Bond Incident Degree Indices

• Bond Incident Degree Indices are degree-based topological descriptors defined by summing a symmetric function over all edges, unifying many classical invariants in graph theory and mathematical chemistry.
• They enable systematic extremal graph analysis with explicit formulas for structured families like polyomino and triangular chain graphs, enhancing both theoretical insights and practical applications.
• Their methodologies, including edge transformations and transfer inequalities, provide actionable tools for exploring molecular structures and optimizing connectivity in diverse graph classes.

A bond incident degree (BID) index is a degree-based topological index associated to a simple graph $G=(V,E)$, defined by summing a symmetric, typically non-negative function $f(d_u, d_v)$ over all edges $uv\in E$, where $d_u$ and $d_v$ are the degrees of the endpoints. This unifies and generalizes numerous classical indices in mathematical chemistry and graph theory, including the Randić, Zagreb, harmonic, Platt, and various connectivity indices. The BID framework provides both a unifying theoretical lens and concrete tools for extremal graph characterization in diverse classes, including polyomino and triangular chain graphs, as well as trees and graphs with prescribed diameters or cycle structures (Ali et al., 2015, Ali et al., 2016, Ali et al., 2017, Hosamani, 11 Mar 2026).

1. Formal Definition and General Properties

Given a simple graph $G=(V,E)$ with maximum degree $\Delta(G)$, the general BID index is defined as

$$\mathrm{BID}_f(G) = \sum_{uv\in E} f(d_u,d_v) = \sum_{1\leq a\leq b\leq \Delta(G)} x_{a,b}(G) \theta_{a,b},$$

where:

• $f(a,b) = \theta_{a,b}$ is a symmetric real function, typically non-negative,
• $x_{a,b}(G)$ counts edges joining vertices of degrees $f(d_u, d_v)$0 and $f(d_u, d_v)$1.

Many classical graph invariants fit this framework via appropriate $f(d_u, d_v)$2:

• Randić index: $f(d_u, d_v)$3,
• First Zagreb: $f(d_u, d_v)$4,
• Second Zagreb: $f(d_u, d_v)$5,
• Harmonic: $f(d_u, d_v)$6,
• General sum-connectivity: $f(d_u, d_v)$7,
• General Platt: $f(d_u, d_v)$8,
• Variable sum exdeg: $f(d_u, d_v)$9,
• Inverse sum indeg (ISI): $uv\in E$0 (Hosamani, 11 Mar 2026).

This abstraction allows systematic study of extremal graphs, closed formulas on specific families, and transfer of proofs across different indices (Ali et al., 2015, Ali et al., 2017).

2. Explicit Formulas for Structured Graph Families

2.1 Polyomino Chains

For a polyomino chain $uv\in E$1 of $uv\in E$2 squares, uniquely decomposed into maximal linear segments $uv\in E$3, with segment lengths $uv\in E$4, indicator vectors $uv\in E$5 describe internal structures (segment lengths and types of edges). The primary closed formula is (Ali et al., 2015): $uv\in E$6 where only indices with degrees 2,3,4 contribute. Specialization to classical indices yields all results of An–Xiong, Deng et al., Yarahmadi et al., Rada, and Ali–Bhatti–Raza as corollaries (Ali et al., 2015).

2.2 Triangular Chain Graphs

For the family $uv\in E$7 of degree $uv\in E$8 triangular chains, $uv\in E$9 decomposes into $d_u$0 segments with segment lengths $d_u$1. Defining indicator variables $d_u$2 (length 3), $d_u$3 (4), $d_u$4 (5), the general formula is (Ali et al., 2016): $d_u$5 with $d_u$6 linear combinations in $d_u$7 as explicit in the original data.

2.3 Extremal Graphs with Degree/Diameter or Cyclomatic Constraints

Systematic graph transformations (edge shifts, pendant transfers, path liftings) show that, under monotonicity and convexity requirements on $d_u$8, maximal BID indices in fixed-diameter trees and unicyclic graphs are uniquely realized by trees/graphs concentrating as much degree as possible on a single central vertex or specific loaded cycles. For ISI, the maximal tree $d_u$9 attaches all but the path vertices to one center, and analogous unicyclic maximizers are $d_v$0 (loaded triangle), $d_v$1 (loaded 3-cycle), and the constructed $d_v$2 for $d_v$3 (Hosamani, 11 Mar 2026).

3. Extremal Graph Characterizations

A central result for BID indices satisfying monotonicity and two key transfer inequalities (Ali–Dimitrov lemma) is that any extremal $d_v$4-graph must have a universal vertex (degree $d_v$5) when maximizing, or minimal degree concentration when minimizing (Ali et al., 2017, Hosamani, 11 Mar 2026). For many indices (sum-connectivity $d_v$6, Platt $d_v$7, variable-sum exdeg $d_v$8), extremal graphs in trees, unicyclic, bicyclic, tricyclic, and tetracyclic classes are:

Graph Class	Unique Maximizer (Monotonic $d_v$9)
Tree ($G=(V,E)$0)	$G=(V,E)$1 (star)
Unicyclic	$G=(V,E)$2 (star plus a cycle edge)
Bicyclic	$G=(V,E)$3 (bicircular star-extension)
Tricyclic	$G=(V,E)$4 or $G=(V,E)$5 (tricyclic star-extensions)
Tetracyclic	$G=(V,E)$6 or $G=(V,E)$7 (tetracyclic star-extensions)

This structure is robust under a variety of alternate index functions $G=(V,E)$8. For BID indices with decreasing $G=(V,E)$9, the corresponding extremal graphs are those with degree as evenly distributed as possible.

4. Unified Treatment of Classical Indices

Special BID kernels correspond to the following classical indices:

• $\Delta(G)$0: Randić, sum-connectivity, and related indices.
• $\Delta(G)$1: First Zagreb.
• $\Delta(G)$2: Second Zagreb.
• $\Delta(G)$3: Harmonic.
• $\Delta(G)$4: Albertson irregularity.
• $\Delta(G)$5: Inverse sum indeg (ISI).

Unified closed formulas immediately yield extremal results and allow for direct comparison, as shown for polyomino and triangular chain graphs. For example, in polyomino chains, $\Delta(G)$6 (linear chain) is extremal for a range of indices, and the sign conditions on auxiliary expressions ($\Delta(G)$7) fully determine the maximizing and minimizing structures (Ali et al., 2015).

5. Methodologies: Proof Techniques and Transformations

The main approaches for extremal results combine:

• Partitioning edge-sets by segment type and counting degree patterns,
• Indicator-based counting (structural $\Delta(G)$8–$\Delta(G)$9 variables encoding segment properties),
• Transformation techniques: edge-shifts, branch relocations, path-lifting operations,
• Transfer inequalities: monotonicity and convexity properties of the kernel $$\mathrm{BID}_f(G) = \sum_{uv\in E} f(d_u,d_v) = \sum_{1\leq a\leq b\leq \Delta(G)} x_{a,b}(G) \theta_{a,b},$$0,
• Sufficient-conditions lemma: the Ali–Dimitrov lemma provides general sufficient conditions for the extremal structure through functional inequalities on $$\mathrm{BID}_f(G) = \sum_{uv\in E} f(d_u,d_v) = \sum_{1\leq a\leq b\leq \Delta(G)} x_{a,b}(G) \theta_{a,b},$$1 (Ali et al., 2017).

These techniques reduce extremal problems to algebraic or combinatorial maximizations over indicator variables or length-vectors, often admitting explicit or recursive solutions in closed form.

6. Applications and Theoretical Implications

The BID framework unifies the study of a vast array of degree-based indices across discrete mathematics and mathematical chemistry. Applications include:

• Quantification of molecular structure: topological indices correlate to chemical and physical properties.
• Extremal chemistry and enumeration: identification of maximally or minimally reactive (or stable) molecular graphs.
• Algorithmic implications: structural insights guide enumeration algorithms and index computation for large molecular families.

A plausible implication is that BID-based techniques will extend to broader classes of molecular graphs (higher-dimensional "animal" graphs, k-polygonal chains) and inform future explorations of graph invariants beyond current classical indices.

7. Connections, Extensions, and Outlook

Results are broadly connected and extend theorems of An–Xiong, Deng et al., Yarahmadi et al., Rada, Ali–Bhatti–Raza, and others (Ali et al., 2015). Further research trends include:

• Generalization to variable-parameter indices,
• Exploration in non-planar and higher-genus families,
• Analytical and algorithmic study of extremal structures under additional constraints.

The BID index paradigm is expected to remain central in the structural study of both theoretical and applied graph classes, supported by the unified and explicit algebraic characterization that it enables (Ali et al., 2015, Ali et al., 2016, Ali et al., 2017, Hosamani, 11 Mar 2026).