---
title: Vertex Energy in Graphs
url: https://www.emergentmind.com/topics/vertex-energy
type: topic
---

# Vertex Energy in Graphs

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 \(\mathcal{E}(G)=\operatorname{Tr}(|A|)=\sum_i \mathcal{E}_G(v_i)\) [1805.05483]. 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 [1210.8273]. 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
\[
\mathcal E(G)=\sum_{j=1}^n |\lambda_j|=\operatorname{Tr}(|A|),
\]
where \(\lambda_1,\dots,\lambda_n\) are the eigenvalues of \(A\). The energy of the vertex \(v_i\) is the \(i\)-th diagonal entry of \(|A|\),
\[
\mathcal E_G(v_i)=|A|_{ii},
\]
and these diagonal entries partition the total energy [1805.05483].

If \(A=U\Lambda U^\top\) with \(U=(u_{ij})\) orthogonal and \(\Lambda=\operatorname{diag}(\lambda_1,\dots,\lambda_n)\), then
\[
\mathcal E_G(v_i)=\sum_{j=1}^n u_{ij}^2\,|\lambda_j|.
\]
Writing \(p_{ij}=u_{ij}^2\), one has \(\sum_j p_{ij}=1\), so vertex energy is the expectation of \(|\lambda|\) under the spectral measure at the basis vector \(e_i\). The same decomposition implies
\[
(A^k)_{ii}=\sum_{j=1}^n p_{ij}\lambda_j^k,
\]
so the moments of the local spectral measure are closed-walk counts at \(v_i\) [2508.11970].

This formulation extends immediately to weighted graphs by replacing the \(\{0,1\}\)-adjacency matrix with a symmetric weighted adjacency matrix. It also extends to uniformly locally finite graphs: if \(A(G)\) is the bounded self-adjoint adjacency operator on \(\ell^2(V)\), then
\[
\mathcal E_G(v)=\langle \delta_v,|A(G)|\delta_v\rangle,
\]
and local convergence of rooted graphs implies convergence of the corresponding vertex energies [1805.05483].

A persistent source of terminological ambiguity is the older extremal quantity
\[
\varphi(G)=\frac{E(G)}{n}=\frac1n\sum_{i=1}^n |\lambda_i|,
\]
called “energy per vertex” or “vertex energy” in the regular-graph literature. This is a graph-level average, not a vertex-indexed local invariant [1210.8273].

## 2. Integral representations and structural consequences

A central analytical tool is the Coulson integral formula for a vertex. If \(\phi(G;z)=\det(zI-A(G))\) and \(G-v_i\) denotes the vertex-deleted graph, then
\[
\mathcal E_G(v_i)=\frac{1}{\pi}\int_{-\infty}^{+\infty}
\left[
1-\frac{\mathbf{i}x\,\phi(G-v_i;\mathbf{i}x)}{\phi(G;\mathbf{i}x)}
\right]dx,
\]
with the integral understood in the Cauchy principal value sense [1809.07833]. This converts a diagonal entry of \(|A|\) into a one-dimensional integral involving only characteristic polynomials.

The formula arises from the resolvent entry \(\bigl((zI-A)^{-1}\bigr)_{ii}\), which admits both a spectral expansion and a determinantal expression. In spectral form, the residues encode \(\sum_j u_{ij}^2|\lambda_j|\); in determinantal form, the same quantity is expressed through the ratio \(\phi(G-v_i;z)/\phi(G;z)\). This equivalence is especially useful when eigenvectors are unavailable or cumbersome to compute [1809.07833].

For bipartite graphs the characteristic polynomial has only even-powered coefficients in the standard form
\[
\phi(G;x)=\sum_{k=0}^{\lfloor n/2\rfloor}(-1)^k b_{2k}x^{n-2k},
\qquad b_{2k}\ge 0.
\]
This coefficient structure yields comparison principles. If \(G\) is bipartite and \(\phi(G-w;x)\succeq \phi(G-v;x)\) coefficientwise in the sense \(b_{2k}(G-w)\ge b_{2k}(G-v)\) for all \(k\), then \(\mathcal E_G(w)\le \mathcal E_G(v)\). One consequence is that in any tree the energy of a leaf is strictly smaller than that of its neighbor [1809.07833].

Bipartiteness also imposes a global splitting law. If \(G\) is bipartite with parts \(V_1,V_2\), then
\[
\sum_{v\in V_1}\mathcal E_G(v)=\sum_{v\in V_2}\mathcal E_G(v)=\tfrac12\mathcal E(G).
\]
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 [1809.07833].

## 3. Bounds, inequalities, and extremal theory

The most basic localized upper bound is the McClelland-type estimate
\[
\mathcal E_G(v_i)\le \sqrt{d_i},
\]
where \(d_i\) is the degree of \(v_i\). Equality holds if and only if the component containing \(v_i\) is a star \(S_{d_i+1}\) with \(v_i\) as its center [1805.05483]. Summing these inequalities recovers the classical upper bound for total energy.

Lower bounds can be derived from short closed walks. If \(M_4(G,v_i)\) is the number of closed walks of length \(4\) based at \(v_i\), then
\[
\mathcal E_G(v_i)\ge \frac{d_i^{3/2}}{\sqrt{M_4(G,v_i)}}.
\]
Using \(M_4(G,v_i)\le d_i^2\Delta\), where \(\Delta\) is the maximum degree, one obtains the coarser but simpler inequality
\[
\mathcal E_G(v_i)\ge \sqrt{\frac{d_i}{\Delta}}.
\]
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 [1805.05483].

A more refined upper bound uses the Perron weight \(p_{i1}=u_{i1}^2\) and the spectral radius \(\lambda_1=\rho(A)\):
\[
\bigl(\mathcal E_G(v_i)-p_{i1}\lambda_1\bigr)^2
\le
(d_i-p_{i1}\lambda_1^2)(1-p_{i1}).
\]
This leads to
\[
\mathcal E_G(v_i)\le p_{i1}\lambda_1+\sqrt{(d_i-p_{i1}\lambda_1^2)(1-p_{i1})},
\]
and further bounds can be expressed in terms of maximum degree and eccentricity [1805.05483].

The alternative quantity \(\varphi(G)=E(G)/n\) has its own extremal theory. For a \(k\)-regular graph with \(k\ge 2\),
\[
\varphi(G)\le \frac{k+(k^2-k)\sqrt{k-1}}{k^2-k+1},
\]
with equality if and only if \(G\) is a disjoint union of incidence graphs of projective planes of order \(k-1\), or, for \(k=2\), a disjoint union of triangles and hexagons [1210.8273]. This result concerns average energy per vertex, not the localized invariant \(\mathcal E_G(v_i)\), 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 [1805.05483].

| Graph family | Vertex energy |
|---|---|
| Complete graph \(K_n\) | \(\mathcal E_{K_n}(v)=\dfrac{2(n-1)}{n}\) |
| Complete bipartite \(K_{r,s}\) | \(\sqrt{s/r}\) on the \(r\)-part; \(\sqrt{r/s}\) on the \(s\)-part |
| Cycle \(C_n\) | \(\dfrac{1}{n}\sum_{j=0}^{n-1}\left|2\cos\frac{2\pi j}{n}\right|\) |
| Hypercube \(Q_n\) | \(\dfrac{2}{2^n}\left\lceil\frac{n}{2}\right\rceil \binom{n}{\lceil n/2\rceil}\) |

For paths \(P_n\), the explicit formula
\[
\mathcal E_{P_n}(i)=
\sum_{j=1}^n
\frac{2}{n+1}\sin^2\!\left(\frac{ij\pi}{n+1}\right)
\left|2\cos\frac{j\pi}{n+1}\right|
\]
shows that endpoint and bulk behavior differ. As \(n\to\infty\), for fixed \(i\) one has \(\mathcal E_{P_n}(i)\to 4/\pi\), while for \(i\) growing proportionally to \(n\) one obtains \(8/(3\pi)\) [1805.05483].

In vertex-transitive graphs, symmetry enforces a uniform distribution of the total energy: each vertex has energy \(\mathcal E(G)/n\). 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 \(1.6\), approximately \(1.4\), approximately \(1.6407\), \(2.25\), and \(1.6\) [2508.11970].

Non-transitive graphs exhibit genuinely inhomogeneous local energy distributions. For the Frucht graph, the twelve vertex energies are
\[
\{1.50636,\ 1.55632,\ 1.45627,\ 1.44865,\ 1.54705,\ 1.52488,\ 1.48642,\ 1.54800,\ 1.43233,\ 1.44129,\ 1.55632,\ 1.56952\},
\]
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 [2508.11970].

## 5. Behavior under graph operations

Vertex energy is sensitive to graph surgery in a parity-dependent way. If a tree \(T\) is coalesced with a bipartite graph \(B\) by identifying a vertex \(v\in V(T)\) with a vertex \(u\in V(B)\), then for any \(w\in V(T)\) the effect on \(\mathcal E_T(w)\) depends only on the parity of the tree distance \(d_T(v,w)\):
\[
d_T(v,w)\ \text{odd}\ \Longrightarrow\ \mathcal E_T(w)>\mathcal E_{T\odot B}(\hat w),
\]
\[
d_T(v,w)\ \text{even}\ \Longrightarrow\ \mathcal E_T(w)<\mathcal E_{T\odot B}(\hat w).
\]
This alternating energy-change theorem gives an exact sign rule for local perturbations induced by joining a tree to a bipartite graph [2402.03225].

The mechanism is proved through a combination of Coulson-type integral formulas and a quasi-order on bipartite graphs defined by the coefficients \(b_{2k}(X)=|[x^{n-2k}]\phi(X;x)|\). By cutting along the unique path from the coalescence vertex to \(w\), one obtains an induction in which the quasi-order alternates direction with path parity; the vertex-energy inequalities then follow from the integral representation [2402.03225].

A minimal example makes the rule transparent. Coalescing two copies of \(P_2\) at one endpoint produces \(P_3\). Before joining, both vertices in \(P_2\) have energy \(1\). After joining, the coalesced middle vertex in \(P_3\) has energy \(\sqrt2\approx 1.414\), while the two vertices at distance \(1\) have energy \(\sqrt2/2\approx 0.707\). Distance \(0\) from the coalescence point is even and the energy increases; distance \(1\) is odd and the energy decreases [2402.03225].

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 [2402.03225].

## 6. Variants, generalizations, and broader usage

The localized construction \(|M|_{ii}\) has been extended well beyond the adjacency matrix. For the Seidel matrix
\[
S(G)=J-I-2A,
\]
the vertex Seidel energy is
\[
\mathcal E_S(v_i)=|S|_{ii}=\sum_j w_{ij}^2|\theta_j|,
\]
where \(\theta_j\) are Seidel eigenvalues. It is invariant under Seidel switching and graph complementation, admits a Coulson-type integral representation, and for \(K_n\) and \(K_{r,s}\) takes the constant value \(2(n-1)/n\) at every vertex [2602.17741].

For the Randić matrix
\[
R_{ij}=
\begin{cases}
1/\sqrt{d_i d_j}, & v_i\sim v_j,\\
0, & \text{otherwise},
\end{cases}
\]
the Randić energy of a vertex is \(\mathcal{RE}_G(v_i)=|R(G)|_{ii}\). It satisfies \(\mathcal{RE}(G)=\sum_i \mathcal{RE}_G(v_i)\), has a Coulson-type integral formula, obeys the universal upper bound \(\mathcal{RE}_G(v_i)\le 1\), 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 [2509.22539].

Graphs with self-loops require a shifted definition. If \(G\) has \(N\) vertices and \(\sigma\) self-loops, one sets \(\mu=\sigma/N\), \(\bar A=A-\mu I\), and defines
\[
\mathcal E_G(v_i)=|\bar A|_{ii}.
\]
For the modified divisor prime graph \(G^*_{Dp}(n)\), which places a self-loop at the vertex \(1\), this construction preserves the decomposition of total energy into vertex contributions and yields explicit formulas in the prime-power and general tensor-product cases [2605.04849].

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 [2302.03785]. 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 [2101.04839; 2211.16768]. In many-body electronic structure, “vertex corrections” in \(GW\) theory concern Hedin’s three-point function \(\Gamma\), again unrelated to graph-theoretic vertex energy [2212.08271]. These parallel usages underscore that, within mathematics, the term has a precise spectral meaning, but across disciplines it is context-dependent.

Source: https://www.emergentmind.com/topics/vertex-energy