Central Graph: Theory & Applications

• Central graph is a graph operation defined by subdividing each edge and connecting nonadjacent original vertices, yielding a distinct two-layer structure.
• It underpins rigorous analysis in spectral theory, resistance distance, and domination studies by enabling explicit block matrix formulations.
• Beyond its classical definition, the term informs related constructions in centrality concepts, graph joins, and machine learning-based semantic networks.

Searching arXiv for recent and relevant papers on “central graph” and closely related graph-theoretic usages. The term central graph has multiple technical meanings in contemporary arXiv literature. In classical graph theory, the standard meaning is an operation on a simple graph $G$: subdivide every edge exactly once and then join every pair of original vertices that were nonadjacent in $G$ (K et al., 2021). This construction, usually denoted $C(G)$, is the dominant usage in recent spectral, resistance-distance, and coloring work (T et al., 2024). A distinct but related line of research uses central to denote graph-theoretic centrality structures such as centers, centered peripheries, central vertices, or central parts (Choi et al., 2017, Pandey et al., 2021, Talukdar, 31 Jan 2025). In applied machine learning and information retrieval, the phrase is also used more loosely for graph representations organized around central entities or centrality-aware operators (Benelallam et al., 2019, Lan et al., 29 Jan 2026, Miao et al., 2021). The graph-theoretic operation $C(G)$ remains the most precise and standardized sense; however, the broader literature shows that “central graph” also functions as a family resemblance term for graph constructions in which metric center, centrality hierarchy, or title-centered semantic organization is the governing principle.

1. Standard graph-theoretic construction

For a simple graph $G$ with $n$ vertices and $m$ edges, the central graph $C(G)$ is obtained by subdividing every edge of $G$ exactly once and then joining all pairs of vertices that were nonadjacent in $G$ (K et al., 2021). Equivalently, if $G$0 has vertex set $G$1 and edge set $G$2, then $G$3 contains the original vertices together with one new vertex for each edge $G$4; these inserted vertices are often described as the vertices corresponding to the edges of $G$5 (T et al., 2024). The construction has $G$6 vertices and $G$7 edges (K et al., 2021).

This definition induces a characteristic two-layer structure. Among the original vertices, adjacency in $G$8 is governed by nonadjacency in $G$9; between original and inserted vertices, incidence is inherited from the edge subdivision; and the inserted vertices themselves form the subdivision layer. In matrix terms, if $C(G)$0 denotes the incidence matrix and $C(G)$1 is the adjacency matrix of the complement, then with a suitable labeling

$C(G)$2

and for an $C(G)$3-regular graph,

$C(G)$4

(K et al., 2021). The same block structure underlies Laplacian and resistance-distance analyses, where the Laplacian is written as

$C(G)$5

with $C(G)$6 the incidence matrix (T et al., 2024).

A useful interpretive point is that $C(G)$7 is neither merely a subdivision graph nor merely a complement-based augmentation. It combines both operations. This suggests why many invariants of $C(G)$8 split into one part controlled by the original spectrum of $C(G)$9 and another part controlled by the inserted edge-vertices.

2. Structural and metric centrality notions distinct from $C(G)$0

A second major usage of “central graph” concerns central vertices and centers of graphs, rather than the operation $C(G)$1. For a connected graph $C(G)$2, the center is the set of vertices of minimum eccentricity,

$C(G)$3

where

$C(G)$4

(Pandey et al., 2021). The same paper treats the median, security center, characteristic center, subgraph core, and core vertices as six distinct “central parts,” and proves that for a connected vertex-transitive graph each of these six central parts is the whole vertex set (Pandey et al., 2021). This is a different notion from the central graph operation, although the overlap in notation is substantial.

Within ring-theoretic graph constructions, the term central vertices appears in the zero-divisor graph $C(G)$5. There the vertices are the nonzero zero-divisors of $C(G)$6, and the paper determines the center completely: $C(G)$7 (Talukdar, 31 Jan 2025). Concretely, the central vertices are exactly the nonzero multiples of $C(G)$8 for primes $C(G)$9 (Talukdar, 31 Jan 2025). The same work also characterizes cut-edges in $G$0, showing that for odd $G$1 there are no cut-edges (Talukdar, 31 Jan 2025).

A further extension is the theory of uniform central graphs (UCGs). For a graph $G$2, the centered periphery is

$G$3

where $G$4 is the center and $G$5 is the set of eccentric vertices of $G$6 (Choi et al., 2017). A graph is a uniform central graph if all central vertices have exactly the same set of eccentric vertices: $G$7 (Choi et al., 2017). The central-peripheral appendage number $G$8 then measures the minimum number of intermediate vertices needed to realize a prescribed center $G$9 and centered periphery $n$0 within a UCG (Choi et al., 2017).

These lines of work should not be conflated. In one case, “central graph” means the graph operation $n$1. In the other, “central” refers to vertices or substructures defined by metric or combinatorial centrality. The shared vocabulary reflects a common concern with graph-theoretic centrality, but the objects are different.

3. Spectral theory of the central graph operation

The spectral theory of $n$2 is particularly tractable when $n$3 is regular. For an $n$4-regular graph with adjacency eigenvalues $n$5, the normalized Laplacian characteristic polynomial of $n$6 is

$n$7

where $n$8 is the polynomial associated with the adjacency spectrum of $n$9 (K et al., 2021). This yields a complete normalized Laplacian description: the eigenvalue $m$0 appears with multiplicity $m$1, while the remaining eigenvalues come from quadratic factors indexed by the adjacency eigenvalues of $m$2 (K et al., 2021).

The same spectral decomposition leads directly to closed formulas for Kemeny’s constant and the degree Kirchhoff index. If $m$3 is $m$4-regular of order $m$5 and size $m$6, then

$m$7

and

$m$8

(K et al., 2021). The structure of these formulas reflects a recurring theme: subdivision vertices generate a large repeated spectral block, while the complement-induced interactions among original vertices inject the dependence on the adjacency spectrum of $m$9.

A different spectral program studies the eccentricity matrix $C(G)$0 for central graphs of regular graphs. For a triangle-free $C(G)$1-regular $C(G)$2-graph $C(G)$3, the eccentricity matrix of $C(G)$4 is written in block form as

$C(G)$5

where $C(G)$6 is the incidence matrix and $C(G)$7 is the adjacency matrix of the line graph $C(G)$8 (Ashokan et al., 2024). From this, the paper derives the $C(G)$9-spectrum, $G$0-energy, inertia, and irreducibility of $G$1 (Ashokan et al., 2024). In particular, for triangle-free regular graphs,

$G$2

and

$G$3

(Ashokan et al., 2024).

This suggests that the central graph operation is unusually amenable to block-matrix methods. Because $G$4 retains a clean separation between original vertices and edge-vertices, spectral quantities that are difficult for generic graph products remain explicit for large regular families.

4. Resistance distance, Kirchhoff index, and random-walk quantities

The resistance-distance theory of central graphs is built from the Laplacian block form and Schur complement identities. For a connected graph $G$5 of order $G$6, the resistance distance between two original vertices $G$7 in $G$8 is

$G$9

(T et al., 2024). For an inserted vertex corresponding to $G$0 and an original vertex $G$1,

$G$2

and for two inserted vertices corresponding to $G$3 and $G$4,

$G$5

(T et al., 2024).

These formulas yield a general Kirchhoff-index expression for $G$6. For a connected graph $G$7 of order $G$8 and size $G$9,

$G$00

where $G$01 is the degree vector of $G$02 (T et al., 2024). The same paper gives a weighted resistance-sum formula for Kemeny’s constant $G$03 (T et al., 2024).

The significance of these results is methodological as much as computational. The central graph converts resistance calculations into a controlled block problem in which the inserted edge-vertices always have degree $G$04, while the original-vertex block is governed by the modified operator $G$05. A plausible implication is that $G$06 serves as a useful testbed for comparing Laplacian-based invariants across graph operations that mix complement structure with subdivision structure.

5. Derived operations based on the central graph

A substantial literature treats $G$07 not as an endpoint but as a scaffold for additional graph operations. Three especially prominent constructions are the central vertex join, central edge join, and several corona-type products.

The central vertex join $G$08 is obtained from $G$09 and $G$10 by joining each vertex of $G$11 with every vertex of $G$12, while the central edge join $G$13 is obtained from $G$14 and $G$15 by joining each vertex corresponding to the edges of $G$16 with every vertex of $G$17 (T et al., 2024). Resistance distance, Kirchhoff index, and Kemeny’s constant for these joins are derived by extending the same Laplacian-Schur-complement framework used for $G$18 itself (T et al., 2024). In the eccentricity-spectrum setting, the analogous operations are analyzed for triangle-free regular $G$19 and regular $G$20, together with formulas for $G$21-spectra, $G$22-Wiener indices, lower bounds for the $G$23-spectral radius, and irreducibility (Ashokan et al., 2024).

Another family of constructions comprises the central vertex corona $G$24, central edge corona $G$25, and central edge neighborhood corona $G$26 (Jahfar et al., 2021). Each begins with $G$27 and then attaches copies of $G$28 either to original vertices, to inserted edge-vertices, or to their neighborhoods (Jahfar et al., 2021). The paper derives adjacency, Laplacian, and signless Laplacian spectra through block determinants, Schur complements, and the coronal identity, and then computes the number of spanning trees and the Kirchhoff index of the resulting graphs (Jahfar et al., 2021).

Two points stand out. First, these constructions inherit the two-layer structure of $G$29, which keeps their algebra manageable. Second, they support systematic cospectral constructions. Both the central-join and central-corona papers explicitly note that their formulas generate infinite families of adjacency-, Laplacian-, signless-Laplacian-, or $G$30-cospectral nonisomorphic graphs (Jahfar et al., 2021, Ashokan et al., 2024).

6. Coloring and domination in central graphs

The central graph operation produces a rigid coloring environment because original vertices become highly adjacent through complement-like edges, while each subdividing vertex has degree exactly $G$31. This structure is central to the theory of total dominator coloring of $G$32 (Kazemnejad et al., 2018). A total dominator coloring is a proper coloring in which each vertex is adjacent to every vertex of some color class; the minimum number of color classes is the total dominator chromatic number $G$33 (Kazemnejad et al., 2018).

For a connected graph $G$34 of order $G$35 with longest path of order $G$36,

$G$37

(Kazemnejad et al., 2018). If $G$38 is connected and not complete, the upper bound improves to

$G$39

(Kazemnejad et al., 2018). The extremal case is characterized exactly: $G$40 (Kazemnejad et al., 2018).

The paper also gives exact formulas for several standard families. For paths,

$G$41

for cycles,

$G$42

and for complete bipartite graphs $G$43 with $G$44,

$G$45

All are established in (Kazemnejad et al., 2018).

These results show that $G$46 is not merely a spectral object. Its mixed complement-subdivision structure imposes strong domination constraints and yields many exact coloring formulas that are unavailable for broader graph products.

7. Broader and nonstandard uses of “central graph”

Outside the strict graph-theoretic operation $G$47, several arXiv papers use “central” in ways that are conceptually related but not equivalent.

In software-ecosystem mining, the Maven Dependency Graph (MDG) is presented as a graph-database representation of Maven Central, and is described as a central graph representation of the ecosystem (Benelallam et al., 2019). The graph is formalized as

$G$48

with artifact nodes, calendar nodes, dependency relationships, and version precedence relationships (Benelallam et al., 2019). This is not a “central graph” in the graph-theoretic sense of $G$49; instead, “central” refers to the role of the graph as the backbone of Maven Central (Benelallam et al., 2019).

In scientific-question-answering systems, CE-GOCD uses paper titles as central entities anchoring subgraph retrieval and community refinement in an academic knowledge graph (Lan et al., 29 Jan 2026). The method identifies a dominant title node in each community via

$G$50

(Lan et al., 29 Jan 2026). Here again, the emphasis is on title-centered semantic organization rather than on the central graph operation (Lan et al., 29 Jan 2026).

In graph representation learning, centrality-constrained graph embedding organizes a drawing so that higher-centrality nodes appear closer to the origin through radial constraints

$G$51

with $G$52 monotone decreasing (Baingana et al., 2013). In skeleton-based action recognition, Central Difference Graph Convolution (CDGC) augments graph convolution by combining ordinary neighborhood aggregation with a central-difference term

$G$53

(Miao et al., 2021). In scale-free graph learning, CenGCN identifies hubs via centrality and rewrites edge weights using centrality- and similarity-based functions (Xia et al., 2022). These are centrality-aware graph operators rather than central graphs in the classical sense.

A plausible implication is that the phrase “central graph” has become polysemous across arXiv fields. The unambiguous graph-theoretic meaning remains $G$54, but applied literatures increasingly use “central” to indicate centrality-respecting geometry, central-entity anchoring, or central-difference message passing.

8. Conceptual synthesis and open boundaries

Across these literatures, three distinct but interacting ideas recur.

First, the classical central graph operation $G$55 is a graph transform with unusually rich exact theory. Its block structure supports explicit normalized Laplacian formulas (K et al., 2021), resistance distances and Kirchhoff indices (T et al., 2024), eccentricity spectra (Ashokan et al., 2024), total dominator colorings (Kazemnejad et al., 2018), and a wide family of derived joins and coronas (Jahfar et al., 2021).

Second, central parts of graphs constitute a separate metric-combinatorial program. Centers, security centers, characteristic centers, subgraph cores, and core vertices can diverge sharply in general connected graphs, though they all collapse to the whole vertex set in connected vertex-transitive graphs (Pandey et al., 2021). Uniform central graphs extend this perspective by fixing both a center and a centered periphery and asking how many intermediate vertices are needed to realize them (Choi et al., 2017). In the special case of the zero-divisor graph of $G$56, the center admits a complete arithmetic characterization (Talukdar, 31 Jan 2025).

Third, the language of centrality has spread into graph databases, graph ML, and knowledge-graph retrieval. These usages preserve the intuition that some vertices or entities function as organizing anchors, but they do not define central graphs by the operation $G$57 (Benelallam et al., 2019, Lan et al., 29 Jan 2026, Miao et al., 2021, Baingana et al., 2013, Xia et al., 2022).

The main source of possible confusion is terminological overlap. In graph theory, $G$58 may denote the central graph of $G$59, while in metric centrality it may denote the center of $G$60 (Pandey et al., 2021). The literature therefore supports a careful distinction: central graph should refer to the subdivision-plus-complement operation unless the context explicitly concerns graph centers or centrality-aware models. This suggests that the term is best understood not as a single universal concept, but as a precise graph operation embedded within a broader family of “central” constructions whose unifying theme is the structural role of vertices, entities, or subgraphs that organize the rest of the network.