M-Polynomial: Graph Invariants

• M-Polynomial is a bivariate graph polynomial that captures the complete degree–pair distribution of edge endpoints, serving as a universal generating function for degree-based indices.
• It enables explicit computation and closed-form expressions for key graph families by leveraging operator calculus and symbolic evaluation.
• The framework underlies many topological indices in chemical graph theory, combinatorics, and network analysis, streamlining complex edge-based enumerations.

The $M$-polynomial is a bivariate graph polynomial that encodes the complete degree–pair distribution of edge endpoints in a graph. Introduced by Deutsch and Klavžar in 2015, it serves as a universal generating function for a broad class of degree-based topological indices, which are key tools in chemical graph theory, combinatorics, and network analysis. The $M$-polynomial admits closed forms for many important graph families and interacts systematically with graph operations such as products, making it an essential algebraic invariant for both theoretical and applied research.

1. Definition and Structural Properties

Let $G=(V,E)$ be a finite simple graph, and for integers $i,j\geq1$, denote

$$m_{ij}(G) = \#\bigl\{ uv\in E : \{\deg(u),\deg(v)\}=\{i,j\} \bigr\},$$

the number of edges whose endpoints have degrees $i$ and $j$. The $M$-polynomial of $G$ is

$M(G;x,y) = \sum_{i\leq j} m_{ij}(G)\, x^i y^j.$

This bivariate generating function is symmetric in $M$0 and $M$1 and encodes all bond degree incidence information. The total edge count is recovered as $M$2. If the graph has maximal degree $M$3, then $M$4 for $M$5 (Deutsch et al., 2014).

2. From $M$6-Polynomial to Degree-Based Indices

Most degree-based topological indices—quantities originally motivated by mathematical chemistry for correlating molecular structure with physical or chemical properties—admit the general form

$M$7

where $M$8 is a function of the endpoint degrees. The $M$9-polynomial method expresses $G=(V,E)$0 as

$G=(V,E)$1

and, for polynomial $G=(V,E)$2, as the evaluation of a corresponding operator combination at $G=(V,E)$3. The main differential and integral operators are as follows:

Operator	Action	Typical Use
$G=(V,E)$4	$G=(V,E)$5	Differentiation in $G=(V,E)$6
$G=(V,E)$7	$G=(V,E)$8	Differentiation in $G=(V,E)$9
$i,j\geq1$0	$i,j\geq1$1	Inverse powers (integral transform)
$i,j\geq1$2	$i,j\geq1$3	"
$i,j\geq1$4	$i,j\geq1$5	Symmetrization for indices depending on $i,j\geq1$6 or similar
$i,j\geq1$7	$i,j\geq1$8	Scalar transformation

Degree-based indices computed from $i,j\geq1$9 via operators include:

• First Zagreb index: $$m_{ij}(G) = \#\bigl\{ uv\in E : \{\deg(u),\deg(v)\}=\{i,j\} \bigr\},$$0
• Second Zagreb index: $$m_{ij}(G) = \#\bigl\{ uv\in E : \{\deg(u),\deg(v)\}=\{i,j\} \bigr\},$$1
• General Randić index: $$m_{ij}(G) = \#\bigl\{ uv\in E : \{\deg(u),\deg(v)\}=\{i,j\} \bigr\},$$2
• Symmetric division index: $$m_{ij}(G) = \#\bigl\{ uv\in E : \{\deg(u),\deg(v)\}=\{i,j\} \bigr\},$$3
• Harmonic and inverse sum indices: use combinations of $$m_{ij}(G) = \#\bigl\{ uv\in E : \{\deg(u),\deg(v)\}=\{i,j\} \bigr\},$$4, $$m_{ij}(G) = \#\bigl\{ uv\in E : \{\deg(u),\deg(v)\}=\{i,j\} \bigr\},$$5, $$m_{ij}(G) = \#\bigl\{ uv\in E : \{\deg(u),\deg(v)\}=\{i,j\} \bigr\},$$6, $$m_{ij}(G) = \#\bigl\{ uv\in E : \{\deg(u),\deg(v)\}=\{i,j\} \bigr\},$$7 (Deutsch et al., 2014, Deutsch et al., 2018, Mehiri, 16 Nov 2025, Nizami et al., 2017).

This formalism enables the simultaneous and systematic computation of a large class of invariants, replacing traditional edgewise enumeration or index-specific counting.

3. Explicit Computation: Examples and Methods

Closed-form $$m_{ij}(G) = \#\bigl\{ uv\in E : \{\deg(u),\deg(v)\}=\{i,j\} \bigr\},$$8-polynomials have been obtained for key graph families via combinatorial, recursive, and linear-algebraic techniques:

• Generalized Möbius Ladder $$m_{ij}(G) = \#\bigl\{ uv\in E : \{\deg(u),\deg(v)\}=\{i,j\} \bigr\},$$9: Partitioning edges by boundary/interior type leads to

$i$0

The line graph $i$1 yields a bivariate polynomial in four monomials centered at degrees 4, 5, 6 (Nizami et al., 2017).

• Bethe Cacti $i$2, $i$3, $i$4: Recursive construction allows closed $i$5-polynomials based on counting by degree class (Deutsch et al., 2018).
• Generalized Hanoi Graphs $i$6: Occupancy-based combinatorial decomposition, refined with Stirling and 2-associated Stirling numbers, enables an explicit bivariate $i$7-polynomial as a sum of diagonal and off-diagonal monomials in the possible degree classes (Mehiri, 16 Nov 2025).
• Product Graphs: Systematic formulas for the $i$8-polynomial under Cartesian, direct, strong, lexicographic, and Sierpiński products follow from degree formulas and double-index sums involving the $i$9 values of the factors, greatly generalizing structural results (Mehiri et al., 11 Mar 2026).
• Gutman's Approach: For bounded-degree planar graphs, $j$0 are determined by solving linear equations derived from vertex and edge counts, possibly extended by Euler's formula for faces (Deutsch et al., 2018).

4. Algorithmic and Computational Significance

Once $j$1 is determined, all degree-based indices are computed by low-order differentiation/integration and evaluation at $j$2. For highly regular or recursively defined graphs, the number of distinct $j$3 is small, yielding swift computation and compact closed forms across parameter ranges (Nizami et al., 2017, Mehiri, 16 Nov 2025).

The method dramatically reduces complexity from $j$4 per invariant (using edgewise summation) to $j$5 symbolic operations after the initial degree-pair count, with particular efficiency for large parametric chemical graph families and regular graph constructions (Barman et al., 16 Feb 2026).

5. The $j$6-Polynomial and Emerging Indices

The $j$7-polynomial framework extends to new families of topological indices. For example, the hyperbolic Sombor index (HSO) is derived by composing operators in sequence: $j$8 where $j$9, $M$0, $M$1, $M$2, and $M$3 act as defined in the associated data. Detailed formulas are provided for standard graphs such as paths, cycles, stars, and for chemical families with tabulated and graphical sensitivity to parameters (Barman et al., 16 Feb 2026).

This suggests a unifying algebraic pipeline for the adaptation and computation of newly proposed degree-based indices, leveraging the operator calculus on $M$4.

6. Applications, Structural Insight, and Outlook

Degree-based indices extracted via $M$5-polynomial methods are fundamental in correlating graph structure to molecular properties such as boiling point, stability, and reactivity in mathematical chemistry. The support (i.e., which degree-pairs occur) of $M$6 itself reveals structural regularity, degree heterogeneity, and edge-type dominance.

Having explicit $M$7-polynomials enables immediate access to the full hierarchy of classical and exotic indices for new graph constructions, provides universal structural parameters for benchmarking and invariance, and allows the study of parametric families across discrete and continuous variables. The computational paradigm and algebraic operator calculus of $M$8-polynomials unify and extend formerly disparate methods in chemical graph theory, combinatorics, and applied network analysis (Deutsch et al., 2014, Nizami et al., 2017, Mehiri, 16 Nov 2025, Mehiri et al., 11 Mar 2026).

The $M$9-polynomial continues to be generalized, including matrix-valued (finite biorthogonal $G$0-matrix) analogues admitting explicit generating functions, recurrence, biorthogonality relations, and matrix differential equations, widening its reach in algebraic and spectral graph theory (Lekesiz, 6 Sep 2025).

• "M-Polynomial and Degree-Based Topological Indices" (Deutsch et al., 2014)
• "The $G$1-Polynomial and Topological Indices of Generalized Möbius Ladder and Its Line Graph" (Nizami et al., 2017)
• "M-Polynomial Revisited: Bethe Cacti and an Extension of Gutman's Approach" (Deutsch et al., 2018)
• "Explicit M-Polynomial and Degree-Based Topological Indices of Generalized Hanoi Graphs" (Mehiri, 16 Nov 2025)
• "M-Polynomial Based Mathematical Formulation of the Hyperbolic Sombor Index" (Barman et al., 16 Feb 2026)
• "M-Polynomial of Product Graphs" (Mehiri et al., 11 Mar 2026)
• "Finite biorthogonal M matrix polynomials" (Lekesiz, 6 Sep 2025)