Sombor Index in Graph Theory

• The Sombor index is a graph invariant defined by summing the Euclidean norms of vertex degree pairs over all edges.
• It is applied in mathematical chemistry and extremal graph theory to analyze structural centralization and branching in various graph families.
• Current research investigates extremal configurations, analytic inequalities, and generalizations, offering practical insights for QSPR studies in chemical informatics.

The Sombor index is a vertex-degree-based topological graph invariant defined for a simple graph by summing the Euclidean norms of the degree pairs at each edge. Introduced in 2021 by Ivan Gutman, the Sombor index rapidly attracted attention in both mathematical chemistry and extremal graph theory, due to its combination of geometric motivation and robust analytic properties. Research has since focused on its behavior across various graph families, extremal configurations, structural characterizations, inequalities, and generalizations.

1. Definition and Basic Properties

Let $G=(V,E)$ be a finite, simple, connected graph. For each vertex $u\in V$, let $d_G(u)$ denote its degree in $G$. The Sombor index of $G$ is defined as

$$\mathrm{SO}(G) = \sum_{uv\in E} \sqrt{d_G(u)^2 + d_G(v)^2}.$$

This definition is symmetric in $u$ and $v$ and depends solely on the degree sequence of $G$ (Du et al., 2022, Andriantiana et al., 2024, Redžić, 2022). For regular graphs of degree $r$, every term equals $u\in V$0, and so $u\in V$1.

A parameterized generalization, the $u\in V$2-Sombor index, is given by: $u\in V$3 with $u\in V$4 recovering the ordinary Sombor index (Wei et al., 2022).

Variants

• Reduced (or "decreasing") Sombor index: $u\in V$5
• Mean Sombor index: $u\in V$6 where $u\in V$7 (Ghods et al., 2021).

The Sombor index is always strictly positive, is monotonic with respect to adding edges, and is sensitive to the presence of high-degree vertices.

2. Extremal Values and Characterization Problems

Much of the literature focuses on identifying extremal graphs, especially in classes specified by order, degree sequence, maximum degree, matching number, cut-vertex count, or presence of cycles.

Extremal Trees and Degree Constraints

• Among all $u\in V$8-vertex trees, Gutman showed that the path $u\in V$9 attains $d_G(u)$0 and the star $d_G(u)$1 attains $d_G(u)$2 (Du et al., 2022, Andriantiana et al., 2024, Redžić, 2022, Wei et al., 2022).
• For a fixed tree degree sequence $d_G(u)$3, the unique minimizer is the greedy tree $d_G(u)$4 constructed by placing the largest available degree as close to the root as possible, while the unique maximizer is the alternating greedy tree $d_G(u)$5 obtained by alternating largest-to-largest and largest-to-smallest matchings between successive levels (Andriantiana et al., 2024, Redžić, 2022, Movahedi, 2022).
• If majorization $d_G(u)$6 holds between tree degree sequences, the maximal Sombor index increases: $d_G(u)$7 (Andriantiana et al., 2024).
• For fixed matching number $d_G(u)$8, the extremal tree is formed by a star with $d_G(u)$9 vertices and $G$0 leaves each extended by one extra pendant, giving a closed-form for $G$1 in terms of $G$2 and $G$3 (Zhou et al., 2021).

Extremal Two-Trees

For the class of two-trees (maximal series-parallel graphs), two canonical constructions govern extremality:

• $G$4: Obtained from $G$5 plus an edge between the high-degree vertices; two vertices have degree $G$6, the rest degree $G$7.
• $G$8: Formed by attaching a new vertex to one leaf and one high-degree vertex of $G$9.

The following extremal properties hold:

• $G$0 among two-trees: Achieved uniquely by $G$1 with $G$2
• Second maximum: Achieved by $G$3, with explicit formula (Du et al., 2022).
• The Sombor coindex (sum over non-edges) is minimal for $G$4, with matching second-minimum for $G$5.

Other Extremal Families

• For graphs with specified cut-vertex count $G$6, the unique $G$7 structure is the broom graph $G$8, a path with a star attached at one end (Hayat et al., 2022).
• For cacti (block graphs with at most one cycle sharing per vertex), concentrating as much degree as possible into a central vertex maximizes $G$9, particularly when all cycles are triangles (Liu, 2021).
• For trees or unicyclic graphs with fixed maximum degree, spider-like structures minimize $$\mathrm{SO}(G) = \sum_{uv\in E} \sqrt{d_G(u)^2 + d_G(v)^2}.$$0, with explicit leg-counts determined by $$\mathrm{SO}(G) = \sum_{uv\in E} \sqrt{d_G(u)^2 + d_G(v)^2}.$$1 and $$\mathrm{SO}(G) = \sum_{uv\in E} \sqrt{d_G(u)^2 + d_G(v)^2}.$$2 (Zhou et al., 2021).

3. Analytic and Combinatorial Principles

The determination of extremal configurations for $$\mathrm{SO}(G) = \sum_{uv\in E} \sqrt{d_G(u)^2 + d_G(v)^2}.$$3 relies on several analytic and combinatorial tools:

• Edge-swap (branch reallocation) and switching lemmas: Local modifications that strictly increase or decrease $$\mathrm{SO}(G) = \sum_{uv\in E} \sqrt{d_G(u)^2 + d_G(v)^2}.$$4 under structural misalignments (Andriantiana et al., 2024, Redžić, 2022, Movahedi, 2022).
• Monotonicity of $$\mathrm{SO}(G) = \sum_{uv\in E} \sqrt{d_G(u)^2 + d_G(v)^2}.$$5: This function is strictly increasing in each variable.
• Karamata convexity: Sequences that majorize others maximize convex functions such as $$\mathrm{SO}(G) = \sum_{uv\in E} \sqrt{d_G(u)^2 + d_G(v)^2}.$$6 (Zhou et al., 2021).
• Majorization: More "spread out" degree sequences increase maximal Sombor index.

Proofs frequently employ induction on graph order, often peeling off leaves or degree-2 vertices and tracking the corresponding change in $$\mathrm{SO}(G) = \sum_{uv\in E} \sqrt{d_G(u)^2 + d_G(v)^2}.$$7.

4. Structural and Chemical Interpretations

From a chemical graph theory perspective, the Sombor index acts as a molecular descriptor sensitive to distribution of branching, presence of articulation points, and overall molecular topology (Hayat et al., 2022, Alikhani et al., 2021).

• Attaching leaves or small substructures to high-degree vertices tends to increase $$\mathrm{SO}(G) = \sum_{uv\in E} \sqrt{d_G(u)^2 + d_G(v)^2}.$$8, reflecting structural centralization/branching.
• For graphs with prescribed degree sequence, $$\mathrm{SO}(G) = \sum_{uv\in E} \sqrt{d_G(u)^2 + d_G(v)^2}.$$9 is minimized when high-degree nodes are as distal from each other as allowed ("greedy" arrangement).
• In cacti and polymer graphs, maximal values arise from the concentration of cycles and branches at a single node, maximizing high-degree interactions (Liu, 2021, Alikhani et al., 2021).

Empirically, strong QSPR correlations have been found between $u$0 and thermochemical properties for various molecular families (Liu, 2022).

5. Inequalities and Closed-Formulas

The Sombor index admits numerous closed-formulas for canonical families and structural bounds:

Graph Family	Sombor Index Formula
Path $u$1	$u$2
Cycle $u$3	$u$4
Star $u$5	$u$6
Complete $u$7	$u$8
Complete bipartite $u$9	$v$0

Simple edge and vertex modifications yield sharp bounds:

• Edge deletion: $v$1 for $v$2 (Ghanbari et al., 2021).
• For a tree on $v$3 vertices, $v$4 (Liu, 2022).

Further, tight bounds relate $v$5 to other degree-based indices (Zagreb, ABC, AG/GA indices, "forgotten" index) as in (Mohammadi et al., 2024), e.g.,

$v$6

with $v$7, equality for regular graphs.

6. Extensions and Generalizations

Research extends the Sombor index in several directions:

• The general Sombor index with power parameter $v$8 allows analysis of more general edge contributions, and relevant extremal graph characterizations change depending on $v$9 (Wei et al., 2022, Hu et al., 2022).
• Sombor-like invariants (denoted $G$0) capture various algebraic and geometric variations of the basic construction, some matching up with classical indices for regular graphs (Ghanbari et al., 2022).
• The Sombor coindex, defined as the sum over non-edges, is meaningful in distinguishing among structurally similar graphs, especially in chordal and series-parallel classes (Du et al., 2022).
• Analytic bounds using variance, geometric mean, and AM-GM inequalities have been developed for estimation and approximation of $G$1 in large or statistical graph settings (Mohammadi et al., 2024).

7. Current Directions and Open Problems

Several themes continue to drive research:

• Extending extremal characterizations to classes beyond trees: unicyclic, bicyclic, cacti, $G$2-trees, and chordal graphs remain of interest, with conjectured analogues of known tree results (Du et al., 2022).
• Complete characterization of maximal Sombor index graphs for prescribed degree sequence, especially for cyclic or multicyclic cases (Wei et al., 2022, Movahedi, 2022).
• Investigation of variants and Sombor-like indices in chemical informatics, including regression studies for QSPR applications (Liu, 2022, Liu et al., 2022).
• Statistical analysis of $G$3 in random chain networks: as chain length grows, the Sombor index distribution approaches normality, enabling ensemble-average predictions (Liu et al., 2022).
• Further development of analytic inequalities involving higher-order moments, AG/GA indices, and SDD indices, with exploration of tightness regimes and extremal sharpness (Mohammadi et al., 2024).

The Sombor index sits at the intersection of geometry-inspired graph invariants, analytic combinatorics, and chemical graph theory, with rigorous structural characterization underpinning both its mathematical and applied significance (Du et al., 2022, Andriantiana et al., 2024, Redžić, 2022, Wei et al., 2022, Movahedi, 2022, Zhou et al., 2021, Alikhani et al., 2021).