Sombor Matrix and Its Spectral Insights

• Sombor matrix is a degree-weighted adjacency matrix defined by √(d_i²+d_j²) for adjacent vertices, incorporating both connectivity and degree information.
• It generalizes to the p-Sombor matrix family and yields explicit spectral results, including characteristic polynomials, energy formulas, and eigenvalue scaling laws.
• The matrix bridges scalar topological indices with spectral graph theory, offering insights for extremal graph studies and potential applications in chemical graph descriptors.

Searching arXiv for recent and foundational papers on the Sombor matrix and related spectral variants. The Sombor matrix is a degree-weighted adjacency matrix associated with a simple graph $G$. For a graph with vertex set $V(G)=\{v_1,\dots,v_n\}$ and vertex degrees $d_i=d(v_i)$, the Sombor matrix $A_{SO}(G)$ is the $n\times n$ real symmetric matrix whose $(i,j)$-entry equals $\sqrt{d_i^2+d_j^2}$ when $v_i$ and $v_j$ are adjacent and $0$ otherwise (Ghanbari, 2021). It is the matrix realization of the Sombor index $V(G)=\{v_1,\dots,v_n\}$0, since the latter is the sum of all nonzero entries of $V(G)=\{v_1,\dots,v_n\}$1 divided by $V(G)=\{v_1,\dots,v_n\}$2 (Ghanbari, 2021). This construction places degree information directly into the off-diagonal edge weights, distinguishing it from the ordinary adjacency matrix, the Randić matrix, and the signless Laplacian (Ghanbari, 2021). In the broader literature, the Sombor matrix is also embedded in the more general family of $V(G)=\{v_1,\dots,v_n\}$3-Sombor matrices, where the edge weight is $V(G)=\{v_1,\dots,v_n\}$4; the classical Sombor matrix is the case $V(G)=\{v_1,\dots,v_n\}$5 (Liu et al., 2021).

1. Definition and matrix-theoretic setting

Let $V(G)=\{v_1,\dots,v_n\}$6 be a simple finite graph with vertex set $V(G)=\{v_1,\dots,v_n\}$7, and let $V(G)=\{v_1,\dots,v_n\}$8 be the degree of $V(G)=\{v_1,\dots,v_n\}$9. The Sombor matrix is defined by

$d_i=d(v_i)$0

This matrix is real and symmetric, hence diagonalizable with real eigenvalues (Ghanbari, 2021). Its adjacency pattern is identical to that of the ordinary adjacency matrix $d_i=d(v_i)$1, but each edge $d_i=d(v_i)$2 carries the weight $d_i=d(v_i)$3 rather than $d_i=d(v_i)$4 (Ghanbari, 2021).

The Sombor matrix differs conceptually from several other degree-based graph matrices. In the adjacency matrix, only adjacency information appears; in the signless Laplacian $d_i=d(v_i)$5, degree information is placed on the diagonal; in Randić-type matrices, edge weights typically involve reciprocal or rational functions of degrees (Ghanbari, 2021). By contrast, the Sombor matrix inserts the Euclidean norm of the endpoint degree vector $d_i=d(v_i)$6 directly into each off-diagonal edge position (Ghanbari, 2021). This places degree heterogeneity and adjacency structure into a single weighted matrix.

The matrix viewpoint extends naturally to the $d_i=d(v_i)$7-Sombor matrix $d_i=d(v_i)$8, defined by

$d_i=d(v_i)$9

for $A_{SO}(G)$0 (Liu et al., 2021). The classical Sombor matrix is exactly $A_{SO}(G)$1 (Liu et al., 2021). This family links the Sombor matrix to other degree-based constructions: $A_{SO}(G)$2 yields the first Zagreb-type weighting, while $A_{SO}(G)$3 yields the inverse sum indeg weighting (Liu et al., 2021).

2. Spectrum, characteristic polynomial, and energy

If $A_{SO}(G)$4 are the eigenvalues of $A_{SO}(G)$5, then they form the Sombor spectrum of $A_{SO}(G)$6 (Ghanbari, 2021). The Sombor characteristic polynomial is

$A_{SO}(G)$7

and the Sombor energy is

$A_{SO}(G)$8

These are the direct analogues of the adjacency characteristic polynomial and graph energy, but computed from the Sombor matrix rather than from $A_{SO}(G)$9 (Ghanbari, 2021).

General bounds on Sombor energy are known in terms of the forgotten index $n\times n$0 and the Sombor index $n\times n$1. Two quoted upper bounds are

$n\times n$2

and

$n\times n$3

These are Sombor analogues of McClelland-type and Koolen–Moulton-type energy bounds (Ghanbari, 2021).

The $n\times n$4-Sombor framework supplies additional spectral infrastructure. For the $n\times n$5-Sombor matrix $n\times n$6, the $n\times n$7-th spectral moment is

$n\times n$8

where $n\times n$9 are the eigenvalues of $(i,j)$0 (Liu et al., 2021). In particular,

$(i,j)$1

and, for $(i,j)$2, this identifies the second spectral moment of the Sombor matrix with a degree-weighted sum over edges (Liu et al., 2021). This suggests that the Sombor matrix admits the same kind of moment-based analysis as classical adjacency-like matrices.

3. Explicit formulas for standard graph families

A substantial part of the Sombor matrix theory consists of explicit characteristic polynomials, spectra, and energies for standard graph classes (Ghanbari, 2021).

For the path $(i,j)$3, endpoints have degree $(i,j)$4 and internal vertices have degree $(i,j)$5, so the Sombor edge weights are $(i,j)$6 at the two terminal edges and $(i,j)$7 on internal edges (Ghanbari, 2021). Defining $(i,j)$8 for the tridiagonal matrix with diagonal $(i,j)$9 and off-diagonal $\sqrt{d_i^2+d_j^2}$0, with recurrence

$\sqrt{d_i^2+d_j^2}$1

the Sombor characteristic polynomial satisfies

$\sqrt{d_i^2+d_j^2}$2

with initial cases

$\sqrt{d_i^2+d_j^2}$3

These formulas are derived by recursive determinant expansion (Ghanbari, 2021).

For the cycle $\sqrt{d_i^2+d_j^2}$4, all vertices have degree $\sqrt{d_i^2+d_j^2}$5, so every edge has weight $\sqrt{d_i^2+d_j^2}$6, and $\sqrt{d_i^2+d_j^2}$7 (Ghanbari, 2021). Its characteristic polynomial is

$\sqrt{d_i^2+d_j^2}$8

with the same $\sqrt{d_i^2+d_j^2}$9 recurrence as for paths (Ghanbari, 2021). Because the Sombor matrix is a scalar multiple of the adjacency matrix, the Sombor eigenvalues of $v_i$0 are simply $v_i$1 times the usual adjacency eigenvalues (Ghanbari, 2021).

For the star $v_i$2, the center has degree $v_i$3, each leaf has degree $v_i$4, and every edge has weight $v_i$5 (Ghanbari, 2021). The matrix is

$v_i$6

and the characteristic polynomial and energy are

$v_i$7

$v_i$8

Hence the spectrum consists of $v_i$9 and $v_j$0 with multiplicity $v_j$1 (Ghanbari, 2021).

For the complete graph $v_j$2, every vertex has degree $v_j$3, so every edge weight is $v_j$4, and

$v_j$5

Its characteristic polynomial and energy are

$v_j$6

$v_j$7

Thus the Sombor spectrum is obtained from the spectrum of $v_j$8 by scalar multiplication (Ghanbari, 2021).

For the complete bipartite graph $v_j$9 with $0$0, every edge joins degrees $0$1 and $0$2, so each weight is $0$3, and

$0$4

The characteristic polynomial and energy are

$0$5

$0$6

Thus the spectrum consists of $0$7 and $0$8 with multiplicity $0$9 (Ghanbari, 2021).

4. Regular graphs, scaling, and the $V(G)=\{v_1,\dots,v_n\}$00-Sombor perspective

A basic structural simplification occurs for regular graphs. If $V(G)=\{v_1,\dots,v_n\}$01 is $V(G)=\{v_1,\dots,v_n\}$02-regular, then every edge has Sombor weight $V(G)=\{v_1,\dots,v_n\}$03, so

$V(G)=\{v_1,\dots,v_n\}$04

Therefore the Sombor spectrum is exactly $V(G)=\{v_1,\dots,v_n\}$05 times the adjacency spectrum, and all spectral quantities of the Sombor matrix reduce to scaled adjacency quantities (Ghanbari, 2021). This explains, for example, why the cycle and complete graph formulas are so explicit.

The $V(G)=\{v_1,\dots,v_n\}$06-Sombor matrix generalizes this statement. For a $V(G)=\{v_1,\dots,v_n\}$07-regular graph,

$V(G)=\{v_1,\dots,v_n\}$08

so the $V(G)=\{v_1,\dots,v_n\}$09-Sombor eigenvalues are $V(G)=\{v_1,\dots,v_n\}$10 times the adjacency eigenvalues (Liu et al., 2021). In particular, for $V(G)=\{v_1,\dots,v_n\}$11,

$V(G)=\{v_1,\dots,v_n\}$12

recovering the Sombor scaling law (Liu et al., 2021).

This scaling principle extends to matrix invariants such as energy and spectral radius. In the $V(G)=\{v_1,\dots,v_n\}$13-Sombor setting, if $V(G)=\{v_1,\dots,v_n\}$14 is the adjacency spectral radius of $V(G)=\{v_1,\dots,v_n\}$15, then the $V(G)=\{v_1,\dots,v_n\}$16-Sombor spectral radius $V(G)=\{v_1,\dots,v_n\}$17 satisfies

$V(G)=\{v_1,\dots,v_n\}$18

with equality if and only if $V(G)=\{v_1,\dots,v_n\}$19 is regular (Liu et al., 2021). Specializing to $V(G)=\{v_1,\dots,v_n\}$20, this gives a general comparison between the Sombor spectral radius and the adjacency spectral radius.

The same paper also establishes that, for a tree $V(G)=\{v_1,\dots,v_n\}$21 on $V(G)=\{v_1,\dots,v_n\}$22 vertices and $V(G)=\{v_1,\dots,v_n\}$23,

$V(G)=\{v_1,\dots,v_n\}$24

with equality if and only if $V(G)=\{v_1,\dots,v_n\}$25 (Liu et al., 2021). Later work refines this by showing that among all trees of order $V(G)=\{v_1,\dots,v_n\}$26, the first three largest $V(G)=\{v_1,\dots,v_n\}$27-Sombor spectral radii are attained uniquely by $V(G)=\{v_1,\dots,v_n\}$28, $V(G)=\{v_1,\dots,v_n\}$29, and $V(G)=\{v_1,\dots,v_n\}$30 (Zheng et al., 2023). This suggests that star-like degree concentration is extremal not only for the scalar Sombor index but also for the spectral radius of Sombor-type matrices.

5. Energy, uniqueness, and computational classifications

The Sombor energy

$V(G)=\{v_1,\dots,v_n\}$31

is one of the central spectral invariants associated with the Sombor matrix (Ghanbari, 2021). Explicit formulas are available for stars, complete graphs, and complete bipartite graphs, while for paths and cycles the energy is computable from the characteristic polynomials even when closed-form root expressions are not written explicitly (Ghanbari, 2021).

The Sombor matrix is also additive over disjoint unions in the expected block-diagonal sense. If $V(G)=\{v_1,\dots,v_n\}$32, then

$V(G)=\{v_1,\dots,v_n\}$33

This yields immediate consequences for 2-regular graphs, which are disjoint unions of cycles, and for graph modifications such as edge deletion that disconnect paths or stars (Ghanbari, 2021).

A detailed computational study was carried out for all 21 cubic graphs of order $V(G)=\{v_1,\dots,v_n\}$34, using Maple to compute Sombor characteristic polynomials and energies (Ghanbari, 2021). The Petersen graph $V(G)=\{v_1,\dots,v_n\}$35, being 3-regular, satisfies

$V(G)=\{v_1,\dots,v_n\}$36

and its characteristic polynomial is

$V(G)=\{v_1,\dots,v_n\}$37

Hence its eigenvalues are

$V(G)=\{v_1,\dots,v_n\}$38

and its Sombor energy is

$V(G)=\{v_1,\dots,v_n\}$39

Within the family of 3-regular graphs of order $V(G)=\{v_1,\dots,v_n\}$40, the Petersen graph has maximum Sombor energy (Ghanbari, 2021).

The same study introduces Sombor energy equivalence: two graphs $V(G)=\{v_1,\dots,v_n\}$41 and $V(G)=\{v_1,\dots,v_n\}$42 are Sombor-energy equivalent if $V(G)=\{v_1,\dots,v_n\}$43 (Ghanbari, 2021). A graph is Sombor energy unique if its equivalence class consists only of itself (Ghanbari, 2021). Among cubic graphs of order $V(G)=\{v_1,\dots,v_n\}$44, six are not Sombor-energy unique; the remaining fifteen are (Ghanbari, 2021). This shows that Sombor energy is a discriminating, but not complete, isomorphism invariant.

The arithmetic nature of Sombor energy is also raised as an open problem. A conjecture states that there is no graph with integer-valued Sombor energy: $V(G)=\{v_1,\dots,v_n\}$45 This is supported by the classes explicitly computed in the paper, including paths, cycles, stars, complete graphs, complete bipartite graphs, 2-regular graphs, and all cubic graphs of order $V(G)=\{v_1,\dots,v_n\}$46 (Ghanbari, 2021).

6. Structural extremality and relations to broader Sombor theory

The Sombor matrix is tightly connected to structural extremal problems for the scalar Sombor index. Because the matrix entries are exactly the edge contributions $V(G)=\{v_1,\dots,v_n\}$47, any result that identifies graphs maximizing or minimizing $V(G)=\{v_1,\dots,v_n\}$48 also identifies graphs that maximize or minimize the total edge-weight of the Sombor matrix.

Several papers in the supplied corpus make this connection explicit. For unicyclic graphs with fixed order $V(G)=\{v_1,\dots,v_n\}$49 and diameter $V(G)=\{v_1,\dots,v_n\}$50, a unique extremal graph $V(G)=\{v_1,\dots,v_n\}$51 maximizes the Sombor index (Liu, 2021). Its structure is highly centralized: a 4-cycle sharing two consecutive vertices with a diametral path, and all remaining vertices attached as pendent vertices to a single high-degree vertex of degree $V(G)=\{v_1,\dots,v_n\}$52 (Liu, 2021). The paper notes that, for someone studying Sombor matrices, $V(G)=\{v_1,\dots,v_n\}$53 is a natural candidate for maximizing spectral quantities such as the spectral radius, because it maximizes the total Sombor edge weight under the same constraints (Liu, 2021). This suggests, but does not prove, a spectral extremal principle.

For trees with a fixed degree sequence, one paper characterizes the extremal trees maximizing the scalar Sombor index through a constructive algorithm based on subtree attachments (Movahedi, 2022). Another paper revisits the same problem from a different extremal direction and shows that, among trees with a given degree sequence, the greedy tree minimizes the Sombor index and the alternating greedy tree maximizes it, with full characterization of all minimizers and maximizers (Andriantiana et al., 2024). Since the Sombor index is half of the sum of the entries of a natural Sombor matrix, these results can be interpreted as describing adjacency patterns that minimize or maximize the total mass of the Sombor matrix for prescribed degree data (Andriantiana et al., 2024).

The same structural phenomenon appears in cacti. Among cacti with $V(G)=\{v_1,\dots,v_n\}$54 vertices and $V(G)=\{v_1,\dots,v_n\}$55 cycles, the unique graph $V(G)=\{v_1,\dots,v_n\}$56 maximizes the Sombor index, and its structure is hub-dominated: all cycles share a common high-degree vertex and all extra vertices attach as pendants to that vertex (Liu, 2021). For cacti with perfect matching, the analogous maximizer is $V(G)=\{v_1,\dots,v_n\}$57 (Liu, 2021). These graphs are natural candidates for extremal Sombor-matrix spectral behavior, although that spectral question is not explicitly settled in the paper.

The paper on $V(G)=\{v_1,\dots,v_n\}$58-cyclic graphs confirms a conjecture that the graph $V(G)=\{v_1,\dots,v_n\}$59, obtained from the star $V(G)=\{v_1,\dots,v_n\}$60 by adding $V(G)=\{v_1,\dots,v_n\}$61 edges from one fixed pendent vertex to $V(G)=\{v_1,\dots,v_n\}$62 other pendent vertices, has the maximum Sombor index among connected $V(G)=\{v_1,\dots,v_n\}$63-cyclic graphs of order $V(G)=\{v_1,\dots,v_n\}$64 (Das et al., 2021). The same graph also maximizes the reduced Sombor index (Das et al., 2021). This suggests that Sombor-matrix extremality often coincides with severe degree concentration around one or two dominant vertices.

These structural results do not by themselves produce spectral formulas for the Sombor matrix, but they delineate the graph families that are “most Sombor-heavy” in the sense of total matrix weight. A plausible implication is that these same families are strong candidates for extremal Sombor spectral radius or energy within the corresponding constrained graph classes.

7. Extensions and related Sombor-type matrices

The Sombor matrix has inspired several variants that alter the edge weight while preserving the basic weighted-adjacency paradigm.

The elliptic Sombor matrix $V(G)=\{v_1,\dots,v_n\}$65 is defined by

$V(G)=\{v_1,\dots,v_n\}$66

Its energy and characteristic polynomials have been computed for several graph classes, and for $V(G)=\{v_1,\dots,v_n\}$67-regular graphs one has

$V(G)=\{v_1,\dots,v_n\}$68

so the elliptic Sombor spectrum is a scalar multiple of the Sombor spectrum on regular graphs (Alikhani et al., 2024). This places the Sombor matrix inside a hierarchy of increasingly degree-sensitive weighted adjacency matrices.

A different normalization appears in the diminished Sombor matrix $V(G)=\{v_1,\dots,v_n\}$69, whose $V(G)=\{v_1,\dots,v_n\}$70-entry is

$V(G)=\{v_1,\dots,v_n\}$71

when $V(G)=\{v_1,\dots,v_n\}$72, and $V(G)=\{v_1,\dots,v_n\}$73 otherwise (Movahedi, 3 Aug 2025). In regular graphs,

$V(G)=\{v_1,\dots,v_n\}$74

so its spectrum is a simple scalar multiple of the adjacency spectrum (Movahedi, 3 Aug 2025). The paper develops sharp bounds for its spectral radius and energy, and characterizes extremal graphs such as complete graphs, complete bipartite graphs, stars, disjoint unions of edges, and edgeless graphs (Movahedi, 3 Aug 2025). This suggests that normalization of Sombor weights changes the extremal geometry but preserves the general spectral-graph-theoretic toolkit.

The Euler–Sombor matrix $V(G)=\{v_1,\dots,v_n\}$75 assigns the edge weight

$V(G)=\{v_1,\dots,v_n\}$76

to each adjacency (Bansode et al., 12 Feb 2025). Its eigenvalues and energy have been computed for complete graphs, cycles, complete bipartite graphs, and stars, and its sum-of-squares identity is

$V(G)=\{v_1,\dots,v_n\}$77

where $V(G)=\{v_1,\dots,v_n\}$78 is the forgotten index and $V(G)=\{v_1,\dots,v_n\}$79 is the second Zagreb index (Bansode et al., 12 Feb 2025). This again mirrors the Sombor matrix program: define a weighted adjacency matrix from a degree-based edge function, then study its characteristic polynomial, spectral radius, energy, and extremal behavior.

Finally, the $V(G)=\{v_1,\dots,v_n\}$80-Sombor matrix unifies all of these constructions at the level of degree aggregation (Liu et al., 2021). A later paper uses it to characterize the first three maximum $V(G)=\{v_1,\dots,v_n\}$81-Sombor spectral radii among trees, unicyclic graphs, and bicyclic graphs (Zheng et al., 2023). This line of work suggests that the classical Sombor matrix is best understood not as an isolated object, but as the $V(G)=\{v_1,\dots,v_n\}$82 representative of a broader family of spectral degree-based matrices.

8. Significance and open directions

The Sombor matrix occupies a natural position between scalar topological indices and spectral graph theory. It converts the edge-wise degree expression $V(G)=\{v_1,\dots,v_n\}$83 into a symmetric weighted adjacency matrix, thereby making available the full apparatus of eigenvalues, characteristic polynomials, energy, spectral moments, and extremal spectral comparisons (Ghanbari, 2021, Liu et al., 2021).

Its most immediate significance is representational. Unlike the adjacency matrix, it reflects both adjacency and degree distribution; unlike the signless Laplacian, it does so through off-diagonal edge weights rather than diagonal degree entries (Ghanbari, 2021). For regular graphs, the matrix collapses to a scalar multiple of $V(G)=\{v_1,\dots,v_n\}$84, showing that it truly extends adjacency-based spectral theory rather than replacing it (Ghanbari, 2021). For irregular graphs, it encodes a nonlinear coupling between degree heterogeneity and connectivity.

The available theory is strongest in three areas. First, for standard graph classes such as paths, cycles, stars, complete graphs, and complete bipartite graphs, exact characteristic polynomials and energy formulas are known (Ghanbari, 2021). Second, in the generalized $V(G)=\{v_1,\dots,v_n\}$85-Sombor setting, there are systematic bounds on spectral radius, energy, Laplacian eigenvalues, spread, and Estrada index (Liu et al., 2021). Third, many extremal combinatorial results identify graph structures that maximize or minimize the scalar Sombor index under degree, diameter, cyclomatic, or matching constraints, thereby providing strong candidates for analogous spectral extremal theorems (Liu, 2021, Liu, 2021, Das et al., 2021, Andriantiana et al., 2024).

Several open directions are explicitly or implicitly indicated. One is the conjecture that no graph has integer-valued Sombor energy (Ghanbari, 2021). Another is the problem of determining further structural extremal graphs for Sombor or $V(G)=\{v_1,\dots,v_n\}$86-Sombor spectral radius beyond the tree, unicyclic, and bicyclic cases already treated (Liu et al., 2021, Zheng et al., 2023). A further plausible direction is to transfer scalar Sombor-index extremal results—such as those for $V(G)=\{v_1,\dots,v_n\}$87, $V(G)=\{v_1,\dots,v_n\}$88, $V(G)=\{v_1,\dots,v_n\}$89, or greedy/alternating greedy trees—into precise statements about the Sombor matrix spectral radius, energy, or Estrada-type quantities. This suggests a mature research program: start from a degree-based edge function, build the corresponding weighted adjacency matrix, and develop a full spectral theory around it.

In that sense, the Sombor matrix is both a concrete matrix invariant and a prototype for a larger class of degree-sensitive spectral constructions. Its theory connects chemical graph descriptors, degree-based extremal combinatorics, and weighted spectral graph theory in a single framework (Ghanbari, 2021, Liu et al., 2021).