degree of vertices in the lexicographic max-product under general conditions; (2) to identify the conditions under which regularity and total regularity are preserved; (3) to provide detailed illustrative examples with complete calculations, tables, and graphical representations; and (4) to compare the structural properties of the lexicographic max-product with related operations, highlighting its distinctive characteristics and application domains. The remainder of this paper is organized as follows. Section 2 reviews essential preliminaries from fuzzy graph theory. Section 3 formally defines the lexicographic max-product and establishes its basic properties. Sections 4 and 5 present the main theorems on vertex degree and total degree, respectively. Section 6 provides detailed illustrative examples with calculations and visualizations. Section 7 discusses the results in relation to existing literature, and Section 8 concludes with a summary and directions for future research.

Preliminaries

This section recalls fundamental definitions and results from fuzzy graph theory that serve as the foundation for our investigation. For comprehensive treatments of fuzzy graph theory, the reader is referred to Mordeson and Nair (2012) [24] and Pal, Samanta, and Ghorai (2020) [30] .

Fuzzy Graphs and Basic Definitions

Definition 2.1 (Rosenfeld, 1975) [35] . A fuzzy graph G = (sigma, mu) is a pair of functions where sigma: V -> [0,1] is a fuzzy subset of a nonempty set V and mu: V x V -> [0,1] is a symmetric fuzzy relation on V such that mu(u, v) <= sigma(u) AND sigma(v) for all u, v in V. The underlying crisp graph of G is denoted G* = (V*, E*), where V* = {u in V: sigma(u) > 0} and E* = {(u, v) in V x V: mu(u, v) > 0}. Definition 2.2 (Rosenfeld, 1975) [35] . A fuzzy graph G = (sigma, mu) is said to be effective if mu(u, v) = sigma(u) AND sigma(v) for all (u, v) in E*. It is complete if mu(u, v) = sigma(u) AND sigma(v) for all distinct u, v in V*. Definition 2.3. A fuzzy graph G = (sigma, mu) is connected if its underlying crisp graph G* is connected. The strength of a path P: v0, V 1 ,., vn is defined as the minimum edge membership along the path, and the strength of connectedness between two vertices u and v, denoted mu-infinity(u, v), is the maximum of the strengths of all paths connecting u and v.

Vertex Degree and Total Degree

Definition 2.4 (Sunitha and Mathew, 2013) [22] . The degree of a vertex v in a fuzzy graph G = (sigma, mu) is defined as deg(v) = Sum_{u!=v} mu(v, u). This definition extends the classical notion of vertex degree to the fuzzy setting by summing the membership degrees of all edges incident with v.

Definition 2.5. The total degree of a vertex v in a fuzzy graph G = (sigma, mu) is defined as td(v) = deg(v) + sigma(v). A fuzzy graph is said to be k-regular if every vertex has degree k, and it is totally k-regular if every vertex has total degree k.

Graph Operations on Fuzzy Graphs

Several product operations have been defined for fuzzy graphs, each preserving different structural properties. The Cartesian product G1 x G2 uses the minimum operator for all edge memberships. The strong product and maximal product employ different combinations of minimum and maximum operators. The lexicographic product, which is the focus of this paper, follows a hierarchical adjacency pattern inherited from crisp graph theory (Radha and Arumugam, 2015; Mordeson and Peng, 1994) [33,26] . The lexicographic max-product G 1 [G 2 ]max of two fuzzy graphs G 1 = (sigma1, mu1) and G 2 = (sigma2, mu2) is defined on the vertex set V 1 x V 2 with the following edge membership functions: (i) For u1 != u2 with u 1 u 2 in E 1 * and any

Lexicographic Max-Product of Fuzzy Graphs 3.1 Formal Definition

Definition 3.1 (Radha and Arumugam, 2015) [33] . Let G1 = (sigma1, mu1) and G2 = (sigma2, mu2) be two fuzzy graphs with underlying crisp graphs G1* = (V1, E1) and G2* = (V2, E2), respectively. The lexicographic max-product of G1 with G2, denoted G1[G2]max = (sigma, mu_max), is defined as follows: The vertex set is V = V1 x V2, and the vertex membership function is sigma(ui, vj) = sigma1(ui) AND sigma2(vj) for all (ui, vj) in V1 x V2. The edge membership function mu_max is defined by: (a) If ui uk in E1 and vj, vl are any vertices in V2 (including vj = vl), then mu_max((ui, vj), (uk, vl)) = mu1(ui, uk) OR sigma2(vj) OR sigma2(vl). (b) If ui = uk and vj vl in E2, then mu_max((ui, vj), (ui, vl)) = sigma1(ui) OR mu2(vj, vl).

Basic Properties

Theorem 3.2. The lexicographic max-product G 1 [G 2 ]max of two fuzzy graphs G 1 and G 2 is a fuzzy graph.

Proof. We must verify that mu_max((ui, vj), (uk, v l )) <= sigma(ui, vj) AND sigma(uk, v l ) for all edges. For case (a), we have mu_max = mu1(ui, uk) OR sigma2(vj) OR sigma2(v l ). Since mu1(ui, uk) <= sigma1(ui) AND sigma1(uk) and sigma2(vj), sigma2(v l ) <= 1, the result follows from the properties of the minimum and maximum operators. For case (b), mu_max = sigma1(ui) OR mu2(vj, v l ) <= sigma1(ui) since mu2(vj, v l ) <= sigma2(vj) AND sigma2(v l ) <= 1. The detailed verification is straightforward.□

Theorem 3.3 (Non-commutativity). The lexicographic max

Proof. This follows from the hierarchical nature of the lexicographic product. In G 1 [G 2 ]max, the connectivity structure of G 1 determines the cross-layer adjacencies, while in G 2 [G 1 ]max, the roles are reversed. Unless G 1 and G 2 are isomorphic with matching membership functions, the resulting products will differ. Example 6.2 in Section 6 illustrates this non-commutativity with concrete numerical values.

Theorem 3.4 (Connectedness

Proof. Since G 1 * is connected, for any two vertices (ui, vj) and (uk, v l ) in V 1 x V 2 , there exists a path from ui to uk in G 1 *. This path induces a sequence of cross-layer edges in G 1 [G 2 ]max, establishing a connection between (ui, vj) and (uk, v l ) regardless of the connectivity of G 2 *.

Relationship with Lexicographic Min-Product

The lexicographic min-product G 1 [G 2 ] min is defined analogously using the minimum (AND) operator for edge memberships. Radha and Arumugam (2015) established that the lexicographic min-product is a spanning fuzzy subgraph of the lexicographic max-product, providing an important structural relationship between these operations.

Proof. For any edge in G 1 [G 2 ]min, the corresponding edge in G 1 [G 2 ]max has greater or equal membership since mu1 AND sigma2 <= mu1 OR sigma2 and sigma1 AND mu2 <= sigma1 OR mu2. Both products share the same vertex set V 1 x V 2 .□

Vertex Degree in Lexicographic Max-Product 4.1 Degree Formula under Standard Conditions

This section presents the main results characterizing the degree of a vertex in the lexicographic max-product. The degree formula depends on the relationship between the vertex membership function sigma1 of G 1 and the edge membership function mu2 of G 2 .

Theorem 4.1 (Radha and Arumugam, 2015) [33] . Let G1 = (sigma1, mu1) and G2 = (sigma2, mu2) be two fuzzy graphs such that sigma1 <= mu2 (i.e., sigma1(ui) <= mu2(vj, vl) for all ui in V1 and vjvl in E2). Then the degree of any vertex (ui, vj) in G1[G2]max is given by: deg_G1[G2]max(ui, vj) = |V2| * deg_G1(ui) + deg_G2(vj). Proof. Consider a vertex (ui, vj) in G 1 [G 2 ]max. Its incident edges arise from two sources: (1) edges corresponding to edges in G 1 incident with ui, and (2) edges corresponding to edges in G 2 incident with vj within the layer ui. For edges of type (1), since sigma1 <= mu2, we have mu_max = mu1 OR sigma2 OR sigma2. Under the condition sigma1 <= mu2, the cross-layer edges from G 1 contribute |V 2 | * deg_G 1 (ui). For edges of type (2), mu_max = sigma1 OR mu2 = mu2, contributing deg_G 2 (vj). Summing both contributions yields the formula.□ 4.2 Degree Formula under Alternative Conditions Theorem 4.2. Let G 1 = (sigma1, mu1) and G 2 = (sigma2, mu2) be two fuzzy graphs such that sigma1 >= mu2. Then the degree of any vertex (ui, vj) in G 1 [G 2 ]max is given by: deg_G 1 [G 2 ]max(ui, vj) = deg_G 1 *(ui) * sigma1(ui) * |V 2 | + deg_G 2 *(vj) * sigma1(ui).

Proof. When sigma1 >= mu2, the edge memberships for cross-layer edges become mu1 OR sigma2, and for intra-layer edges become sigma1. The calculation follows by aggregating the contributions from each type of edge under this modified condition. Proof. When G 2 has no edges, only cross-layer edges from G 1 contribute to the degree. Each edge incident with ui in G 1 generates |V 2 | edges in the product, yielding the formula.

Total Degree in Lexicographic Max-Product 5.1 Total Degree Formula

Building upon the degree formulas established in Section 4, we now derive expressions for the total degree of vertices in the lexicographic max-product. Recall that the total degree of a vertex v in a fuzzy graph is td(v) = deg(v) + sigma(v). Proof. By definition, td(ui, vj) = deg(ui, vj) + sigma(ui, vj). From Theorem 4.1, deg(ui, vj) = |V 2 | * deg_G 1 (ui) + deg_G 2 (vj). By definition of the lexicographic max-product, sigma(ui, vj) = sigma1(ui) AND sigma2(vj). Adding these expressions yields the result.□

Regularity Conditions

A fuzzy graph is regular if all vertices have the same degree, and totally regular if all vertices have the same total degree. The following theorems characterize when the lexicographic max-product preserves these properties. Theorem 5.2. Let G 1 and G 2 be fuzzy graphs such that sigma1 <= mu2. The lexicographic max-product G 1 [G 2 ]max is regular if and only if |V 2 | * deg_G 1 (ui) + deg_G 2 (vj) is constant for all (ui, vj) in V 1 x V 2 .

Proof. From Theorem 4.1, the degree of (ui, vj) depends on deg_G 1 (ui) and deg_G 2 (vj). For G 1 [G 2 ]max to be regular, this expression must be independent of the choice of ui and vj. This requires specific relationships between the degree sequences of G 1 and G 2 .□ Proof. Total regularity requires td(ui, vj) to be constant. Under the given conditions, td(ui, vj) = |V 2 | * (k1 -c1) + k2 + c1 AND sigma2(vj), where c1 is the constant value of sigma1. For this to be constant, additional constraints on sigma2 may be required.

Illustrative Examples

This section provides detailed numerical examples to illustrate the theoretical results established in the preceding sections. Complete calculations, tabular presentations, and graphical visualizations are provided to validate the degree and total degree formulas for the lexicographic max-product.

Example: Computing Degrees in G1[G2]max

Consider the fuzzy graph G 1 = (sigma1, mu1) with vertex set V 1 = {u1, u2, u3} and the fuzzy graph G 2 = (sigma2, mu2) with vertex set V 2 = {V 1 , V 2 }, defined as follows: For G 1: sigma1(u1) = 0.5, sigma1(u2) = 0.7, sigma1(u3) = 0.6; mu1(u1, u2) = 0.4, mu1(u2, u3) = 0.5, mu1(u1, u3) = 0.3.

Computing Vertex Memberships in the Product

The vertex set of G 1 [G 2 ]max is V 1 x V 2 = {(u1, V 1 ), (u1, V 2 ), (u2, V 1 ), (u2, V 2 ), (u3, V 1 ), (u3, V 2 )}. The vertex membership function is sigma(ui, vj) = sigma1(ui) AND sigma2(vj):

Computing Edge Memberships in the Product

The edge memberships in G 1 [G 2 ]max are computed using the maximum operator. For cross-layer edges (corresponding to edges in G 1 ): mu_max ((u1, V 1 ), (u2, V 1 )) = mu1(u1, u2) OR sigma2(V 1 ) OR sigma2(V 1 ) = 0.4 OR 0.4 OR 0.4 = 0.4. mu_max((u1, V 1 ), (u2, V 2 )) = 0.4 OR 0.4 OR 0.6 = 0.6. mu_max ((u2, V 1 ), (u3, V 2 )) = 0.5 OR 0.4 OR 0.6 = 0.6. For intra-layer edges (corresponding to edges in G 2 ): mu_max

Computing Total Degrees

The total degree of each vertex is obtained by adding its vertex membership to its degree:

Non-commutativity Demonstration

To demonstrate that the lexicographic max-product is not commutative, we construct G 2 [G 1 ]max and compare it with

, but the adjacency structure differs significantly. Figure 7 reveals that the total degree consistently exceeds the vertex degree by exactly the vertex membership value, confirming the definitional relationship td(v) = deg(v) + sigma(v). The parallel structure of the two curves indicates that variations in total degree are primarily driven by variations in vertex degree, with vertex membership providing a uniform offset within each G 1 layer. for determining vertex degrees in the lexicographic maxproduct under different conditions on the factor graphs' membership functions. These formulas reveal that the degree of a vertex in the product depends linearly on its degree in the first factor graph (scaled by the size of the second factor's vertex set) and additively on its degree in the second factor graph.

Comparative Summary

The condition sigma1 <= mu2, under which the simplest degree formula holds, has an interesting interpretation: when the vertex memberships of the first factor graph are dominated by the edge memberships of the second factor graph, the maximum operator in the cross-layer edges selects the edge memberships of G 1 , while the maximum operator in the intra-layer edges selects the edge memberships of G 2 . This condition is satisfied in many practical applications where the first factor represents a coarse-grained backbone structure with relatively weak vertex memberships, and the second factor represents a fine-grained interaction pattern with stronger edge memberships.

Comparison with Related Work

Our findings are consistent with and extend several related studies in the literature. Syamala [2] has further generalized the lexicographic product to accommodate more expressive membership representations. The degree formulas in the qrung orthopair setting reduce to those presented here when the membership and non-membership functions satisfy the standard fuzzy constraint.

Application Domains

The lexicographic max-product has significant potential applications in several domains. In hierarchical social network analysis, the first factor graph can represent organizational levels (e.g., departments in a company) while the second factor represents interpersonal relationships within each level. The maximum operator naturally models situations where influence flows primarily through hierarchical channels, with the strongest connection pathway determining the effective relationship strength (Alqahtani, 2025) [5] .

In transportation network modeling, the lexicographic maxproduct can represent multi-modal transportation systems where the first factor corresponds to geographic regions and the second factor corresponds to transportation modes within each region. The maximum operator captures the idea that the most reliable route between two points determines the effective connectivity (Naeem et al. 2021) [27] . For decision-making under uncertainty, the lexicographic max-product provides a framework for aggregating criteria hierarchies. The first factor represents primary decision dimensions, while the second factor represents sub-criteria within each dimension. The maximum operator reflects optimistic aggregation, where the strongest supporting evidence for a decision alternative determines its overall evaluation (Banitalebi and Borzooei, 2023) [8] .

Limitations and Future Directions

Several limitations of the present study suggest directions for future research. First, the degree formulas require specific conditions (such as sigma1 <= mu2) to hold; developing more general formulas that apply under arbitrary membership conditions remains an open problem. Second, the investigation of spectral properties, including the energy and Laplacian energy of the lexicographic max-product, has not been addressed in this paper and represents a promising avenue for future work (Romdhini et al. 2024) [34] . Third, the application of lexicographic max-products to dynamic fuzzy graphs, where membership functions evolve over time, would be valuable for modeling temporal networks. Fourth, the development of efficient algorithms for computing the lexicographic max-product of large-scale fuzzy graphs is essential for practical applications. Finally, the extension of the present results to m-polar fuzzy graphs and complex fuzzy graphs would further generalize the theoretical framework.

Conclusion

This paper has presented a comprehensive investigation of the lexicographic max-product of fuzzy graphs and its structural implications for vertex degree and total degree. The main contributions of this work are summarized as follows: First, we established explicit formulas for computing the degree of vertices in the lexicographic max-product G 1 [G 2 ]max under different conditions on the membership functions of the factor graphs. When sigma1 <= mu2, the degree formula takes the simple form deg(ui, vj) = |V 2 | * deg_G 1 (ui) + deg_G 2 (vj), revealing the additive contribution of each factor graph to the product's vertex degrees. Under alternative conditions, modified formulas were derived and proved.

Second, we derived corresponding formulas for the total degree of vertices in the lexicographic max-product, incorporating both the connectivity-based degree and the vertex membership values. The relationship between regularity in the factor graphs and regularity in the product was characterized, identifying conditions under which the lexicographic max-product preserves regular and totally regular properties. Third, detailed illustrative examples with complete numerical calculations, tabular presentations, and graphical visualizations were provided to validate the theoretical results. The examples demonstrated the non-commutativity of the operation, the preservation of connectedness, and the degree distribution patterns in the product graph. Fourth, a comparative analysis with the lexicographic minproduct and other graph products highlighted the distinctive features of the lexicographic max-product, including its hierarchical structure, maximum-based aggregation, and suitability for modeling systems where dominant relationships propagate through network layers. The findings contribute to the theoretical foundations of fuzzy graph theory and provide computational tools for analyzing complex hierarchical fuzzy networks. Future work will extend these results to dynamic fuzzy graphs, investigate spectral properties of the lexicographic max-product, and develop efficient algorithms for large-scale computations. Applications to hierarchical decision-making, multi-modal transportation systems, and layered social networks will be explored in subsequent studies.

Acknowledgments

The author gratefully acknowledges the contributions of researchers in fuzzy graph theory whose foundational work made this investigation possible. Special thanks to the reviewers for their constructive feedback and suggestions.

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