Topological index
A topological index is a numerical parameter derived from the graph-theoretic representation of a molecular structure, serving as a graph invariant that encodes topological features such as connectivity and branching, independent of vertex labeling or spatial embedding. The concept originated in 1947 when chemist Harold Wiener introduced the Wiener index, defined as the sum of the shortest path distances between all pairs of vertices in a molecular graph, to correlate the structural properties of paraffin hydrocarbons with their boiling points.[1] This pioneering work marked the beginning of topological indices in cheminformatics, with subsequent developments including the Randić index (1975), which weights paths by the product of vertex degrees to better capture branching, and the Balaban index (1982), incorporating both distances and degrees for enhanced discriminatory power. Over time, hundreds of such indices have been proposed, categorized broadly into distance-based (e.g., Wiener), degree-based (e.g., Zagreb indices), and eigenvalue-based types. Topological indices play a central role in quantitative structure-activity relationship (QSAR) and quantitative structure-property relationship (QSPR) modeling, enabling the prediction of physicochemical, biological, and pharmacological properties of compounds without extensive experimental data. For instance, they are used to forecast drug efficacy, toxicity, and solubility by mapping molecular graphs to scalar values that capture shape and size effects. Their utility extends beyond chemistry to network analysis in biology and materials science, where they quantify complexity in protein interaction graphs or polymer structures.[2] Despite their simplicity, topological indices must be selected carefully, as correlations with properties vary, and advanced machine learning now integrates them with other descriptors for more accurate predictions.Overview
Definition and purpose
Topological indices are numerical invariants derived from the topology of a molecular graph, which represents the connectivity of atoms in a molecule without explicit consideration of their spatial arrangement. In this graph-theoretic framework, atoms serve as vertices and chemical bonds as edges, typically in a hydrogen-suppressed form where hydrogen atoms are omitted to simplify the structure while preserving essential bonding information. These indices capture structural features such as branching, cyclicity, and overall size, providing a compact encoding of molecular architecture that remains unchanged under graph isomorphisms. The primary purpose of topological indices is to establish quantitative relationships between a molecule's two-dimensional connectivity and its physicochemical, biological, or pharmacological properties, bypassing the need for computationally intensive three-dimensional conformational analysis. This approach allows for efficient processing of vast chemical libraries in drug discovery and materials design, where rapid prediction of traits like solubility, reactivity, or bioactivity is crucial. By transforming qualitative structural data into measurable scalars, these indices enable statistical modeling that correlates graph-derived features directly with experimental outcomes. A representative example is the Wiener index, introduced as an early topological descriptor that sums the shortest path distances between all pairs of vertices in the molecular graph, effectively quantifying molecular size and branching extent. For instance, branched alkanes exhibit lower Wiener index values compared to their linear isomers, reflecting increased compactness. In cheminformatics, topological indices like this one serve as foundational tools, integrating graph theory principles with chemical sciences to support predictive analytics and virtual screening workflows.Historical development
The roots of topological indices trace back to foundational developments in graph theory, initiated by Leonhard Euler's 1736 solution to the Seven Bridges of Königsberg problem, which established key concepts like paths and connectivity in networks. In the 19th century, Arthur Cayley advanced these ideas by enumerating tree structures to represent alkane isomers, laying groundwork for applying graph invariants to chemical structures.[3] However, the explicit use of such invariants as numerical descriptors for molecular properties emerged in the mid-20th century, marking the birth of topological indices in mathematical chemistry. The application of topological indices to chemistry began in 1947 with Harold Wiener's introduction of the Wiener index, defined as the sum of shortest path lengths between all pairs of vertices in a molecular graph, to predict boiling points of paraffins. This work pioneered distance-based measures derived from path counts. Building on this, the 1960s saw the formal introduction of distance matrices in chemical graph theory, enabling systematic computation of topological distances for structure-property correlations, as part of a broader resurgence in graph-theoretic tools for organic chemistry.[4] In 1971, Haruo Hosoya proposed the pi index, counting matchings in graphs to characterize saturated hydrocarbons. The 1970s and 1980s marked rapid expansion, with Milan Randić's 1975 connectivity index incorporating vertex degrees to quantify molecular branching, significantly improving quantitative structure-activity relationship (QSAR) models. Concurrently, Ivan Gutman and Nenad Trinajstić introduced the Zagreb indices in 1972, summing squared vertex degrees to relate graph topology to pi-electron energies in alternant hydrocarbons. The proliferation continued into the 1980s and 1990s, exemplified by Alexandru T. Balaban's 1982 J index, a distance-based descriptor balancing path lengths and cyclomatic numbers for enhanced discrimination among isomers. Eigenvalue-based indices emerged in this era, notably the Schultz molecular topological index in 1990, derived from the product of degrees and distances via the distance matrix. Recent studies from the 2020s have combined topological indices with DFT-derived properties to improve QSPR predictions for chemotherapy drugs, such as modeling thermodynamic properties of anticancer agents.[5]Theoretical foundations
Graph theory basics for molecules
In chemical graph theory, molecules are modeled as molecular graphs, which are undirected simple graphs consisting of a set of vertices and edges without loops or multiple edges between the same pair of vertices. Vertices represent non-hydrogen atoms (often called heavy atoms), while edges symbolize covalent bonds connecting these atoms. This hydrogen-suppressed representation, known as the hydrogen-depleted molecular graph, simplifies analysis while capturing the core skeletal structure of organic compounds. For molecules with multiple bonds, such as double or triple bonds, weighted molecular graphs are employed, where edge weights correspond to bond orders (e.g., weight 2 for a double bond). Key structural features of molecular graphs are quantified using basic graph invariants and matrices. The degree of a vertex, denoted \delta(v), is the number of edges incident to it, reflecting the valency or number of covalent bonds attached to the corresponding atom. The adjacency matrix A is a square symmetric binary matrix of order n (where n is the number of vertices), with entries A_{ij} = 1 if vertices i and j are adjacent (connected by an edge) and A_{ij} = 0 otherwise; the diagonal elements are zero since graphs are loop-free. Complementing this, the distance matrix D is also an n \times n symmetric matrix, where the entry D_{ij} gives the length of the shortest path (in terms of the minimum number of edges) between vertices i and j, providing a measure of topological separation within the molecule. These matrices serve as foundational tools for deriving higher-order topological descriptors. Paths and cycles form essential substructures in molecular graphs, illustrating connectivity and branching. A simple path is a sequence of distinct vertices connected by edges with no repetitions except possibly at the endpoints, representing linear chains of atoms; for instance, in the molecular graph of n-pentane (C5H12), the entire structure is a simple path of five vertices. Cycles are simple closed paths where the starting and ending vertices coincide, common in ring-containing molecules like benzene, but absent in acyclic alkanes, which form tree graphs. Branch points, or vertices with degree \delta(v) \geq 3, indicate sites of structural divergence; in branched alkanes such as isopentane (2-methylbutane), a central carbon atom serves as a branch point connecting three paths, contrasting with the linear arrangement in unbranched counterparts. These elements highlight the skeletal topology of molecular graphs, particularly in hydrocarbon series. Graph isomorphism addresses the equivalence of molecular structures under relabeling of atoms, ensuring that topological analyses remain consistent regardless of vertex numbering. Two molecular graphs are isomorphic if there exists a bijective mapping (permutation) between their vertex sets that preserves adjacency relations, meaning connected vertices map to connected vertices and vice versa. This structural invariance is crucial for topological indices, as it guarantees that numerically equivalent graphs—representing the same molecular topology—yield identical index values, independent of arbitrary labeling schemes. For example, all linear alkanes of the same carbon count share isomorphic path graphs, underscoring the focus on intrinsic connectivity over extrinsic representations.Calculation methods
The calculation of topological indices begins with representing a molecule as an undirected graph, where atoms (typically non-hydrogen) are vertices and covalent bonds are edges, often assuming unit bond lengths for simplicity. The general approach involves constructing key matrices from this graph: the adjacency matrix A, a symmetric n \times n matrix (for n vertices) with A_{ij} = 1 if vertices i and j are adjacent and 0 otherwise; the degree matrix \Delta, a diagonal matrix with \Delta_{ii} equal to the degree d_i (number of edges incident to vertex i); or the distance matrix D, where D_{ij} is the length of the shortest path between i and j. Indices are then derived through operations such as matrix summations, traces, or eigenvalue decompositions of these structures.[6][7] A representative distance-based index is the Wiener index W, defined as the sum of shortest-path distances over all unordered pairs of vertices: W = \sum_{1 \leq i < j \leq n} d(i,j), where d(i,j) is the (i,j)-entry of the distance matrix. This index, introduced by Wiener to correlate molecular branching with physical properties like boiling points, quantifies overall molecular size and compactness.[1] To compute W for a linear alkane chain, such as n-butane (C_4H_{10}), model the carbon skeleton as a path graph with vertices labeled 1--2--3--4. First, build the distance matrix: D = \begin{pmatrix} 0 & 1 & 2 & 3 \\ 1 & 0 & 1 & 2 \\ 2 & 1 & 0 & 1 \\ 3 & 2 & 1 & 0 \end{pmatrix}. The off-diagonal upper triangle sums to d(1,2) + d(1,3) + d(1,4) + d(2,3) + d(2,4) + d(3,4) = 1 + 2 + 3 + 1 + 2 + 1 = 10, so W = 10. For a general linear alkane with n carbon atoms (path graph P_n), distances follow a pattern where each possible separation k (from 1 to n-1) occurs (n - k) times, yielding the closed-form W(P_n) = \frac{n(n^2 - 1)}{6}; for n=4, this confirms W = 10.[1][8] Matrix methods extend to degree-based and spectral indices. For instance, the first Zagreb index M_1, a simple vertex-degree sum, is M_1 = \sum_{i=1}^n d_i^2, computed directly from the degree sequence without full matrix construction: extract degrees from the graph (or row sums of A), square each, and sum. In n-butane's path graph, degrees are d_1 = d_4 = 1 and d_2 = d_3 = 2, so M_1 = 1^2 + 2^2 + 2^2 + 1^2 = 1 + 4 + 4 + 1 = 10. This index, originally linked to \pi-electron energy in hydrocarbons, emphasizes vertex connectivity. More advanced indices employ the Laplacian matrix L = \Delta - A, a positive semidefinite matrix whose eigenvalues \lambda_1 = 0 \leq \lambda_2 \leq \cdots \leq \lambda_n encode graph connectivity; for example, some indices sum non-zero eigenvalues or use cofactors of L for spanning tree counts, with computations involving standard eigenvalue solvers on L. The Laplacian's role in chemical graph theory stems from its ability to capture diffusion-like processes on molecular structures.[6][7] Implementations in software streamline these matrix constructions and computations for large datasets. The open-source RDKit library supports computation of topological indices through its tools and extensions, such as custom functions for the Wiener index and the Mordred package for Zagreb indices and others, from molecular input files like SMILES strings.[9][10] Similarly, the commercial Dragon software computes over 4,800 descriptors, with a dedicated block of 119 topological indices derived from adjacency and distance matrices, supporting batch processing for QSAR workflows.[11][12]Classification
Global and local indices
Topological indices are broadly classified into global and local categories based on the scope of the structural information they encode. Global indices yield a single scalar value that summarizes the topology of the entire molecular graph, often reflecting overall features such as branching, compactness, or total connectivity. These indices are derived by aggregating local vertex or edge invariants across the graph, providing a holistic measure suitable for comparing distinct molecules. In contrast, local indices compute values specific to individual vertices (atoms) or edges (bonds), capturing the immediate neighborhood or contributions from particular structural elements within a single molecule. This distinction allows global indices to address intermolecular variations, while local indices mitigate intramolecular degeneracy by differentiating equivalent atoms or bonds.[13] A classic example of a global index is the Randić connectivity index, introduced to quantify molecular branching and defined as \chi(G) = \sum_{(u,v) \in E(G)} (d_u d_v)^{-1/2}, where d_u and d_v are the degrees of adjacent vertices u and v in the graph G. This index has been widely applied in quantitative structure-activity relationship (QSAR) modeling to correlate with properties like boiling points and enthalpies of formation in alkanes. For local indices, the per-bond contribution to the Atom-Bond Connectivity (ABC) index serves as an illustrative case, given by \sqrt{(d_u + 1)(d_v + 1)} for each edge, which emphasizes the local degree-based environment around bonds and sums to the global ABC index. The ABC framework originated from efforts to model entropy in alkanes and has been extended to predict strain energies and stability. Global indices excel in applications requiring an overall molecular descriptor, such as predicting bulk physicochemical properties like solubility or octanol-water partition coefficients, where the aggregate topology correlates with macroscopic behavior in large datasets of organic compounds. Local indices, however, are invaluable for site-specific analyses, enabling the identification of reactive atoms or bonds—for instance, in predicting electrophilic attack sites or metabolic hotspots in drug molecules—by providing atom- or bond-resolved insights that global measures overlook. This complementary use enhances the discriminatory power in QSAR and quantitative structure-property relationship (QSPR) studies, with local variants reducing ambiguity in complex structures like substituted benzenes.[13]Distance-based indices
Distance-based topological indices are graph-theoretic invariants calculated using the shortest path distances between pairs of vertices in the molecular graph, which represent inter-atomic distances in the molecule. These indices quantify structural features such as molecular size, branching, and compactness, making them useful for correlating graph topology with physicochemical properties. The Wiener index, introduced in 1947 as the first such descriptor, is defined as the sum of the shortest path distances over all unordered pairs of vertices: W(G) = \sum_{u < v} d(u,v) where d(u,v) is the length of the shortest path between vertices u and v in the connected graph G. This index measures the overall "spread" or extensiveness of the molecular structure and was originally applied to predict boiling points of alkanes. To address limitations of the Wiener index in distinguishing branched isomers, the Balaban index J was developed in 1982, incorporating valence adjustments via distance sums and a correction for molecular cyclicity. It is given by J(G) = \frac{m}{\mu + 1} \sum_{(u,v) \in E(G)} (D_u D_v)^{-1/2} where m is the number of edges, \mu = m - n + 1 is the cyclomatic number (with n the number of vertices for a connected graph), E(G) the edge set, and D_u = \sum_w d(u,w) the distance sum from vertex u to all other vertices w. This index provides higher discrimination power for isomeric structures, particularly in quantitative structure-activity relationship studies.[14] The Harary index, proposed in 1993, complements these by using reciprocal distances to emphasize closer vertex pairs, defined as H(G) = \sum_{u < v} \frac{1}{d(u,v)}. It balances the Wiener index's sensitivity to long paths, offering a measure more attuned to local connectivity while still global in scope. These indices can be extended to local versions by restricting sums to distances from a specific vertex, aligning with the broader distinction between global and local topological descriptors. Distance-based indices are particularly straightforward to compute for acyclic graphs like alkanes, where shortest paths can be enumerated efficiently without cycle detection.Eigenvalue-based and other advanced types
Eigenvalue-based topological indices derive from the eigenvalues of graph matrices, such as the adjacency matrix or distance matrix, offering insights into the spectral properties and overall structural complexity of molecular graphs. These indices leverage linear algebra to capture global characteristics that are not easily enumerated through simpler counting methods, making them particularly useful for correlating with physicochemical properties like stability and reactivity. For instance, the eigenvalues of the adjacency matrix reflect the graph's connectivity spectrum, while those of the distance matrix (Wiener matrix) can model aspects like molecular polarity by quantifying path-related invariances.[15][16] A seminal example is the Schultz molecular topological index (MTI), defined as the sum of all eigenvalues of the adjacency matrix, which provides a measure of the graph's total "energy" in spectral terms and correlates with molecular weight and complexity. Introduced by Schultz in 1989, this index has been applied in quantitative structure-property relationships (QSPR) for diverse chemical datasets.[17] Another notable application involves eigenvalues of the Wiener matrix to assess polarity, where spectral sums or moments extend the classical Wiener polarity index—originally the count of vertex pairs at distance 3—to capture finer topological nuances in polar compounds.[1] Other advanced topological indices encompass valence-based and information-theoretic variants, which address limitations in homonuclear graphs by incorporating atomic valences or probabilistic structural information. Valence-based indices, such as the adjusted Randić connectivity index, generalize the original Randić index by using effective valences (accounting for heteroatoms and lone pairs) instead of degrees, enhancing applicability to real molecules like those with oxygen or nitrogen. These were formalized by Kier and Hall in the early 1990s as valence connectivity indices to improve correlations in QSAR models for biological activity.[18] Information-theoretic indices, drawing from Shannon entropy, quantify graph complexity through measures like the entropy of the vertex degree distribution, where probabilities are assigned based on degree frequencies to yield a single value representing structural uncertainty or diversity. Such indices, explored since the 2000s, aid in distinguishing isomers and predicting entropy-related properties in chemical networks.[19][20] The following table lists 12 common advanced topological indices, focusing on eigenvalue-based, valence-adjusted, and related types, with brief descriptions and introduction years:| Index | Description | Year Introduced | Citation |
|---|---|---|---|
| Zagreb M1 | Sum of the squares of vertex degrees, capturing local connectivity density. | 1972 | [21] |
| Zagreb M2 | Sum over edges of the product of degrees of adjacent vertices, emphasizing edge contributions. | 1972 | [21] |
| Hosoya π (Z-index) | Sum of the number of matchings of lengths 1 and 2, quantifying branching and cyclicity. | 1971 | [22] |
| Randić χ (adjusted valence version) | Sum over edges of (valence_u * valence_v)^{-0.5}, extending to heteroatoms for better QSAR fit. | 1975 (original); 1991 (valence) | [18] |
| Schultz MTI | Sum of eigenvalues of the adjacency matrix, reflecting spectral energy. | 1989 | [17] |
| Balaban J | Distance-weighted variant of Randić index, balancing branchiness and size. | 1982 | [14] |
| Gálvez topological charge G1 | Absolute sum of off-diagonal elements in the Gálvez matrix (adjacency times reciprocal squared distances), modeling charge transfer. | 1993 | [23] |
| Wiener polarity WP | Number of vertex pairs at graph distance 3, indicating polar path motifs. | 1947 | [1] |
| Estrada EE | Sum of exponentials of adjacency matrix eigenvalues, akin to molecular "vibrational" spectrum. | 2000 | [24] |
| Degree entropy | Shannon entropy of the probability distribution over vertex degrees, measuring structural irregularity. | 2008 | [20] |
| Spectral radius μ | Largest eigenvalue of the adjacency matrix, indicating maximum connectivity strength. | 1973 (in chemical context) | [15] |
| Valence Zagreb M1^v | Valence-adjusted sum of squares of atomic valences, for heteromolecular graphs. | 1986 | [18] |