An Eccentricity-Based Estrada Index for Finite Simple Connected Graphs

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1. Introduction

Graph theory has emerged as one of the most influential mathematical disciplines due to its extensive applications in chemistry, biology, computer science, communication networks, and social sciences [3, 8]. A graph provides an abstract representation of relationships among objects and offers a convenient framework for quantitative analysis. The study of graph invariants known as topological indices began with the pioneering work of Wiener in 1947 [7]. The Wiener index was introduced to correlate boiling points of paraffin hydrocarbons with molecular structures. Since then, numerous degree-based and distance-based topological indices have been proposed and successfully applied in quantitative structure– property relationships (QSPR) and quantitative structure–activity relationships (QSAR) [4, 5]. One of the most important spectral invariants is the Estrada index, introduced by Ernesto Estrada [1]. This index has become an effective tool in analyzing molecular folding, protein structures, complex networks, and communication systems [2]. The Estrada index reflects the degree of compactness and the extent of connectivity present in a graph. Motivated by the significance of eccentricity in graph analysis and the growing interest in distance-based graph invariants [5, 6], we propose a modified Estrada index based on vertex eccentricities. The proposed invariant combines structural and distance-related information and provides an alternative perspective for studying graph properties.

2. Preliminaries

Definition 2.4 (Classical Estrada Index). Let λ1, λ2,., λn be the eigenvalues of the adjacency matrix of a graph � with n vertices. The Estrada index of � is defined by [1, 2].

3. Proposed Eccentricity-Based Estrada Index

Inspired by the classical Estrada index [1, 2], and distance-based graph invariants [4, 5], we define the modified Estrada index as This index depends solely on the eccentricity distribution of the graph.

3.1. Basic Properties

The following properties follow directly from the definition of the proposed index:

5. Exact Values for Standard Graphs

Hence the proof is complete.

This completes the proof.

In ??????6, the greatest distance from any vertex to another vertex is 3. Hence, the eccen-tricity of every vertex is[3, 8].

Hence, the theorem is verified. The example illustrates how the uniform eccentricity distribution of cycle graphs leads to a simple closed-form expression for the index [5].

for each vertex �??????, we obtain Therefore, Hence the theorem is proved.

The distances of each vertex from the two end vertices are shown below [3, 8]:

6. Modified Estrada Index of Total Graphs

The effect of graph transformations on the proposed eccentricity-based Estrada index is investigated. The study of graph transformations and their influence on topological indices has been an active area of research in graph theory and network analysis [3, 8, 5].

Hence the theorem is proved.

is verified for the graph P3. This example illustrates how graph transformations may increase the value of topological indices by introducing additional structural information into the graph [5, 6].

7. Modified Estrada Index of Eccentric Graphs

This section studies the relationship between EEε(G) and EEε(Ge), where Ge denotes the eccentric graph of G. The study of graph transformations and their associated topological indices has attracted considerable attention in graph theory and network analysis [3, 8, 5].

Based on computational experiments performed on several classes of graphs, we for-mulate the following conjecture.

Conjecture 6.1. Let G be a connected graph. Then EEε(Ge) ≤ EEε(G).

The conjecture is motivated by the behavior of eccentricity-based graph invariants and the role of exponential weighting in Estrada-type indices [1, 2, 6].

The eccentricities of the vertices are ε(v1) = 3, ε(v2) = 2, ε(v3) = 2, ε(v4) = 3, which follow directly from the distance structure of the path graph[3, 8].

The eccentric graph P4, e consists of the edges v1v4, v2v4, v1v3, since these pairs are mutually eccentric vertices in P4. Thus P4, e is the path v2 − v4 − v1 − v3.

This example shows that equality can occur in the conjecture. Similar phenomena have been observed when studying the behavior of graph invariants under graph transformations and derived graph constructions[5, 6].

8. Results

In this section, explicit formulae for the eccentricity-based Estrada index of several stan-dard graph classes are obtained. Numerical results are presented to illustrate the derived formulae and facilitate comparisons among different graph families[3, 8].

Table 1 presents the computed values of the eccentricity-based Estrada index for sev-eral standard graph classes. These values illustrate the dependence of the index on the structural properties and eccentricity distribution of the underlying graph, which are fun-damental concepts in graph theory and network analysis [2, 3].

7.1. Applications of the Eccentricity-Based Estrada Index

The numerical values presented in Table 1 demonstrate the applicability of the eccentricity-based Estrada index in characterizing the structural properties of graphs through vertex eccentricities. Since the index is based on the exponential contribution of eccentricities, it provides valuable information regarding the global connectivity and compactness of a network, extending ideas originating from the classical Estrada index [1, 2].

The computed results have the following applications:

• Graph Structure Characterization The index effectively distinguishes graph classes having the same order but different structural arrangements[3, 8]. For ex-ample, among the graphs of order four, the complete graph K4 possesses the smallest index value due to its minimum eccentricities, whereas the path graph P4 exhibits the largest value because of its larger vertex eccentricities.

• Network Compactness Analysis Smaller index values indicate highly compact and well-connected networks. The value obtained for K4 reflects its maximum con-nectivity, while larger values for P4 indicate a less compact structure with greater distances between vertices [2].

• Comparison of Different Graph Families The results facilitate comparisons among complete graphs, path graphs, cycle graphs, and wheel graphs. Such com-parisons help identify how graph topology influences eccentricity-based measures [5].

• Chemical Graph Theory In molecular graph modeling, vertices and edges repre-sent atoms and chemical bonds, respectively. The eccentricity-based Estrada index can serve as a molecular descriptor for studying physicochemical properties, biologi-cal activities, and structure–property relationships of chemical compounds [7, 1, 4, 6].

• Communication and Transportation Networks The index can be used to evaluate the efficiency of communication, transportation, and social networks. Net-works with lower index values generally exhibit shorter communication paths and improved accessibility among vertices [2].

• Network Design and Optimization Since the index is sensitive to changes in vertex eccentricities, it can be employed in the design of efficient networks where minimizing communication delays and maximizing connectivity are important ob-jectives [2, 5].

• Benchmarking New Graph Invariants The obtained values for standard graph classes provide benchmark results that can be used for validating and comparing newly proposed eccentricity-based topological indices [5, 6].

From Table 1, it is observed that the eccentricity-based Estrada index increases as the graph becomes less compact and the vertex eccentricities increase. Therefore, the index serves as a useful quantitative measure for analyzing the overall structural complexity and connectivity of graph-based systems[2, 5].

9. Application in Function Approximation

The eccentricity-based Estrada index contains an exponential weighting of the vertex eccentricities. Since exponential func-tions play a fundamental role in approximation theory, numerical analysis, and machine learning, the proposed index may be viewed as a graph-based exponential aggregation operator. The use of exponential weighting is inspired by the classical Estrada index and its applications in network analysis [1, 2]. In function approximation, weighted exponential sums are frequently employed to rep-resent, estimate, or approximate complex functions. The quantity ∑�??????(�)�∈??????(??????)

assigns larger weights to vertices having greater eccentricity and smaller weights to vertices that are more centrally located in the graph. Consequently, the index captures the global structural distribution of vertices and can serve as a compact numerical descriptor of a network [2].

where the exponential term acts as a weighting factor determined by the topological position of the vertex. Such representations can be useful in graph-based learning and approximation frameworks.

The exponential weighting mechanism provides greater sensitivity to peripheral ver-tices, making the index suitable for applications where boundary or extremal structural characteristics are important. Therefore, the eccentricity-based Estrada index and its vari-ants may be employed in the construction of graph similarity measures, feature extraction techniques, and approximation models for complex networks [5, 6]. Potential applications include:

1. Data Clustering: The index can be used as a graph descriptor to compare differ-ent datasets represented as graphs and to identify clusters with similar structural characteristics [6].

2. Pattern Recognition: Exponentially weighted

eccentricity information can be incorporated into feature vectors for recognizing and classifying graph-based patterns [6].

3. Artificial Intelligence and Machine Learning: Graph

neural networks and other learning algorithms may utilize eccentricity-based exponential weights as in-formative topological features for prediction and classification tasks [6]. 4. Network Optimization: The index can assist in identifying structurally important vertices and evaluating network configurations in communication, transportation, and social networks [2].

5. Graph Similarity Analysis: Networks having close

values of EEε(G) may ex-hibit similar structural behavior, making the index useful in graph comparison and retrieval problems [5].

6. Complex Network Modeling: The index may be

employed as a quantitative measure in the analysis of biological, technological, and information networks where peripheral vertices significantly influence system behavior [4, 6]. Thus, the proposed eccentricity-based Estrada index provides a bridge between graph-theoretical structure and exponential approximation techniques, offering promising oppor-tunities for applications in data science, network analysis, and computational intelligence [1, 2, 6].

10. Conclusion

In this paper, we introduced an eccentricity-based Estrada index, which incorporates vertex eccentricities through an exponential weighting scheme. Fun-damental properties of the index were established, and explicit formulae were derived for several standard graph classes, including complete, cycle, path, and wheel graphs. The behavior of the index under graph transformations was also investigated, yielding a result for total graphs and a conjectured relationship for eccentric graphs.

The obtained results show that the proposed index effectively captures graph struc-ture, connectivity, and eccentricity distribution, making it a useful descriptor for complex networks. Potential applications in graph theory, network analysis, chemical graph the-ory, machine learning, and function approximation were discussed. Future work includes establishing sharp bounds, characterizing extremal graphs, exploring relationships with other graph invariants, and investigating further applications in network science and data analytics.