Diminished Sombor Index (DSO)

• DSO is a normalized degree-based descriptor that weighs each edge by the ratio of the Euclidean norm of vertex degrees to their sum, distinguishing it from the ordinary Sombor index.
• Recent studies establish sharp bounds comparing DSO with classical indices, impacting extremal graph problems and QSPR analyses in chemical graph theory.
• DSO extends to spectral theory via the diminished Sombor matrix, enabling precise evaluations of spectral radius and energy, with applications to structured graphs.

The diminished Sombor index (DSO) is a degree-based topological index for a simple graph $G$, defined by

$$DSO(G)=\sum_{uv\in E(G)}\frac{\sqrt{d_u^2+d_v^2}}{d_u+d_v},$$

where $d_u$ and $d_v$ are the degrees of the endpoints of the edge $uv$. In the recent literature it is treated as a normalized Sombor-type descriptor, with developments in extremal graph theory, chemical graph theory, and spectral graph theory. The index has been studied through sharp inequalities with classical graph invariants, through extremal problems on molecular graphs and molecular trees, and through the diminished Sombor matrix, spectral radius, and energy; one paper also reports notable linear correlation with density, melting point, and critical volume for octane isomers under a Pearson-correlation screening criterion $|corr|>0.6$ (Movahedi, 3 Aug 2025, Movahedi, 3 Aug 2025, Alotaibi et al., 15 Sep 2025, Guo et al., 15 Dec 2025).

1. Definition and degree-pair formalism

The defining feature of DSO is that each edge contributes a normalized Euclidean degree term,

$\frac{\sqrt{d_u^2+d_v^2}}{d_u+d_v}.$

This places DSO in the Sombor family while distinguishing it from the ordinary Sombor index

$SO(G)=\sum_{uv\in E(G)}\sqrt{d_u^2+d_v^2},$

which omits the denominator and therefore weights high-degree edges more directly (Guo et al., 15 Dec 2025, Movahedi, 3 Aug 2025).

For extremal analysis, the literature frequently rewrites DSO in terms of degree pairs. One paper introduces

$f(i,j)=\frac{\sqrt{i^2+j^2}}{i+j}$

and denotes by $m_{i,j}(G)$ the number of edges whose endpoints have degrees $$DSO(G)=\sum_{uv\in E(G)}\frac{\sqrt{d_u^2+d_v^2}}{d_u+d_v},$$0 and $$DSO(G)=\sum_{uv\in E(G)}\frac{\sqrt{d_u^2+d_v^2}}{d_u+d_v},$$1. In that notation, DSO becomes an edge-type sum over the degree-pair profile of the graph. This degree-pair decomposition is central in recent proofs for molecular graphs, because once the admissible degrees are restricted to the molecular range $$DSO(G)=\sum_{uv\in E(G)}\frac{\sqrt{d_u^2+d_v^2}}{d_u+d_v},$$2, comparisons among the values $$DSO(G)=\sum_{uv\in E(G)}\frac{\sqrt{d_u^2+d_v^2}}{d_u+d_v},$$3 can be organized explicitly (Alotaibi et al., 15 Sep 2025).

The same local-weight viewpoint underlies the diminished Sombor matrix. For a graph $$DSO(G)=\sum_{uv\in E(G)}\frac{\sqrt{d_u^2+d_v^2}}{d_u+d_v},$$4 with vertex set $$DSO(G)=\sum_{uv\in E(G)}\frac{\sqrt{d_u^2+d_v^2}}{d_u+d_v},$$5, the diminished Sombor matrix $$DSO(G)=\sum_{uv\in E(G)}\frac{\sqrt{d_u^2+d_v^2}}{d_u+d_v},$$6 is defined by

$$DSO(G)=\sum_{uv\in E(G)}\frac{\sqrt{d_u^2+d_v^2}}{d_u+d_v},$$7

Thus $$DSO(G)=\sum_{uv\in E(G)}\frac{\sqrt{d_u^2+d_v^2}}{d_u+d_v},$$8 is a weighted adjacency matrix whose nonzero off-diagonal entries are precisely the DSO edge weights (Movahedi, 3 Aug 2025).

2. Inequalities and relationships with classical indices

A substantial part of the 2025 DSO literature establishes sharp bounds in terms of classical topological indices. The most systematic treatment gives inequalities involving the Zagreb, Albertson, Harmonic, Randić, geometric-arithmetic, symmetric division degree, forgotten, and related indices (Movahedi, 3 Aug 2025).

One basic comparison is with the Albertson index $$DSO(G)=\sum_{uv\in E(G)}\frac{\sqrt{d_u^2+d_v^2}}{d_u+d_v},$$9 and the geometric-arithmetic index $d_u$0: $d_u$1 where $d_u$2 and $d_u$3 are the maximum and minimum degrees. The lower-bound equality holds iff $d_u$4, while the upper-bound equality holds iff $d_u$5 consists of components, each of which is a regular graph. A related upper bound is

$d_u$6

and for a connected graph this yields

$d_u$7

with equality iff $d_u$8 is regular (Movahedi, 3 Aug 2025).

Further sharp estimates connect DSO to harmonic and Zagreb data: $d_u$9 and, using only the size $d_v$0,

$d_v$1

Lower bounds are also obtained through $d_v$2,

$d_v$3

with

$d_v$4

and through a mixed Zagreb–inverse-sum-indeg–harmonic expression,

$d_v$5

In each of these formulations, equality is characterized by regularity or by unions of regular components, except for the harmonic-type upper bound, where equality requires $d_v$6 to be constant over all edges (Movahedi, 3 Aug 2025).

The index is also sharply compared with the ordinary Sombor index: $d_v$7 with equality iff $d_v$8 is regular. Analogous two-sided inequalities are proved for

$d_v$9

for

$uv$0

for

$uv$1

and for the forgotten and multiplicative forgotten indices,

$uv$2

A recurring structural pattern is that most sharp equality cases are attained by regular graphs or unions of regular components (Movahedi, 3 Aug 2025).

3. Minimum DSO on fixed-order molecular graphs with cyclomatic number at least 3

A connected graph of maximum degree at most $uv$3 is called a molecular graph. In that setting, the cyclomatic number is the smallest number of edges whose removal makes the graph acyclic. The fixed-order minimization problem for DSO on molecular graphs is solved for all orders $uv$4 and cyclomatic numbers $uv$5 satisfying

$uv$6

in “On the Diminished Sombor Index of Fixed-Order Molecular Graphs With Cyclomatic Number at Least 3” (Alotaibi et al., 15 Sep 2025).

The primary motivation is a conjecture from a paper of Movahedi, Gutman, Redžepović, and Furtula concerning connected graphs with cyclomatic number $uv$7. That conjecture proposed as minimizers graphs “obtained by connecting two disjoint cycles by two edges, so that a quadrangle is formed.” The molecular-graph result confirms the minimization phenomenon but shows that the exact extremal structure is slightly different from that conjectured shape (Alotaibi et al., 15 Sep 2025).

The main theorem gives a complete characterization of all minimizers. Using the stated edge-type counts together with $uv$8, the minimum value is

$uv$9

Equality holds in exactly two situations. If $|corr|>0.6$0, equality holds iff $|corr|>0.6$1 is $|corr|>0.6$2-regular. If $|corr|>0.6$3, equality holds iff

$|corr|>0.6$4

and

$|corr|>0.6$5

Accordingly, the extremal graphs are either cubic graphs or graphs with only degrees $|corr|>0.6$6 and $|corr|>0.6$7, with exactly two $|corr|>0.6$8-$|corr|>0.6$9 edges, exactly $\frac{\sqrt{d_u^2+d_v^2}}{d_u+d_v}.$0 edges of type $\frac{\sqrt{d_u^2+d_v^2}}{d_u+d_v}.$1-$\frac{\sqrt{d_u^2+d_v^2}}{d_u+d_v}.$2, and exactly $\frac{\sqrt{d_u^2+d_v^2}}{d_u+d_v}.$3 edges of type $\frac{\sqrt{d_u^2+d_v^2}}{d_u+d_v}.$4-$\frac{\sqrt{d_u^2+d_v^2}}{d_u+d_v}.$5. The final minimizers have no vertices of degree $\frac{\sqrt{d_u^2+d_v^2}}{d_u+d_v}.$6 (Alotaibi et al., 15 Sep 2025).

The proof has two central ingredients. First, a structural lemma shows that a minimizing graph cannot contain a pendent edge incident with a degree-$\frac{\sqrt{d_u^2+d_v^2}}{d_u+d_v}.$7 vertex. This is proved by local edge rearrangement: if such a configuration exists, an edge is moved from a branching region toward the pendant vertex, and the DSO value strictly decreases, contradicting extremality. Second, once that configuration is excluded, degree-counting identities and inequalities among the values $\frac{\sqrt{d_u^2+d_v^2}}{d_u+d_v}.$8 for $\frac{\sqrt{d_u^2+d_v^2}}{d_u+d_v}.$9 are used to derive the lower bound and to characterize the equality cases. The paper further employs a table of auxiliary positive coefficients $SO(G)=\sum_{uv\in E(G)}\sqrt{d_u^2+d_v^2},$0 to show that any deviation from the candidate extremal structure strictly increases DSO (Alotaibi et al., 15 Sep 2025).

In chemical-graph-theoretic terms, the restriction $SO(G)=\sum_{uv\in E(G)}\sqrt{d_u^2+d_v^2},$1 reflects valence constraints of atoms in many organic molecules. The result is therefore both extremal and chemically meaningful within the molecular class (Alotaibi et al., 15 Sep 2025).

4. Maximum DSO on molecular trees with a perfect matching

A complementary extremal direction is studied for molecular trees with a perfect matching. Here a molecular tree is a tree whose maximum degree is at most $SO(G)=\sum_{uv\in E(G)}\sqrt{d_u^2+d_v^2},$2. For molecular trees of order $SO(G)=\sum_{uv\in E(G)}\sqrt{d_u^2+d_v^2},$3 with a perfect matching, the maximum DSO is determined exactly, and the extremal trees are fully characterized (Guo et al., 15 Dec 2025).

The paper begins from a chemical-motivated perspective. It treats DSO as a chemical descriptor and tests its applicability to octane isomers using Pearson correlation coefficients with experimental physicochemical properties. The reported correlations retained under $SO(G)=\sum_{uv\in E(G)}\sqrt{d_u^2+d_v^2},$4 include at least density, melting point, and critical volume (Guo et al., 15 Dec 2025).

The extremal analysis is based on three structural lemmas. If $SO(G)=\sum_{uv\in E(G)}\sqrt{d_u^2+d_v^2},$5 is a molecular tree with maximum DSO among trees of order $SO(G)=\sum_{uv\in E(G)}\sqrt{d_u^2+d_v^2},$6 with a perfect matching $SO(G)=\sum_{uv\in E(G)}\sqrt{d_u^2+d_v^2},$7, then every matched edge is pendant: $SO(G)=\sum_{uv\in E(G)}\sqrt{d_u^2+d_v^2},$8 A second lemma states that if $SO(G)=\sum_{uv\in E(G)}\sqrt{d_u^2+d_v^2},$9, then every neighbor $f(i,j)=\frac{\sqrt{i^2+j^2}}{i+j}$0 has degree in $f(i,j)=\frac{\sqrt{i^2+j^2}}{i+j}$1. Thus a degree-$f(i,j)=\frac{\sqrt{i^2+j^2}}{i+j}$2 vertex in an extremal tree is not adjacent to degree-$f(i,j)=\frac{\sqrt{i^2+j^2}}{i+j}$3 or degree-$f(i,j)=\frac{\sqrt{i^2+j^2}}{i+j}$4 vertices. A third lemma gives

$f(i,j)=\frac{\sqrt{i^2+j^2}}{i+j}$5

All three lemmas are proved by local rewiring arguments that strictly increase DSO when a forbidden configuration is present (Guo et al., 15 Dec 2025).

These constraints motivate three extremal families. Starting from $f(i,j)=\frac{\sqrt{i^2+j^2}}{i+j}$6, the family of trees in which all vertices have degree $f(i,j)=\frac{\sqrt{i^2+j^2}}{i+j}$7 or $f(i,j)=\frac{\sqrt{i^2+j^2}}{i+j}$8, the paper defines $f(i,j)=\frac{\sqrt{i^2+j^2}}{i+j}$9 by inserting exactly one vertex into every $m_{i,j}(G)$0-$m_{i,j}(G)$1 edge, into all but one such edge, or into all but two such edges, respectively. For each $m_{i,j}(G)$2, $m_{i,j}(G)$3 is then obtained from a tree in $m_{i,j}(G)$4 by adding one new leaf to every existing vertex. These families are exactly the extremal structures for the three congruence classes of $m_{i,j}(G)$5 modulo $m_{i,j}(G)$6: equality in the corresponding piecewise linear upper bound holds precisely for $m_{i,j}(G)$7, $m_{i,j}(G)$8, or $m_{i,j}(G)$9 according as $$DSO(G)=\sum_{uv\in E(G)}\frac{\sqrt{d_u^2+d_v^2}}{d_u+d_v},$$00 (Guo et al., 15 Dec 2025).

The proof then passes to degree counts. If $$DSO(G)=\sum_{uv\in E(G)}\frac{\sqrt{d_u^2+d_v^2}}{d_u+d_v},$$01 denotes the number of vertices of degree $$DSO(G)=\sum_{uv\in E(G)}\frac{\sqrt{d_u^2+d_v^2}}{d_u+d_v},$$02, then

$$DSO(G)=\sum_{uv\in E(G)}\frac{\sqrt{d_u^2+d_v^2}}{d_u+d_v},$$03

and the pendant-edge lemma forces $$DSO(G)=\sum_{uv\in E(G)}\frac{\sqrt{d_u^2+d_v^2}}{d_u+d_v},$$04. Combined with the other structural restrictions, this yields

$$DSO(G)=\sum_{uv\in E(G)}\frac{\sqrt{d_u^2+d_v^2}}{d_u+d_v},$$05

The explicit DSO values are then computed from the edge-type constants

$$DSO(G)=\sum_{uv\in E(G)}\frac{\sqrt{d_u^2+d_v^2}}{d_u+d_v},$$06

where $$DSO(G)=\sum_{uv\in E(G)}\frac{\sqrt{d_u^2+d_v^2}}{d_u+d_v},$$07 (Guo et al., 15 Dec 2025).

5. Diminished Sombor matrix, spectral radius, and energy

The diminished Sombor matrix extends DSO from a scalar invariant to a weighted spectral object. Its eigenvalues are real, are denoted

$$DSO(G)=\sum_{uv\in E(G)}\frac{\sqrt{d_u^2+d_v^2}}{d_u+d_v},$$08

and satisfy

$$DSO(G)=\sum_{uv\in E(G)}\frac{\sqrt{d_u^2+d_v^2}}{d_u+d_v},$$09

because the diagonal of $$DSO(G)=\sum_{uv\in E(G)}\frac{\sqrt{d_u^2+d_v^2}}{d_u+d_v},$$10 is zero. The characteristic polynomial is

$$DSO(G)=\sum_{uv\in E(G)}\frac{\sqrt{d_u^2+d_v^2}}{d_u+d_v},$$11

the largest eigenvalue $$DSO(G)=\sum_{uv\in E(G)}\frac{\sqrt{d_u^2+d_v^2}}{d_u+d_v},$$12 is the diminished Sombor spectral radius, and the diminished Sombor energy is

$$DSO(G)=\sum_{uv\in E(G)}\frac{\sqrt{d_u^2+d_v^2}}{d_u+d_v},$$13

Trace identities include

$$DSO(G)=\sum_{uv\in E(G)}\frac{\sqrt{d_u^2+d_v^2}}{d_u+d_v},$$14

and

$$DSO(G)=\sum_{uv\in E(G)}\frac{\sqrt{d_u^2+d_v^2}}{d_u+d_v},$$15

Thus the second trace is twice the sum of squared DSO edge weights (Movahedi, 3 Aug 2025).

For connected $$DSO(G)=\sum_{uv\in E(G)}\frac{\sqrt{d_u^2+d_v^2}}{d_u+d_v},$$16-regular graphs, every edge has weight

$$DSO(G)=\sum_{uv\in E(G)}\frac{\sqrt{d_u^2+d_v^2}}{d_u+d_v},$$17

so

$$DSO(G)=\sum_{uv\in E(G)}\frac{\sqrt{d_u^2+d_v^2}}{d_u+d_v},$$18

Accordingly, the diminished Sombor spectrum of a regular graph is exactly $$DSO(G)=\sum_{uv\in E(G)}\frac{\sqrt{d_u^2+d_v^2}}{d_u+d_v},$$19 times its adjacency spectrum (Movahedi, 3 Aug 2025).

Several standard families admit explicit spectra. For the complete graph,

$$DSO(G)=\sum_{uv\in E(G)}\frac{\sqrt{d_u^2+d_v^2}}{d_u+d_v},$$20

For the cycle,

$$DSO(G)=\sum_{uv\in E(G)}\frac{\sqrt{d_u^2+d_v^2}}{d_u+d_v},$$21

For the complete bipartite graph,

$$DSO(G)=\sum_{uv\in E(G)}\frac{\sqrt{d_u^2+d_v^2}}{d_u+d_v},$$22

and for the star $$DSO(G)=\sum_{uv\in E(G)}\frac{\sqrt{d_u^2+d_v^2}}{d_u+d_v},$$23,

$$DSO(G)=\sum_{uv\in E(G)}\frac{\sqrt{d_u^2+d_v^2}}{d_u+d_v},$$24

The paper also proves that if $$DSO(G)=\sum_{uv\in E(G)}\frac{\sqrt{d_u^2+d_v^2}}{d_u+d_v},$$25 has $$DSO(G)=\sum_{uv\in E(G)}\frac{\sqrt{d_u^2+d_v^2}}{d_u+d_v},$$26 distinct eigenvalues, then $$DSO(G)=\sum_{uv\in E(G)}\frac{\sqrt{d_u^2+d_v^2}}{d_u+d_v},$$27; that $$DSO(G)=\sum_{uv\in E(G)}\frac{\sqrt{d_u^2+d_v^2}}{d_u+d_v},$$28 has exactly two distinct diminished Sombor eigenvalues iff $$DSO(G)=\sum_{uv\in E(G)}\frac{\sqrt{d_u^2+d_v^2}}{d_u+d_v},$$29; and that

$$DSO(G)=\sum_{uv\in E(G)}\frac{\sqrt{d_u^2+d_v^2}}{d_u+d_v},$$30

(Movahedi, 3 Aug 2025).

The spectral radius is bounded sharply by DSO and by the degree-based quantity

$$DSO(G)=\sum_{uv\in E(G)}\frac{\sqrt{d_u^2+d_v^2}}{d_u+d_v},$$31

$$DSO(G)=\sum_{uv\in E(G)}\frac{\sqrt{d_u^2+d_v^2}}{d_u+d_v},$$32

The left equality holds iff $$DSO(G)=\sum_{uv\in E(G)}\frac{\sqrt{d_u^2+d_v^2}}{d_u+d_v},$$33 is regular, and the right equality holds iff $$DSO(G)=\sum_{uv\in E(G)}\frac{\sqrt{d_u^2+d_v^2}}{d_u+d_v},$$34. There are also comparison bounds with the adjacency spectral radius $$DSO(G)=\sum_{uv\in E(G)}\frac{\sqrt{d_u^2+d_v^2}}{d_u+d_v},$$35,

$$DSO(G)=\sum_{uv\in E(G)}\frac{\sqrt{d_u^2+d_v^2}}{d_u+d_v},$$36

again with equality iff $$DSO(G)=\sum_{uv\in E(G)}\frac{\sqrt{d_u^2+d_v^2}}{d_u+d_v},$$37 is regular (Movahedi, 3 Aug 2025).

For the diminished Sombor energy, the paper proves

$$DSO(G)=\sum_{uv\in E(G)}\frac{\sqrt{d_u^2+d_v^2}}{d_u+d_v},$$38

as well as

$$DSO(G)=\sum_{uv\in E(G)}\frac{\sqrt{d_u^2+d_v^2}}{d_u+d_v},$$39

For connected graphs, the lower equality in the latter bound holds iff $$DSO(G)=\sum_{uv\in E(G)}\frac{\sqrt{d_u^2+d_v^2}}{d_u+d_v},$$40 is complete bipartite; the upper equality holds for a graph with no edges or with all vertices of degree one. The paper also derives a closed form for complete bipartite graphs,

$$DSO(G)=\sum_{uv\in E(G)}\frac{\sqrt{d_u^2+d_v^2}}{d_u+d_v},$$41

and shows that among $$DSO(G)=\sum_{uv\in E(G)}\frac{\sqrt{d_u^2+d_v^2}}{d_u+d_v},$$42 with $$DSO(G)=\sum_{uv\in E(G)}\frac{\sqrt{d_u^2+d_v^2}}{d_u+d_v},$$43, this quantity is minimized at the star and maximized at the balanced complete bipartite graph (Movahedi, 3 Aug 2025).

6. Variants, terminology, and neighboring Sombor constructions

Recent work places DSO inside a broader family of normalized Sombor-type indices. The hyperbolic Sombor index is defined by

$$DSO(G)=\sum_{uv\in E(G)}\frac{\sqrt{d_u^2+d_v^2}}{d_u+d_v},$$44

while the complementary diminished Sombor index (CDSO) is

$$DSO(G)=\sum_{uv\in E(G)}\frac{\sqrt{d_u^2+d_v^2}}{d_u+d_v},$$45

In this framework, HSO arises from DSO by replacing the denominator $$DSO(G)=\sum_{uv\in E(G)}\frac{\sqrt{d_u^2+d_v^2}}{d_u+d_v},$$46 with $$DSO(G)=\sum_{uv\in E(G)}\frac{\sqrt{d_u^2+d_v^2}}{d_u+d_v},$$47, and CDSO replaces it with $$DSO(G)=\sum_{uv\in E(G)}\frac{\sqrt{d_u^2+d_v^2}}{d_u+d_v},$$48. The basic edgewise comparison

$$DSO(G)=\sum_{uv\in E(G)}\frac{\sqrt{d_u^2+d_v^2}}{d_u+d_v},$$49

holds with equality iff $$DSO(G)=\sum_{uv\in E(G)}\frac{\sqrt{d_u^2+d_v^2}}{d_u+d_v},$$50 is regular. The same paper emphasizes that neither HSO nor CDSO is monotone under edge addition in general, and it provides explicit sufficient conditions for both increase and decrease under the operation $$DSO(G)=\sum_{uv\in E(G)}\frac{\sqrt{d_u^2+d_v^2}}{d_u+d_v},$$51 (Albalahi et al., 28 Oct 2025).

A persistent source of confusion is terminological rather than mathematical. Several papers in the broader Sombor literature use names that are close to “diminished Sombor index” for different formulas. One 2021 paper on silicon-carbide graphs uses the term “decreasing Sombor index” and the notation $$DSO(G)=\sum_{uv\in E(G)}\frac{\sqrt{d_u^2+d_v^2}}{d_u+d_v},$$52 for

$$DSO(G)=\sum_{uv\in E(G)}\frac{\sqrt{d_u^2+d_v^2}}{d_u+d_v},$$53

and explicitly notes that it does not define the present DSO separately (Ghods et al., 2021). Another paper on the general $$DSO(G)=\sum_{uv\in E(G)}\frac{\sqrt{d_u^2+d_v^2}}{d_u+d_v},$$54-KA index treats

$$DSO(G)=\sum_{uv\in E(G)}\frac{\sqrt{d_u^2+d_v^2}}{d_u+d_v},$$55

as a “modified Sombor index,” corresponding to $$DSO(G)=\sum_{uv\in E(G)}\frac{\sqrt{d_u^2+d_v^2}}{d_u+d_v},$$56, and again does not use the 2025 DSO definition (Hernández et al., 2021). Likewise, work on extremal cacti with respect to the Sombor index studies the ordinary Sombor index and the reduced Sombor index but does not define DSO (Liu, 2021). The contemporary DSO literature therefore uses a specific normalized edge weight $$DSO(G)=\sum_{uv\in E(G)}\frac{\sqrt{d_u^2+d_v^2}}{d_u+d_v},$$57, and this should not be conflated with the reduced/decreasing or reciprocal/modified Sombor variants (Movahedi, 3 Aug 2025).

Within chemical graph theory, the DSO literature is tied especially to molecular graphs, where the bound $$DSO(G)=\sum_{uv\in E(G)}\frac{\sqrt{d_u^2+d_v^2}}{d_u+d_v},$$58 encodes valence restrictions, and to QSPR/QSAR-style reasoning through degree-based descriptors. Within pure graph theory, the dominant themes are now clear: sharp inequalities against classical indices, extremal graph characterization under structural constraints such as cyclomatic number or perfect matching, and matrix-based spectral theory built directly from the DSO edge weights (Alotaibi et al., 15 Sep 2025, Guo et al., 15 Dec 2025, Movahedi, 3 Aug 2025).