Vertex Energy in Graphs

• Vertex energy is defined as the localized diagonal invariant from |A|, partitioning total graph energy across vertices.
• Key methodologies involve spectral decompositions and Coulson-type integrals to relate local energy to closed-walk counts and graph topology.
• The concept exhibits dual usage, serving as both a local invariant and an average measure in regular graphs, informing extremal and perturbation analyses.

Vertex energy is a spectral invariant that localizes graph energy at individual vertices. For a graph $G$ with adjacency matrix $A$, the localized definition introduced by Arizmendi and Juárez-Romero sets $\mathcal{E}_G(v_i)=|A|_{ii}$, where $|A|=(AA^*)^{1/2}$, so that the total graph energy satisfies $$\mathcal{E}(G)=\operatorname{Tr}(|A|)=\sum_i \mathcal{E}_G(v_i)$$ (Arizmendi et al., 2018). The same phrase, however, is also used in another established sense: for a graph on $n$ vertices, the “energy per vertex” or “vertex energy” may denote the average $\varphi(G)=E(G)/n$, especially in the extremal theory of regular graphs (Dam et al., 2012). The term therefore has a dual usage, with the localized diagonal invariant now dominant in the modern spectral-graph-theoretic literature.

1. Definition, spectral meaning, and scope

Let $G=(V,E)$ be a simple undirected graph with adjacency matrix $A\in M_n(\mathbb{R})$. The global energy of $G$ is

$A$0

where $A$1 are the eigenvalues of $A$2. The energy of the vertex $A$3 is the $A$4-th diagonal entry of $A$5,

$A$6

and these diagonal entries partition the total energy (Arizmendi et al., 2018).

If $A$7 with $A$8 orthogonal and $A$9, then

$\mathcal{E}_G(v_i)=|A|_{ii}$0

Writing $\mathcal{E}_G(v_i)=|A|_{ii}$1, one has $\mathcal{E}_G(v_i)=|A|_{ii}$2, so vertex energy is the expectation of $\mathcal{E}_G(v_i)=|A|_{ii}$3 under the spectral measure at the basis vector $\mathcal{E}_G(v_i)=|A|_{ii}$4. The same decomposition implies

$\mathcal{E}_G(v_i)=|A|_{ii}$5

so the moments of the local spectral measure are closed-walk counts at $\mathcal{E}_G(v_i)=|A|_{ii}$6 (Nagesh et al., 16 Aug 2025).

This formulation extends immediately to weighted graphs by replacing the $\mathcal{E}_G(v_i)=|A|_{ii}$7-adjacency matrix with a symmetric weighted adjacency matrix. It also extends to uniformly locally finite graphs: if $\mathcal{E}_G(v_i)=|A|_{ii}$8 is the bounded self-adjoint adjacency operator on $\mathcal{E}_G(v_i)=|A|_{ii}$9, then

$|A|=(AA^*)^{1/2}$0

and local convergence of rooted graphs implies convergence of the corresponding vertex energies (Arizmendi et al., 2018).

A persistent source of terminological ambiguity is the older extremal quantity

$|A|=(AA^*)^{1/2}$1

called “energy per vertex” or “vertex energy” in the regular-graph literature. This is a graph-level average, not a vertex-indexed local invariant (Dam et al., 2012).

2. Integral representations and structural consequences

A central analytical tool is the Coulson integral formula for a vertex. If $|A|=(AA^*)^{1/2}$2 and $|A|=(AA^*)^{1/2}$3 denotes the vertex-deleted graph, then

$|A|=(AA^*)^{1/2}$4

with the integral understood in the Cauchy principal value sense (Arizmendi et al., 2018). This converts a diagonal entry of $|A|=(AA^*)^{1/2}$5 into a one-dimensional integral involving only characteristic polynomials.

The formula arises from the resolvent entry $|A|=(AA^*)^{1/2}$6, which admits both a spectral expansion and a determinantal expression. In spectral form, the residues encode $|A|=(AA^*)^{1/2}$7; in determinantal form, the same quantity is expressed through the ratio $|A|=(AA^*)^{1/2}$8. This equivalence is especially useful when eigenvectors are unavailable or cumbersome to compute (Arizmendi et al., 2018).

For bipartite graphs the characteristic polynomial has only even-powered coefficients in the standard form

$|A|=(AA^*)^{1/2}$9

This coefficient structure yields comparison principles. If $$\mathcal{E}(G)=\operatorname{Tr}(|A|)=\sum_i \mathcal{E}_G(v_i)$$0 is bipartite and $$\mathcal{E}(G)=\operatorname{Tr}(|A|)=\sum_i \mathcal{E}_G(v_i)$$1 coefficientwise in the sense $$\mathcal{E}(G)=\operatorname{Tr}(|A|)=\sum_i \mathcal{E}_G(v_i)$$2 for all $$\mathcal{E}(G)=\operatorname{Tr}(|A|)=\sum_i \mathcal{E}_G(v_i)$$3, then $$\mathcal{E}(G)=\operatorname{Tr}(|A|)=\sum_i \mathcal{E}_G(v_i)$$4. One consequence is that in any tree the energy of a leaf is strictly smaller than that of its neighbor (Arizmendi et al., 2018).

Bipartiteness also imposes a global splitting law. If $$\mathcal{E}(G)=\operatorname{Tr}(|A|)=\sum_i \mathcal{E}_G(v_i)$$5 is bipartite with parts $$\mathcal{E}(G)=\operatorname{Tr}(|A|)=\sum_i \mathcal{E}_G(v_i)$$6, then

$$\mathcal{E}(G)=\operatorname{Tr}(|A|)=\sum_i \mathcal{E}_G(v_i)$$7

More generally, any vertex cover in a bipartite graph carries at least half of the total energy, while any independent set carries at most half (Arizmendi et al., 2018).

3. Bounds, inequalities, and extremal theory

The most basic localized upper bound is the McClelland-type estimate

$$\mathcal{E}(G)=\operatorname{Tr}(|A|)=\sum_i \mathcal{E}_G(v_i)$$8

where $$\mathcal{E}(G)=\operatorname{Tr}(|A|)=\sum_i \mathcal{E}_G(v_i)$$9 is the degree of $n$0. Equality holds if and only if the component containing $n$1 is a star $n$2 with $n$3 as its center (Arizmendi et al., 2018). Summing these inequalities recovers the classical upper bound for total energy.

Lower bounds can be derived from short closed walks. If $n$4 is the number of closed walks of length $n$5 based at $n$6, then

$n$7

Using $n$8, where $n$9 is the maximum degree, one obtains the coarser but simpler inequality

$\varphi(G)=E(G)/n$0

These estimates make the dependence on local density explicit: large local degree tends to increase vertex energy, while concentration of short walks suppresses the lower bound (Arizmendi et al., 2018).

A more refined upper bound uses the Perron weight $\varphi(G)=E(G)/n$1 and the spectral radius $\varphi(G)=E(G)/n$2: $\varphi(G)=E(G)/n$3 This leads to

$\varphi(G)=E(G)/n$4

and further bounds can be expressed in terms of maximum degree and eccentricity (Arizmendi et al., 2018).

The alternative quantity $\varphi(G)=E(G)/n$5 has its own extremal theory. For a $\varphi(G)=E(G)/n$6-regular graph with $\varphi(G)=E(G)/n$7,

$\varphi(G)=E(G)/n$8

with equality if and only if $\varphi(G)=E(G)/n$9 is a disjoint union of incidence graphs of projective planes of order $G=(V,E)$0, or, for $G=(V,E)$1, a disjoint union of triangles and hexagons (Dam et al., 2012). This result concerns average energy per vertex, not the localized invariant $G=(V,E)$2, but it remains an important part of the terminology.

4. Exact values and representative families

For highly symmetric graphs, vertex energy is often constant across vertices because the local spectral measures coincide. Standard examples admit closed forms (Arizmendi et al., 2018).

Graph family	Vertex energy
Complete graph $G=(V,E)$3	$G=(V,E)$4
Complete bipartite $G=(V,E)$5	$G=(V,E)$6 on the $G=(V,E)$7-part; $G=(V,E)$8 on the $G=(V,E)$9-part
Cycle $A\in M_n(\mathbb{R})$0	$A\in M_n(\mathbb{R})$1
Hypercube $A\in M_n(\mathbb{R})$2	$A\in M_n(\mathbb{R})$3

For paths $A\in M_n(\mathbb{R})$4, the explicit formula

$A\in M_n(\mathbb{R})$5

shows that endpoint and bulk behavior differ. As $A\in M_n(\mathbb{R})$6, for fixed $A\in M_n(\mathbb{R})$7 one has $A\in M_n(\mathbb{R})$8, while for $A\in M_n(\mathbb{R})$9 growing proportionally to $G$0 one obtains $G$1 (Arizmendi et al., 2018).

In vertex-transitive graphs, symmetry enforces a uniform distribution of the total energy: each vertex has energy $G$2. This principle is explicit in computations for the Desargues graph, Tutte–Coxeter graph, Heawood graph, Shrikhande graph, and Petersen graph, whose per-vertex energies are respectively $G$3, approximately $G$4, approximately $G$5, $G$6, and $G$7 (Nagesh et al., 16 Aug 2025).

Non-transitive graphs exhibit genuinely inhomogeneous local energy distributions. For the Frucht graph, the twelve vertex energies are

$G$8

computed by combining the spectrum with a moment-matching method based on closed walks. This illustrates how local symmetry breaking produces nontrivial but still tightly constrained variation (Nagesh et al., 16 Aug 2025).

5. Behavior under graph operations

Vertex energy is sensitive to graph surgery in a parity-dependent way. If a tree $G$9 is coalesced with a bipartite graph $A$00 by identifying a vertex $A$01 with a vertex $A$02, then for any $A$03 the effect on $A$04 depends only on the parity of the tree distance $A$05: $A$06

$A$07

This alternating energy-change theorem gives an exact sign rule for local perturbations induced by joining a tree to a bipartite graph (Arizmendi et al., 2024).

The mechanism is proved through a combination of Coulson-type integral formulas and a quasi-order on bipartite graphs defined by the coefficients $A$08. By cutting along the unique path from the coalescence vertex to $A$09, one obtains an induction in which the quasi-order alternates direction with path parity; the vertex-energy inequalities then follow from the integral representation (Arizmendi et al., 2024).

A minimal example makes the rule transparent. Coalescing two copies of $A$10 at one endpoint produces $A$11. Before joining, both vertices in $A$12 have energy $A$13. After joining, the coalesced middle vertex in $A$14 has energy $A$15, while the two vertices at distance $A$16 have energy $A$17. Distance $A$18 from the coalescence point is even and the energy increases; distance $A$19 is odd and the energy decreases (Arizmendi et al., 2024).

Related corollaries treat the reverse operation of deleting an edge and repeated coalescences at the same tree vertex. In the latter setting, the sequence of vertex energies is monotone, increasing at even distance and decreasing at odd distance, with bounds determined by the tree obtained after removing the coalescence vertex (Arizmendi et al., 2024).

6. Variants, generalizations, and broader usage

The localized construction $A$20 has been extended well beyond the adjacency matrix. For the Seidel matrix

$A$21

the vertex Seidel energy is

$A$22

where $A$23 are Seidel eigenvalues. It is invariant under Seidel switching and graph complementation, admits a Coulson-type integral representation, and for $A$24 and $A$25 takes the constant value $A$26 at every vertex (Popat et al., 19 Feb 2026).

For the Randić matrix

$A$27

the Randić energy of a vertex is $A$28. It satisfies $A$29, has a Coulson-type integral formula, obeys the universal upper bound $A$30, and among connected graphs the maximum is attained at the center of a star while the minimum is attained at a pendant vertex of a star (Gogoi et al., 26 Sep 2025).

Graphs with self-loops require a shifted definition. If $A$31 has $A$32 vertices and $A$33 self-loops, one sets $A$34, $A$35, and defines

$A$36

For the modified divisor prime graph $A$37, which places a self-loop at the vertex $A$38, this construction preserves the decomposition of total energy into vertex contributions and yields explicit formulas in the prime-power and general tensor-product cases (Makadiya et al., 6 May 2026).

The phrase “vertex energy” is also used outside spectral graph theory in ways that are mathematically unrelated. In three-dimensional biological vertex models, the relevant object is an energy functional governing cell shape, adhesion, and surface contractility rather than a graph-spectral invariant (Khan et al., 2023). In large liquid scintillator detectors such as JUNO, “vertex and energy reconstruction” refers to simultaneous estimation of interaction position and visible energy from PMT charge and time information (Qian et al., 2021, Huang et al., 2022). In many-body electronic structure, “vertex corrections” in $A$39 theory concern Hedin’s three-point function $A$40, again unrelated to graph-theoretic vertex energy (Weng et al., 2022). These parallel usages underscore that, within mathematics, the term has a precise spectral meaning, but across disciplines it is context-dependent.