Diminished Sombor Matrix: Spectral Insights

• The diminished Sombor matrix is defined as a normalized degree-weighted analogue of the adjacency matrix, using the weight (sqrt(d²_u + d²_v))/(d_u + d_v) to emphasize relative degree profiles.
• It enables precise spectral analysis by providing bounded edge weights that yield exact spectra for standard graphs and clear trace identities linked to graph invariants.
• The matrix framework effectively sharpens spectral-radius and energy bounds, facilitating structural characterizations and extremal results in spectral graph theory.

The diminished Sombor matrix $M_{DS}(G)$ is a degree-weighted analogue of the adjacency matrix of a simple graph $G$, obtained by assigning to each edge $uv$ the normalized Sombor weight $\frac{\sqrt{d_u^2+d_v^2}}{d_u+d_v}$. It is the matrix counterpart of the diminished Sombor index and differs from the ordinary Sombor matrix by dividing the Euclidean degree term by the degree sum, a normalization that suppresses raw degree growth and emphasizes the relative degree profile of adjacent vertices (Movahedi, 3 Aug 2025).

1. Definition and normalization

For a simple graph $G$ with vertex set $V(G)=\{v_1,\dots,v_n\}$, degrees $d_i=d(v_i)$, and edge set $E(G)$, the diminished Sombor matrix is

$M_{DS}(G)=\big((M_{DS}(G))_{ij}\big)_{n\times n},$

where

$(M_{DS}(G))_{ij}= \begin{cases} \dfrac{\sqrt{d_i^2+d_j^2}}{d_i+d_j}, & v_iv_j\in E(G),\[6pt] 0, & \text{otherwise.} \end{cases}$

Thus $G$0 has the same zero–nonzero pattern as the ordinary adjacency matrix $G$1, but replaces each nonzero adjacency entry by the edge weight

$G$2

The associated scalar invariant is the diminished Sombor index

$G$3

so $G$4 is the matrix version of $G$5 in the same sense that the Sombor matrix is the matrix version of the ordinary Sombor index (Movahedi, 3 Aug 2025).

The normalization is structurally significant. The scalar-index theory shows that for positive degrees on an edge,

$G$6

with the lower extreme achieved when $G$7. Consequently, the nonzero entries of $G$8 lie in a narrow bounded interval, rather than growing unboundedly with degree as in the ordinary Sombor weight $G$9 (Movahedi, 3 Aug 2025).

2. Spectral invariants and trace identities

Because the edge weight is symmetric in the endpoints, $uv$0 is real symmetric. Its eigenvalues are therefore real, and may be written in nonincreasing order as

$uv$1

The largest eigenvalue $uv$2 is called the diminished Sombor spectral radius, and the diminished Sombor energy is defined by

$uv$3

Since the diagonal entries are zero,

$uv$4

The second trace identity is

$uv$5

equivalently,

$uv$6

A useful reformulation involves the Gutman–Milovanović index

$uv$7

In particular,

$uv$8

This identity is repeatedly used to compress spectral-radius and energy bounds into a form depending on $uv$9 and $\frac{\sqrt{d_u^2+d_v^2}}{d_u+d_v}$0. The paper also compares this quantity with the geometric-arithmetic index

$\frac{\sqrt{d_u^2+d_v^2}}{d_u+d_v}$1

which enters one of the later spectral-radius corollaries (Movahedi, 3 Aug 2025).

3. Exact spectra for standard graph families

A central structural fact is that regular graphs collapse the diminished Sombor matrix to a scalar multiple of the adjacency matrix. If $\frac{\sqrt{d_u^2+d_v^2}}{d_u+d_v}$2 is connected and $\frac{\sqrt{d_u^2+d_v^2}}{d_u+d_v}$3-regular, then every edge has weight

$\frac{\sqrt{d_u^2+d_v^2}}{d_u+d_v}$4

so

$\frac{\sqrt{d_u^2+d_v^2}}{d_u+d_v}$5

Hence the spectrum of $\frac{\sqrt{d_u^2+d_v^2}}{d_u+d_v}$6 is exactly $\frac{\sqrt{d_u^2+d_v^2}}{d_u+d_v}$7 times the adjacency spectrum (Movahedi, 3 Aug 2025).

For the complete graph $\frac{\sqrt{d_u^2+d_v^2}}{d_u+d_v}$8,

$\frac{\sqrt{d_u^2+d_v^2}}{d_u+d_v}$9

Therefore

$G$0

For the cycle $G$1,

$G$2

For $G$3, this yields eigenvalues $G$4.

For the complete bipartite graph $G$5, all nonzero entries have common weight

$G$6

and the spectrum is

$G$7

Hence

$G$8

The star $G$9 is the special case

$V(G)=\{v_1,\dots,v_n\}$0

so

$V(G)=\{v_1,\dots,v_n\}$1

For the path $V(G)=\{v_1,\dots,v_n\}$2, the matrix is tridiagonal with endpoint weight

$V(G)=\{v_1,\dots,v_n\}$3

and internal weight

$V(G)=\{v_1,\dots,v_n\}$4

The paper does not diagonalize $V(G)=\{v_1,\dots,v_n\}$5 explicitly, but gives the characteristic polynomial. For $V(G)=\{v_1,\dots,v_n\}$6,

$V(G)=\{v_1,\dots,v_n\}$7

where

$V(G)=\{v_1,\dots,v_n\}$8

For $V(G)=\{v_1,\dots,v_n\}$9, this gives

$d_i=d(v_i)$0

4. Structural characterizations

The diminished Sombor spectrum admits several rigidity results. If all diminished Sombor eigenvalues have the same modulus,

$d_i=d(v_i)$1

then and only then

$d_i=d(v_i)$2

The first graph has all eigenvalues $d_i=d(v_i)$3, while the second decomposes into $d_i=d(v_i)$4 blocks with eigenvalues $d_i=d(v_i)$5 and $d_i=d(v_i)$6 (Movahedi, 3 Aug 2025).

There is also a complete classification of connected graphs with only two distinct diminished Sombor eigenvalues: if $d_i=d(v_i)$7 is connected of order $d_i=d(v_i)$8, then $d_i=d(v_i)$9 has exactly two distinct eigenvalues if and only if

$E(G)$0

The converse direction uses an induced $E(G)$1, whose corresponding principal submatrix has three distinct eigenvalues, together with interlacing.

A further structural theorem links graph distance to spectral complexity. If $E(G)$2 is connected and $E(G)$3 has $E(G)$4 distinct eigenvalues, then

$E(G)$5

The proof uses a Perron–Frobenius polynomial argument: a suitable polynomial in $E(G)$6 becomes a positive rank-one matrix, forcing some power $E(G)$7 with $E(G)$8 to have strictly positive $E(G)$9-entry for every pair $M_{DS}(G)=\big((M_{DS}(G))_{ij}\big)_{n\times n},$0. This yields a walk, and hence a path, of length at most $M_{DS}(G)=\big((M_{DS}(G))_{ij}\big)_{n\times n},$1 between every two vertices (Movahedi, 3 Aug 2025).

5. Spectral-radius bounds and complement inequalities

The principal spectral-radius theorem states that if $M_{DS}(G)=\big((M_{DS}(G))_{ij}\big)_{n\times n},$2 is connected with $M_{DS}(G)=\big((M_{DS}(G))_{ij}\big)_{n\times n},$3 vertices and $M_{DS}(G)=\big((M_{DS}(G))_{ij}\big)_{n\times n},$4 edges, then

$M_{DS}(G)=\big((M_{DS}(G))_{ij}\big)_{n\times n},$5

Equality in the lower bound holds if and only if $M_{DS}(G)=\big((M_{DS}(G))_{ij}\big)_{n\times n},$6 is regular, and equality in the upper bound holds if and only if $M_{DS}(G)=\big((M_{DS}(G))_{ij}\big)_{n\times n},$7. The lower bound comes from the Rayleigh quotient with the all-ones vector,

$M_{DS}(G)=\big((M_{DS}(G))_{ij}\big)_{n\times n},$8

while the upper bound uses $M_{DS}(G)=\big((M_{DS}(G))_{ij}\big)_{n\times n},$9, Cauchy–Schwarz, and the trace identity (Movahedi, 3 Aug 2025).

A corollary gives

$(M_{DS}(G))_{ij}= \begin{cases} \dfrac{\sqrt{d_i^2+d_j^2}}{d_i+d_j}, & v_iv_j\in E(G),\[6pt] 0, & \text{otherwise.} \end{cases}$0

The inequality follows from

$(M_{DS}(G))_{ij}= \begin{cases} \dfrac{\sqrt{d_i^2+d_j^2}}{d_i+d_j}, & v_iv_j\in E(G),\[6pt] 0, & \text{otherwise.} \end{cases}$1

However, the printed equality condition attached to this corollary is inconsistent with the connectedness assumption and with positivity of the spectral radius for a connected nontrivial graph; the inequality itself is meaningful, but the equality statement is not correctly stated.

The diminished Sombor spectral radius is also compared directly with the adjacency spectral radius $(M_{DS}(G))_{ij}= \begin{cases} \dfrac{\sqrt{d_i^2+d_j^2}}{d_i+d_j}, & v_iv_j\in E(G),\[6pt] 0, & \text{otherwise.} \end{cases}$2: $(M_{DS}(G))_{ij}= \begin{cases} \dfrac{\sqrt{d_i^2+d_j^2}}{d_i+d_j}, & v_iv_j\in E(G),\[6pt] 0, & \text{otherwise.} \end{cases}$3 with equality if and only if $(M_{DS}(G))_{ij}= \begin{cases} \dfrac{\sqrt{d_i^2+d_j^2}}{d_i+d_j}, & v_iv_j\in E(G),\[6pt] 0, & \text{otherwise.} \end{cases}$4 is regular. This follows from the entrywise comparison

$(M_{DS}(G))_{ij}= \begin{cases} \dfrac{\sqrt{d_i^2+d_j^2}}{d_i+d_j}, & v_iv_j\in E(G),\[6pt] 0, & \text{otherwise.} \end{cases}$5

on each edge.

The paper also derives Nordhaus–Gaddum-type bounds for the sum of the diminished Sombor spectral radii of a graph and its complement. For a graph of order $(M_{DS}(G))_{ij}= \begin{cases} \dfrac{\sqrt{d_i^2+d_j^2}}{d_i+d_j}, & v_iv_j\in E(G),\[6pt] 0, & \text{otherwise.} \end{cases}$6,

$(M_{DS}(G))_{ij}= \begin{cases} \dfrac{\sqrt{d_i^2+d_j^2}}{d_i+d_j}, & v_iv_j\in E(G),\[6pt] 0, & \text{otherwise.} \end{cases}$7

with equality if and only if $(M_{DS}(G))_{ij}= \begin{cases} \dfrac{\sqrt{d_i^2+d_j^2}}{d_i+d_j}, & v_iv_j\in E(G),\[6pt] 0, & \text{otherwise.} \end{cases}$8. For connected $(M_{DS}(G))_{ij}= \begin{cases} \dfrac{\sqrt{d_i^2+d_j^2}}{d_i+d_j}, & v_iv_j\in E(G),\[6pt] 0, & \text{otherwise.} \end{cases}$9 of order $G$00 and size $G$01,

$G$02

with equality if and only if $G$03 is complete (Movahedi, 3 Aug 2025).

6. Energy theory and extremal graphs

The diminished Sombor energy

$G$04

satisfies a basic pair of bounds: $G$05 The upper inequality is an immediate application of Cauchy–Schwarz to the non-Perron eigenvalues.

The central energy theorem is trace-based: $G$06 Equality on the right holds if and only if

$G$07

For connected $G$08, equality on the left holds if and only if $G$09 is complete bipartite (Movahedi, 3 Aug 2025).

Using

$G$10

the same estimate becomes

$G$11

This formulation is especially convenient in extremal problems.

Complete bipartite graphs are the equality cases for the connected lower bound, and their energy is explicit: $G$12 For fixed order $G$13, this yields

$G$14

with equality on the left if and only if $G$15, and on the right if and only if

$G$16

Thus, among complete bipartite graphs of fixed order, the star has the smallest diminished Sombor energy and the balanced complete bipartite graph has the largest.

7. Broader context, proof methods, and textual issues

The spectral theory of the diminished Sombor matrix is built with standard tools from spectral graph theory: Rayleigh quotient estimates, Perron–Frobenius theory for nonnegative irreducible matrices, trace identities, repeated use of Cauchy–Schwarz, interlacing, a closed-walk divisibility lemma in the bipartite equality case, and degree-ratio inequalities based on $G$17 and $G$18. The same source also contains several typographical inconsistencies: malformed defining formulas, repeated theorem numbering, an incorrect equality statement in the corollary involving $G$19, and later bounds whose intended forms are recoverable from the derivations (Movahedi, 3 Aug 2025).

The matrix construction is rooted in the scalar diminished Sombor index literature. The paper on the diminished Sombor index does not explicitly define a matrix, but it establishes the same edge weight

$G$20

as a normalized degree-based descriptor and develops its relationships with indices such as $G$21, $G$22, $G$23, $G$24, $G$25, $G$26, and $G$27 (Movahedi, 3 Aug 2025). A broader matrix-theoretic background is provided by the earlier $G$28-Sombor matrix framework, which studies weighted adjacency matrices with edge weights $G$29 and develops moment formulas, Laplacian theory, spectral spread, energy, Estrada index, and Nordhaus–Gaddum results (Liu et al., 2021). This suggests that the diminished Sombor matrix may be viewed as a normalized Sombor-type weighted adjacency operator whose spectral theory inherits the general architecture of Sombor-matrix methods while remaining tightly linked to the bounded edge-weight geometry of $G$30.

In that setting, the distinctive feature of $G$31 is its normalization. For regular graphs it reduces exactly to

$G$32

so the diminished Sombor spectrum is a scaled adjacency spectrum. Outside the regular case, the weights remain sensitive to degree imbalance but stay bounded. The resulting theory combines exact solvability on standard graph families with sharp extremal bounds and clean graph characterizations, making the diminished Sombor matrix a well-controlled normalized variant of Sombor-type spectral graph operators.