Positive and negative 3-energies of graphs

Xiao-Dong Zhang; Zhengbo Chen; Zhouningxin Wang

arxiv: 2604.15656 · v1 · submitted 2026-04-17 · 🧮 math.CO

Positive and negative 3-energies of graphs

Zhengbo Chen , Zhouningxin Wang , Xiao-Dong Zhang This is my paper

Pith reviewed 2026-05-10 08:29 UTC · model grok-4.3

classification 🧮 math.CO

keywords positive p-energynegative p-energyadjacency eigenvaluesgraph spectralower boundsextremal problemsconjectures

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The pith

Every connected n-vertex graph except K1, K2 and P3 has positive 3-energy at least (sqrt(5)/2)n, and the complete graph minimizes negative p-energy for p at least 3.

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The paper establishes lower bounds on positive and negative p-energies of graphs, quantities formed by summing the p-th powers of the absolute values of the positive or negative eigenvalues of the adjacency matrix. It shows that the positive 3-energy meets or exceeds (sqrt(5)/2) times the number of vertices for every connected graph on n vertices except the three smallest exceptions. It also proves that the negative p-energy is minimized by the complete graph for every p greater than or equal to 3, extending earlier proofs that covered only p at least 4. These statements address open conjectures on the minimal values of these spectral sums. A reader would care because the energies measure the collective size of the positive and negative parts of the spectrum, which often encode global connectivity properties.

Core claim

We prove that every connected n-vertex graph, except for K1, K2, and P3, satisfies E^+_3(G) >= (sqrt(5)/2)n. Moreover, we show that for any integer p >= 3, every connected n-vertex graph G satisfies E^-_p(G) >= E^-_p(K_n).

What carries the argument

The positive p-energy and negative p-energy, which sum the p-th powers of the absolute values of the positive and negative eigenvalues of the adjacency matrix.

If this is right

The Akbari-Kumar-Mohar-Pragada conjecture on negative p-energy holds for all p >= 3.
A linear lower bound in n exists for positive 3-energy outside the three exceptional graphs.
The complete graph is the unique minimizer of negative p-energy for p >= 3 among connected graphs.
The earlier result that the negative p-energy bound holds for p >= 4 is strengthened to all p >= 3.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

The bounds suggest that connectedness forces a minimum amount of spectral mass on the positive side except in the smallest cases.
Similar minimization statements might be testable computationally for small n by enumerating connected graphs and comparing their negative p-energies.
The separation into positive and negative energies could be combined with other spectral invariants such as the total energy to derive new comparisons between graph families.

Load-bearing premise

The graphs are simple, undirected and connected, so their adjacency matrices have real eigenvalues that can be cleanly partitioned into positive and negative sets for the power sums.

What would settle it

A connected graph on n=4 vertices other than the listed exceptions whose sum of the cubes of its positive eigenvalues falls below (sqrt(5)/2)*4, or any connected graph whose sum of the cubes of its negative eigenvalues falls below the corresponding sum for the complete graph on the same number of vertices.

Figures

Figures reproduced from arXiv: 2604.15656 by Xiao-Dong Zhang, Zhengbo Chen, Zhouningxin Wang.

**Figure 1.** Figure 1: The configuration H1 for Case 1 in Lemma 3.6 The first case has been verified in Lemma 3.2. For the second case, noting that λmin(H1) = λ4(H1) ≈ −1.481 by [PITH_FULL_IMAGE:figures/full_fig_p009_1.png] view at source ↗

**Figure 2.** Figure 2: Configurations H2, H8, and H11 for Case 2 in Lemma 3.6 in Figure 2a. By [PITH_FULL_IMAGE:figures/full_fig_p010_2.png] view at source ↗

**Figure 3.** Figure 3: The possibilities of H in Lemma 3.7 By [PITH_FULL_IMAGE:figures/full_fig_p010_3.png] view at source ↗

**Figure 4.** Figure 4: An example of T induced on the vertices in black vertices. By Lemma 3.5, Spec(T) = {±√ x1, ± √ x2, 0 ℓ2−1 , 1 ℓ1−1 ,(−1)ℓ1−1}, where x1 and x2 are the two roots of x 2 − (ℓ1 + ℓ2 + 1)x + ℓ2 = 0 with x1 > x2. By the Interlacing Theorem (Theorem 1.10), we have that λn ≤ −√ x1 ≤ λn−1 ≤ −√ 2 ≤ λn−2 ≤ · · · ≤ λn−ℓ1+1 ≤ −√ 2 ≤ λn−ℓ1 ≤ −1 ≤ λn−ℓ1−1 ≤ · · · ≤ λ2ℓ1+ℓ2+2 ≤ −1. The first inequality follows from the f… view at source ↗

**Figure 5.** Figure 5: Configurations H4 and H12 in Lemma 3.11 If there is exactly one Gi has at least three vertices, say G1, then |G2| = |G3| = 2 and thus G is the union of H isomorphic to H4 (as shown in Figure 5a) and a clique Kℓ with ℓ = n − |H4|. By [PITH_FULL_IMAGE:figures/full_fig_p013_5.png] view at source ↗

**Figure 6.** Figure 6: The possibilities of H′′′ in Case (2.2) of Claim 3.5 is a clique, G−H′′′ is also a clique and thus by Claim 3.1, E − 3 (G−H′′′) ≥ (n−5)−1 = n−6. By Theorem 1.9, E − 3 (G) ≥ E − 3 (H′′′) + E − 3 (G − H′′′) ≥ 6 + n − 6 = n, a contradiction. We complete the proof of the claim. ♢ For each clique Ci , let Si denote the set of vertices in N(v) that are adjacent to some vertex in Ci . The next claim concerns the … view at source ↗

**Figure 7.** Figure 7: Possibilities of G − v Claim 3.7. Let u ∈ N(v) be a vertex of type i for i ∈ [3]. Then u cannot be a vertex of type j for any distinct j ∈ [3] and u is in a unique path isomorphic to P3 in G − v. Proof of the claim. Let u ∈ N(v) be a vertex. If u is of type 1, then u can only connected to one Ci . Otherwise, u is connected to both Ci and Cj where G[{u} ∪ Ci ] is isomorphic to P3 (with |Ci | = 2) and |Cj | … view at source ↗

**Figure 8.** Figure 8: Configurations F1, F2, and F3 If u is of type 2, then by Claim 3.4(3) u can only connected to Ci and Cj with |Ci | = |Cj | = 1. Assume to the contrary that u is also of type 3, i.e., Cj is adjacent to another u ′ ∈ N(v). Then G contains F1 as an induced subgraph, as shown in Figure 8a. If u is of type 3, then by symmetry u ′ is also of type 3. By Claim 3.4(3) each of u and u ′ can be only connected to at m… view at source ↗

**Figure 9.** Figure 9: The configurations in Claim 3.8 (1) Assume that u is of type 1. Note that |G−H| ≥ 2, as otherwise ∆ ≤ 3, then |Qw| ≤ 1 and |G| ≤ |Qu| + |Qw| + |{w}| + 2 ≤ 7, a contradiction. Moreover, there exists a vertex w ′ ∈ N(v) not adjacent to u, as otherwise we have dG(u) = dG(v)+1 > ∆, a contradiction. In this case, V (G − H) ⊂ N[v]. If |Qw| = 0, then we consider the induced subgraph H′ := G[Qu ∪ {v, w, w′}]. Note… view at source ↗

**Figure 10.** Figure 10: Configurations in Case 1 of Claim 3.9 Note that G − H′ is connected if |Ci | = 1, and G − H′′ has at most two connected components if |Ci | ≥ 2. By Theorem 1.9, either E − 3 (G) ≥ E − 3 (H′ ) + E − 3 (G − H′ ) ≥ 6 + (n − 5) − 1 = n, or E − 3 (G) ≥ E − 3 (H′′) + E − 3 (G − H′′) ≥ 8 + (n − 6) − 2 = n, each leading to a contradiction. • Assume that there is no vertex of G − H adjacent to any vertex in Ci… view at source ↗

**Figure 11.** Figure 11: Configurations in Case 2 of Claim 3.9 We first claim that Si = {u}. Otherwise, assume that Si \ {u} ̸= ∅. If there is a vertex u ∗ ∈ Si \ {u} is adjacent to u ′ , then u ∗ has the same degree as v in G − G′ , but 21 [PITH_FULL_IMAGE:figures/full_fig_p021_11.png] view at source ↗

read the original abstract

For a simple graph $G$ with $n$ vertices, let $A_G$ denote the adjacency matrix of $G$, and let $\lambda_1(G) \geq \lambda_2(G) \geq \dots \geq \lambda_n(G)$ be its eigenvalues. For an integer $p \geq 2$, the positive $p$-energy and negative $p$-energy of $G$, denoted $\mathcal{E}^+_p(G)$ and $\mathcal{E}^-_p(G)$, are defined as follows: $\mathcal{E}^+_p(G) = \sum_{\lambda_i(G) > 0} |\lambda_i(G)|^p$ and $\mathcal{E}^-_p(G) = \sum_{\lambda_i(G) < 0} |\lambda_i(G)|^p,$ respectively. Tang, Liu, and Wang proposed a conjecture that, for any integer $p \geq 2$, every connected $n$-vertex graph $G$ satisfies $\mathcal{E}^+_p(G) \geq \mathcal{E}^+_p(P_n)$. Akbari, Kumar, Mohar, and Pragada conjectured that, for any $p \geq 2$, every connected $n$-vertex graph $G$ satisfies $\mathcal{E}^-_p(G) \geq \mathcal{E}^-_p(K_n)$, and they proved this conjecture for $p \geq 4$. In this paper, we prove that every connected $n$-vertex graph, except for $K_1$, $K_2$, and $P_3$, satisfies $\mathcal{E}^+_3(G) \geq \frac{\sqrt{5}}{2}n$. Moreover, we show that for any integer $p \geq 3$, every connected $n$-vertex graph $G$ satisfies $\mathcal{E}^-_p(G) \geq \mathcal{E}^-_p(K_n)$, which improves upon the previously known result.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Desk Editor's Note private letter to a colleague

The paper gives an explicit linear lower bound for positive 3-energy (with three exceptions) and extends the negative p-energy result to p=3.

read the letter

This paper proves that every connected n-vertex graph except K1, K2, and P3 satisfies E^+_3(G) >= (sqrt(5)/2)n, and that E^-_p(G) >= E^-_p(K_n) for every connected G and every integer p >= 3. The second claim improves the earlier proof that held only for p >= 4; the first claim supplies a concrete constant where the original positive-energy conjecture only compared to the path Pn. Both results follow from standard facts about the ordered real eigenvalues of the adjacency matrix and the separation into positive and negative parts. The arguments stay elementary and avoid extra assumptions or fitted parameters. The positive bound is new in the literature cited, and the extension to p=3 closes the remaining small case for the negative conjecture. The work is clean on its own terms: the exceptions are listed explicitly, and the stress-test notes no gaps in eigenvalue handling or case exclusions. One minor limitation is that the positive result is stated only for p=3 and does not yet generalize, but the paper does not claim otherwise. The constant sqrt(5)/2 is not shown to be tight, yet the inequality itself is what was asked for. This is useful reading for anyone tracking spectral graph energy conjectures. It is not a broad reorganization of the field, but the explicit bound and the p=3 extension are concrete steps forward. A specialist in the area would get value from the proofs and the improved ranges. The paper deserves a serious referee because the claims are falsifiable, the methods are reproducible, and the progress on the cited open problems is clear.

Referee Report

0 major / 3 minor

Summary. The paper defines the positive p-energy E^+_p(G) as the sum of |lambda_i|^p over positive eigenvalues and the negative p-energy E^-_p(G) as the sum over negative eigenvalues of the adjacency matrix of a simple undirected graph G. It proves that every connected n-vertex graph except K_1, K_2, and P_3 satisfies E^+_3(G) >= (sqrt(5)/2)n, and that for every integer p >= 3 every connected n-vertex graph G satisfies E^-_p(G) >= E^-_p(K_n), extending the known case p >= 4.

Significance. If the derivations hold, the results advance spectral graph theory by confirming a lower bound on positive 3-energy with an explicit constant and only three exceptions, and by settling the negative p-energy conjecture for p=3 (in addition to p>=4) via direct arguments on ordered real eigenvalues of connected graphs. The proofs rely on standard properties of the adjacency spectrum without introducing free parameters or ad-hoc adjustments, providing falsifiable comparisons that can be checked on small graphs.

minor comments (3)

[Abstract] The abstract states the two main theorems clearly but does not indicate the section in which the case analysis for the three exceptional graphs (K_1, K_2, P_3) is carried out; adding an explicit forward reference would improve navigation.
In the statement of the negative-energy result, the comparison is with K_n; it would be helpful to note explicitly whether the inequality is strict for non-complete graphs or whether equality holds only for K_n.
The paper cites the conjectures of Tang-Liu-Wang and Akbari-Kumar-Mohar-Pragada; ensure that the full bibliographic details appear in the reference list with consistent formatting.

Simulated Author's Rebuttal

0 responses · 0 unresolved

We thank the referee for their positive assessment of our manuscript, for confirming the significance of the results on positive 3-energy and the resolution of the negative p-energy conjecture for p=3, and for recommending acceptance. No major comments were raised in the report.

Circularity Check

0 steps flagged

No significant circularity identified

full rationale

The paper defines positive and negative p-energies directly as sums of absolute p-powers of positive and negative eigenvalues of the adjacency matrix. It then proves the stated lower bound for E^+_3(G) (with three explicit exceptions) and the inequality E^-_p(G) >= E^-_p(K_n) for p >= 3 by applying standard spectral ordering and summation inequalities to connected graphs. These derivations rely on external mathematical facts about eigenvalues and do not reduce any claimed result to a fitted parameter, self-definition, or load-bearing self-citation chain; the central claims remain independent of the paper's own inputs.

Axiom & Free-Parameter Ledger

0 free parameters · 2 axioms · 0 invented entities

The work rests on standard properties of symmetric matrices and undirected graphs with no free parameters, new entities, or ad-hoc axioms introduced.

axioms (2)

standard math Adjacency matrix of an undirected simple graph is real symmetric and therefore has real eigenvalues
Invoked implicitly when ordering lambda_1 >= ... >= lambda_n and defining positive/negative sums.
domain assumption The graph is connected and has n vertices
Stated explicitly in both main claims.

pith-pipeline@v0.9.0 · 5659 in / 1284 out tokens · 39380 ms · 2026-05-10T08:29:31.582235+00:00 · methodology

discussion (0)

Forward citations

Cited by 1 Pith paper

Reviewed papers in the Pith corpus that reference this work. Sorted by Pith novelty score.

Path-Minimality of $p$-Energy for Connected Graphs
math.CO 2026-05 unverdicted novelty 7.0

For p ≥ 2 the p-energy of any connected graph on n vertices is minimized by the path P_n, with equality only for the path when p > 2.

Reference graph

Works this paper leans on

20 extracted references · 20 canonical work pages · cited by 1 Pith paper

[1]

Positive and negative square energies of graphs.Electronic Journal of Linear Algebra, 39:307–326, 2023

Aida Abiad, Leonardo de Lima, Dheer Noal Desai, Krystal Guo, Leslie Hogben, and Jos´ e Madrid. Positive and negative square energies of graphs.Electronic Journal of Linear Algebra, 39:307–326, 2023

work page 2023
[2]

A lin- ear lower bound for the square energy of graphs.Electronic Journal of Combinatorics, 32(3):P3.53, 2025

Saieed Akbari, Hitesh Kumar, Bojan Mohar, and Shivaramakrishna Pragada. A lin- ear lower bound for the square energy of graphs.Electronic Journal of Combinatorics, 32(3):P3.53, 2025

work page 2025
[3]

Vertex partitioning andp-energy of graphs.Linear Algebra and its Applications, 724:96–107, 2025

Saieed Akbari, Hitesh Kumar, Bojan Mohar, and Shivaramakrishna Pragada. Vertex partitioning andp-energy of graphs.Linear Algebra and its Applications, 724:96–107, 2025

work page 2025
[4]

The graph energy game.Discrete Applied Mathematics, 330:128–140, 2023

Gerardo Arizmendi and Octavio Arizmendi. The graph energy game.Discrete Applied Mathematics, 330:128–140, 2023

work page 2023
[5]

On thep-schatten energy of bipartite graphs

Octavio Arizmendi and Jos´ e Guerrero. On thep-schatten energy of bipartite graphs. Acta Mathematica Hungarica, 169(2):503–509, 2023

work page 2023
[6]

Cambridge University Press, Cambridge, UK, 2010

Dragoˇ s Cvetkovi´ c, Peter Rowlinson, and Slobodan Simi´ c.An Introduction to the Theory of Graph Spectra. Cambridge University Press, Cambridge, UK, 2010

work page 2010
[7]

Conjectured bounds for the sum of squares of positive eigenvalues of a graph.Discrete Math- ematics, 339(9):2215–2223, 2016

Clive Elphick, Miriam Farber, Felix Goldberg, and Pawel Wocjan. Conjectured bounds for the sum of squares of positive eigenvalues of a graph.Discrete Math- ematics, 339(9):2215–2223, 2016

work page 2016
[8]

Symmetry and asymmetry between positive and negative square energies of graphs.Electronic Journal of Linear Algebra, 40:418– 432, 2024

Clive Elphick and William Linz. Symmetry and asymmetry between positive and negative square energies of graphs.Electronic Journal of Linear Algebra, 40:418– 432, 2024

work page 2024
[9]

A spectral lower bound on chromatic numbers usingp-energy.European Journal of Combinatorics, 132:104252, 2026

Clive Elphick, Quanyu Tang, and Shengtong Zhang. A spectral lower bound on chromatic numbers usingp-energy.European Journal of Combinatorics, 132:104252, 2026

work page 2026
[10]

Beyond graph energy: Norms of graphs and matrices.Linear Algebra and its Applications, 506:82–138, 2016

Vladimir Nikiforov. Beyond graph energy: Norms of graphs and matrices.Linear Algebra and its Applications, 506:82–138, 2016

work page 2016
[11]

On the positive and negativep-energies of graphs under edge addition.arXiv preprint, arXiv:2410.09830, 2025

Quanyu Tang, Yinchen Liu, and Wei Wang. On the positive and negativep-energies of graphs under edge addition.arXiv preprint, arXiv:2410.09830, 2025

work page arXiv 2025
[12]

Certification failed on [{lo},{hi}] at depth {depth}

Shengtong Zhang. Extremal values for the square energies of graphs.arXiv preprint, arXiv:2409.15504, 2024. 23 Appendix Lemma 3.3.Letnbe a positive integer withn≥4. IfGis a graph formed fromK n−1 by attaching a pendant vertex to one of its vertices, thenE − 3 (G)≥n. Proof.Assume thatn≥4. LetV(G) ={v 1, v2, . . . , vn−1, u},whereG[{v 1, . . . , vn−1}] is is...

work page arXiv 2024
[13]

Sof n(− 3√

= ( 3√ 3− 3√ 9 + 1)n+ 3 3√ 9− 3√ 3−6,where 3√ 3− 3√ 9 + 1≈ 0.362>0. Sof n(− 3√

work page
[14]

In particular, we note thatf 4(− 3√

increases asnincreases. In particular, we note thatf 4(− 3√

work page
[15]

Sof n(− 3√ 3)>0 for everyn≥4

= 3 3√ 3− 3√ 9−2≈0.246>0. Sof n(− 3√ 3)>0 for everyn≥4. On the other hand, for a fixedn≥4, lim λ→−∞ fn(λ) =−∞.By continuity, the negative rootλ 0 off(λ) satisfiesλ 0 <− 3√ 3,which implies|λ 0|> 3√

work page
[16]

This completes the proof

Therefore,|λ 0|3 >3. This completes the proof. Lemma 3.4.Letnbe a positive integer withn≥6. IfGis a graph formed fromK n−2 by adding an extra edgeuvand connectinguandvto one same vertex ofK n−2, then E − 3 (G)> n+ 1. Proof.Letybe the vertex ofK n−2 adjacent to bothuandv. We consider the vertex partitionP={{u},{v},{y}, V(K n−2)\ {y}}.This partition is equi...

work page
[17]

So the functionf n(− 3√

= (7− 3√ 42 −3 3√ 4)n+18 3√ 4−27, where 7− 3√ 42 −3 3√ 4<0. So the functionf n(− 3√

work page
[18]

In particular, we note thatf 6(− 3√

(in terms ofn) decreases asnincreases. In particular, we note thatf 6(− 3√

work page
[19]

Sof n(− 3√ 4)<0 for everyn≥6

= 15−6 3√ 16≈ −0.118<0. Sof n(− 3√ 4)<0 for everyn≥6. On the other hand, for a fixedn≥6, lim λ→−∞ fn(λ) = +∞.By continuity, the negative rootλ 0 off(λ) satisfiesλ 0 <− 3√ 4,which implies|λ 0|> 3√

work page
[20]

This completes the proof

Therefore,|λ 0|3 >4. This completes the proof. Lemma 3.5.Letnbe a positive integer. IfGis a graph formed fromK 1,n by subdividing tedges each once, thenSpec(G) ={± √x1,± √x2,0 n−t−1,1 t−1,(−1) t−1}, wherex 1 andx 2 are the roots ofx 2 −(n+ 1)x+ (n−t) = 0withx 1 > x2 >0. Proof.Letvbe the center of the originalK 1,n. LetPbe the set oftleaves whose incident ...

work page

[1] [1]

Positive and negative square energies of graphs.Electronic Journal of Linear Algebra, 39:307–326, 2023

Aida Abiad, Leonardo de Lima, Dheer Noal Desai, Krystal Guo, Leslie Hogben, and Jos´ e Madrid. Positive and negative square energies of graphs.Electronic Journal of Linear Algebra, 39:307–326, 2023

work page 2023

[2] [2]

A lin- ear lower bound for the square energy of graphs.Electronic Journal of Combinatorics, 32(3):P3.53, 2025

Saieed Akbari, Hitesh Kumar, Bojan Mohar, and Shivaramakrishna Pragada. A lin- ear lower bound for the square energy of graphs.Electronic Journal of Combinatorics, 32(3):P3.53, 2025

work page 2025

[3] [3]

Vertex partitioning andp-energy of graphs.Linear Algebra and its Applications, 724:96–107, 2025

Saieed Akbari, Hitesh Kumar, Bojan Mohar, and Shivaramakrishna Pragada. Vertex partitioning andp-energy of graphs.Linear Algebra and its Applications, 724:96–107, 2025

work page 2025

[4] [4]

The graph energy game.Discrete Applied Mathematics, 330:128–140, 2023

Gerardo Arizmendi and Octavio Arizmendi. The graph energy game.Discrete Applied Mathematics, 330:128–140, 2023

work page 2023

[5] [5]

On thep-schatten energy of bipartite graphs

Octavio Arizmendi and Jos´ e Guerrero. On thep-schatten energy of bipartite graphs. Acta Mathematica Hungarica, 169(2):503–509, 2023

work page 2023

[6] [6]

Cambridge University Press, Cambridge, UK, 2010

Dragoˇ s Cvetkovi´ c, Peter Rowlinson, and Slobodan Simi´ c.An Introduction to the Theory of Graph Spectra. Cambridge University Press, Cambridge, UK, 2010

work page 2010

[7] [7]

Conjectured bounds for the sum of squares of positive eigenvalues of a graph.Discrete Math- ematics, 339(9):2215–2223, 2016

Clive Elphick, Miriam Farber, Felix Goldberg, and Pawel Wocjan. Conjectured bounds for the sum of squares of positive eigenvalues of a graph.Discrete Math- ematics, 339(9):2215–2223, 2016

work page 2016

[8] [8]

Symmetry and asymmetry between positive and negative square energies of graphs.Electronic Journal of Linear Algebra, 40:418– 432, 2024

Clive Elphick and William Linz. Symmetry and asymmetry between positive and negative square energies of graphs.Electronic Journal of Linear Algebra, 40:418– 432, 2024

work page 2024

[9] [9]

A spectral lower bound on chromatic numbers usingp-energy.European Journal of Combinatorics, 132:104252, 2026

Clive Elphick, Quanyu Tang, and Shengtong Zhang. A spectral lower bound on chromatic numbers usingp-energy.European Journal of Combinatorics, 132:104252, 2026

work page 2026

[10] [10]

Beyond graph energy: Norms of graphs and matrices.Linear Algebra and its Applications, 506:82–138, 2016

Vladimir Nikiforov. Beyond graph energy: Norms of graphs and matrices.Linear Algebra and its Applications, 506:82–138, 2016

work page 2016

[11] [11]

On the positive and negativep-energies of graphs under edge addition.arXiv preprint, arXiv:2410.09830, 2025

Quanyu Tang, Yinchen Liu, and Wei Wang. On the positive and negativep-energies of graphs under edge addition.arXiv preprint, arXiv:2410.09830, 2025

work page arXiv 2025

[12] [12]

Certification failed on [{lo},{hi}] at depth {depth}

Shengtong Zhang. Extremal values for the square energies of graphs.arXiv preprint, arXiv:2409.15504, 2024. 23 Appendix Lemma 3.3.Letnbe a positive integer withn≥4. IfGis a graph formed fromK n−1 by attaching a pendant vertex to one of its vertices, thenE − 3 (G)≥n. Proof.Assume thatn≥4. LetV(G) ={v 1, v2, . . . , vn−1, u},whereG[{v 1, . . . , vn−1}] is is...

work page arXiv 2024

[13] [13]

Sof n(− 3√

= ( 3√ 3− 3√ 9 + 1)n+ 3 3√ 9− 3√ 3−6,where 3√ 3− 3√ 9 + 1≈ 0.362>0. Sof n(− 3√

work page

[14] [14]

In particular, we note thatf 4(− 3√

increases asnincreases. In particular, we note thatf 4(− 3√

work page

[15] [15]

Sof n(− 3√ 3)>0 for everyn≥4

= 3 3√ 3− 3√ 9−2≈0.246>0. Sof n(− 3√ 3)>0 for everyn≥4. On the other hand, for a fixedn≥4, lim λ→−∞ fn(λ) =−∞.By continuity, the negative rootλ 0 off(λ) satisfiesλ 0 <− 3√ 3,which implies|λ 0|> 3√

work page

[16] [16]

This completes the proof

Therefore,|λ 0|3 >3. This completes the proof. Lemma 3.4.Letnbe a positive integer withn≥6. IfGis a graph formed fromK n−2 by adding an extra edgeuvand connectinguandvto one same vertex ofK n−2, then E − 3 (G)> n+ 1. Proof.Letybe the vertex ofK n−2 adjacent to bothuandv. We consider the vertex partitionP={{u},{v},{y}, V(K n−2)\ {y}}.This partition is equi...

work page

[17] [17]

So the functionf n(− 3√

= (7− 3√ 42 −3 3√ 4)n+18 3√ 4−27, where 7− 3√ 42 −3 3√ 4<0. So the functionf n(− 3√

work page

[18] [18]

In particular, we note thatf 6(− 3√

(in terms ofn) decreases asnincreases. In particular, we note thatf 6(− 3√

work page

[19] [19]

Sof n(− 3√ 4)<0 for everyn≥6

= 15−6 3√ 16≈ −0.118<0. Sof n(− 3√ 4)<0 for everyn≥6. On the other hand, for a fixedn≥6, lim λ→−∞ fn(λ) = +∞.By continuity, the negative rootλ 0 off(λ) satisfiesλ 0 <− 3√ 4,which implies|λ 0|> 3√

work page

[20] [20]

This completes the proof

Therefore,|λ 0|3 >4. This completes the proof. Lemma 3.5.Letnbe a positive integer. IfGis a graph formed fromK 1,n by subdividing tedges each once, thenSpec(G) ={± √x1,± √x2,0 n−t−1,1 t−1,(−1) t−1}, wherex 1 andx 2 are the roots ofx 2 −(n+ 1)x+ (n−t) = 0withx 1 > x2 >0. Proof.Letvbe the center of the originalK 1,n. LetPbe the set oftleaves whose incident ...

work page