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Edge-addition monotonicity of positive p-energy fails for every p >= 1
Koyar Afrasyab
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Summary
This paper disproves the conjecture that positive p-energy exhibits edge-addition monotonicity for any real exponent p >= 1. While previous work established nonmonotonicity for 1 <= p < 3, this study proves that for every real p > 2, there exist infinitely many connected graphs where adding an edge decreases the positive p-energy. The authors construct counterexamples using chains of clique blocks joined by regular bipartite graphs, providing explicit rational certificates for p=4 (38 vertices) and analyzing a 17-vertex counterexample at p=3.
Entities (8)
Relation Signals (6)
This Paper → disproves → Monotonicity for p >= 3
confidence 98% · We disprove this conjectured high-exponent extension completely: for every real p > 2 there are infinitely many connected graphs G and nonedges uv such that E+_p(G + uv) < E+_p(G).
Tang, Liu, and Wang → provednonmonotonicityfor → 1 <= p < 3
confidence 95% · Tang, Liu, and Wang then introduced positive p-energy, proved nonmonotonicity for every 1 <= p < 3
Chain of clique blocks → usedtoconstruct → Counterexamples
confidence 95% · The construction is a chain of clique blocks joined by regular bipartite graphs.
Tang, Liu, and Wang → conjecturedmonotonicityfor → p >= 3
confidence 90% · explicitly conjectured that monotonicity should hold for p >= 3.
p=4 certificate → demonstrates → Nonmonotonicity
confidence 90% · We also give a fully rational 38-vertex certificate at p = 4
Guo → proposedconjecture → Edge-addition monotonicity of positive square energy
confidence 90% · Guo conjectured that the positive square energy s+ = E+_2 should inherit the familiar edge-addition monotonicity
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Abstract
Abstract:At a 2021 AIM workshop, Guo conjectured that the positive square energy s+ = E+_2 should inherit the familiar edge-addition monotonicity of the spectral radius, rho(G + uv) >= rho(G). That conjecture was subsequently shown to fail at p = 2. Tang, Liu, and Wang then introduced positive p-energy, proved nonmonotonicity for every 1 <= p < 3, and in version 3 of their preprint (26 March 2025) explicitly conjectured that monotonicity should hold for p >= 3. We disprove this conjectured high-exponent extension completely: for every real p > 2 there are infinitely many connected graphs G and nonedges uv such that E+_p(G + uv) < E+_p(G). Together with the Tang-Liu-Wang counterexamples below 3, this shows that no exponent p >= 1 restores the spectral-radius-style monotonicity: positive p-energy can decrease under the addition of an edge for every real p >= 1. The construction is a chain of clique blocks joined by regular bipartite graphs. An equitable quotient converges to Q = I + cA(P_k). For noninteger p, a binomial-series sign argument for a fractional power of I - cA(P_k) gives the required negative endpoint entry. At integer exponents, choosing c across the first spectral threshold leaves exactly one negative eigenvalue, and path locality forces the positive spectral contribution to have negative sign. We also give a fully rational 38-vertex certificate at p = 4 and determine the complete failure interval of a fixed 17-vertex counterexample at p = 3.
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- Source: https://arxiv.org/abs/2609.13476v1
- Canonical: https://arxiv.org/abs/2609.13476v1
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Edge-addition monotonicity of positive p-energy fails for every p≥1p≥ 1 Koyar Afrasyab Abstract For a graph G with adjacency eigenvalues λi _i, define the positive p-energy by ℰp+(G)=∑λi>0λip.E_p^+(G)= _ _i>0 _i^p. At a 2021 AIM workshop, Guo conjectured that the positive square energy s+=ℰ2+s^+=E_2^+ should inherit the familiar edge-addition monotonicity of the spectral radius, ρ(G+uv)≥ρ(G)ρ(G+uv)≥ρ(G). That conjecture was subsequently shown to fail at p=2p=2. Tang, Liu, and Wang then introduced positive p-energy, proved nonmonotonicity for every 1≤p<31≤ p<3, and in version 3 of their preprint (26 March 2025) explicitly conjectured that monotonicity should hold for p≥3p≥ 3. We disprove this conjectured high-exponent extension completely: for every real p>2p>2 there are infinitely many connected graphs G and nonedges uvuv such that ℰp+(G+uv)<ℰp+(G).E_p^+(G+uv)<E_p^+(G). Together with the Tang–Liu–Wang counterexamples below 33, this shows that no exponent p≥1p≥ 1 restores the spectral-radius-style monotonicity: positive p-energy can decrease under the addition of an edge for every real p≥1p≥ 1. The construction is a chain of clique blocks joined by regular bipartite graphs. An equitable quotient converges to Q=I+cA(Pk)Q=I+cA(P_k). For noninteger p, a binomial-series sign argument for a fractional power of I−cA(Pk)I-cA(P_k) gives the required negative endpoint entry. At integer exponents, choosing c across the first spectral threshold leaves exactly one negative eigenvalue, and path locality forces the positive spectral contribution to have negative sign. We also give a fully rational 3838-vertex certificate at p=4p=4 and determine the complete failure interval of a fixed 1717-vertex counterexample at p=3p=3. 1 Introduction For a simple graph G and a real number p≥1p≥ 1, let ℰp+(G)=∑λi(G)>0λi(G)p.E_p^+(G)= _ _i(G)>0 _i(G)^p. The surrounding literature begins with classical graph energy and its Schatten-norm extensions; see, for example, Nikiforov [11, 12] and Arizmendi–Guerrero [13]. The separate positive and negative square energies were developed in connection with the Elphick–Farber–Goldberg–Wocjan conjecture [5]; subsequent work includes Abiad et al. [4], Elphick–Linz [6], and the vertex-partitioning inequalities of Akbari et al. [7]. The square-energy conjecture itself was recently resolved by Liu, Tang, and Zhang [8]. Positive and negative p-energies have also appeared in recent extremal and coloring results [9, 10]. A more direct lineage for the present problem comes from spectral-radius monotonicity. If uv∉E(G)uv∉ E(G), then Perron–Frobenius monotonicity gives ρ(G+uv)≥ρ(G)ρ(G+uv)≥ρ(G). At the 2021 AIM workshop on spectral graph and hypergraph theory, Guo proposed that the positive square energy should satisfy the analogous inequality s+(G+uv)≥s+(G),s^+(G+uv)≥ s^+(G), explicitly as a strengthening of this spectral-radius phenomenon [3]. The conjecture is false: Abiad et al. gave an explicit graph for which deleting one edge increases s+s^+ [4, Example 2.6], equivalently an edge addition that decreases it. Tang, Liu, and Wang [1] generalized the problem from s+=ℰ2+s^+=E_2^+ to positive p-energy. They constructed counterexamples for every 1≤p<31≤ p<3. Importantly, version 3 of their arXiv preprint did more than pose an open question: after reporting that their constructions did not refute monotonicity for p≥3p≥ 3, they explicitly reformulated Guo’s conjecture as Conjecture 3, asserting that ℰp+(G+uv)≥ℰp+(G)(p≥3),E_p^+(G+uv) _p^+(G) (p≥ 3), for every graph and every missing edge [2, Conjecture 3]. The published version phrases the remaining regime as an open monotonicity problem [1]. Thus the issue is a conjectured extension of spectral-radius monotonicity, not merely an isolated inequality question. We disprove that conjectured extension completely. The local monotonicity problem is distinct from the global extremal problem of minimizing p-energy or positive p-energy among connected graphs; recent path-minimality results are given in [14, 15]. Theorem 1. For every real p>2p>2, there are infinitely many connected simple graphs G and nonedges uvuv such that ℰp+(G+uv)<ℰp+(G).E_p^+(G+uv)<E_p^+(G). Corollary 1. For every real p≥1p≥ 1, positive p-energy is not monotone under edge addition. Proof. For 1≤p<31≤ p<3 this is the result of Tang–Liu–Wang [1]. For p≥3p≥ 3 it follows from Theorem 1; the overlap 2<p<32<p<3 is harmless. ∎ The key point is that the phenomenon has no upper threshold. In particular, p=4p=4 is not a monotonicity boundary, and neither is any larger exponent. This is compatible with recent global path-minimality results for positive p-energy [15]: a graph parameter can have a global minimum at a path while still decreasing under a particular local edge addition. An earlier three-block family also gives counterexamples throughout 2<p<42<p<4. To make the quantifiers explicit, that auxiliary argument has the form ∀p∈(2,4)∃K=K(p)∃N=N(p,K)∀n≥N:Δn,K(p)<0.∀ p∈(2,4)\;∃ K=K(p)\;∃ N=N(p,K)\;∀ n≥ N: _n,K(p)<0. Thus the block-ratio parameter K is chosen after fixing p; no single K works uniformly all the way up to 44. In fact the nonuniformity becomes acute as p↑4p 4 (rather than as p↓2p 2). The chain construction below makes that auxiliary family unnecessary for the main theorem. 2 The chain construction and the quotient limit Fix an integer k≥2k≥ 2. Let C1,…,CkC_1,…,C_k be pairwise disjoint cliques of order n. Between CiC_i and Ci+1C_i+1 place any d-regular bipartite graph, for 1≤i<k1≤ i<k, and place no edges between nonconsecutive blocks. Add two vertices u,vu,v, join u to every vertex of C1C_1, and join v to every vertex of CkC_k. Let Hn,d,k(0)H_n,d,k^(0) be the resulting graph, and set Hn,d,k(1)=Hn,d,k(0)+uv.H_n,d,k^(1)=H_n,d,k^(0)+uv. For every integer 0≤d≤n0≤ d≤ n, a d-regular bipartite graph with two parts of size n exists, for example as a union of d edge-disjoint perfect matchings of Kn,nK_n,n. For d≥1d≥ 1 the graph Hn,d,k(0)H_n,d,k^(0) is connected. The partition u,C1,…,Ck,v\u\,C_1,…,C_k,\v\ is equitable (see, e.g., [17] for equitable partitions and quotient spectra). In the normalized block-constant basis, and with t∈[0,1]t∈[0,1] temporarily denoting the weight of uvuv, the quotient matrix is Bt(n,d)=(0ne1⊤tne1(n−1)Ik+dA(Pk)nektnek⊤0).B_t(n,d)= pmatrix0& ne_1 &t\\ ne_1&(n-1)I_k+dA(P_k)& ne_k\\ t& ne_k &0 pmatrix. (1) The orthogonal complement of the block-constant subspace is invariant: a regular bipartite adjacency matrix sends zero-sum vectors to zero-sum vectors, clique adjacency acts as −I-I there, and the endpoint vertices couple only to block-constant vectors. The restriction to this orthogonal complement is identical for t=0t=0 and t=1t=1. Hence it cancels from every energy difference. For a real symmetric matrix Q=∑jθjwjwj⊤Q= _j _jw_jw_j and a real s, define Q+s:=∑θj>0θjswjwj⊤.Q_+^s:= _ _j>0 _j^sw_jw_j . When s=0s=0, this is the orthogonal projector onto the positive eigenspace. Lemma 1 (Quotient perturbation limit). Fix k≥2k≥ 2. Let d=dnd=d_n satisfy dn/n→c∈(0,1]d_n/n→ c∈(0,1], and put Q=Ik+cA(Pk).Q=I_k+cA(P_k). Assume that Q has no zero eigenvalue. For every fixed real p>2p>2, limn→∞ℰp+(Hn,dn,k(1))−ℰp+(Hn,dn,k(0))np−2=2p(Q+p−3)1k. _n→∞ E_p^+(H_n,d_n,k^(1))-E_p^+(H_n,d_n,k^(0))n^p-2=2p\,(Q_+^p-3)_1k. (2) Proof. Write A=A(Pk)A=A(P_k) and Qn=(1−1n)Ik+dnnA,B~t=1nBt(n,dn),Sn=0⊕Qn⊕0.Q_n= (1- 1n )I_k+ d_nnA, B_t= 1nB_t(n,d_n), S_n=0 Q_n 0. All estimates below are uniform for t∈[0,1]t∈[0,1]; the implicit constants may depend on k and Q, but not on t or n. From (1), using the Frobenius norm to bound the operator norm, sup0≤t≤1‖B~t−Sn‖≤4n+2n2≤3n. _0≤ t≤ 1\| B_t-S_n\|≤ 4n+ 2n^2≤ 3 n. (3) Moreover Qn→Q_n→ Q in operator norm. We shall use only the elementary Weyl eigenvalue bound and the Hellmann–Feynman identity for real symmetric matrices; see, e.g., Horn–Johnson [16]. Let θ1,…,θk _1,…, _k be the eigenvalues of Q. They are simple because the eigenvalues of A(Pk)A(P_k) are simple, and they are all nonzero by hypothesis. Set γ=14minminj|θj|,mini≠j|θi−θj|>0.γ= 14 \ _j| _j|, _i≠ j| _i- _j| \>0. For all sufficiently large n, the eigenvalues θj,n _j,n of QnQ_n satisfy |θj,n−θj|<γ/2| _j,n- _j|<γ/2, and the perturbation in (3) is <γ/2<γ/2. Weyl’s inequality therefore gives, for every t∈[0,1]t∈[0,1], exactly one eigenvalue μj,t(n) _j,t(n) of B~t B_t in a fixed γ-neighborhood of each θj _j, while the remaining two eigenvalues lie in a neighborhood of 00 whose radius tends to 00. In particular, supt∈[0,1]|μj,t(n)−θj|⟶0. _t∈[0,1]| _j,t(n)- _j| 0. (4) The separated eigenvalues μj,t(n) _j,t(n) are simple for all sufficiently large n. Fix j, and let (xt,yt,zt)(x_t,y_t,z_t) be a unit eigenvector of B~t B_t for μj,t(n) _j,t(n), where yt∈ℝky_t ^k. The endpoint and core equations are μj,txt _j,tx_t =n−1/2(yt)1+(t/n)zt, =n^-1/2(y_t)_1+(t/n)z_t, (5) (Qn−μj,tI)yt (Q_n- _j,tI)y_t =−n−1/2(xte1+ztek), =-n^-1/2(x_te_1+z_te_k), (6) μj,tzt _j,tz_t =n−1/2(yt)k+(t/n)xt. =n^-1/2(y_t)_k+(t/n)x_t. (7) By the definition of γ and (4), |μj,t|≥2γ| _j,t|≥ 2γ for large n. Equations (5) and (7) thus give |xt|+|zt|=O(n−1/2)|x_t|+|z_t|=O(n^-1/2) (8) uniformly in t. The matrices QnQ_n and Q are affine functions of A(Pk)A(P_k) and hence have the same orthonormal eigenvectors; write wjw_j for a unit eigenvector belonging to θj _j. By (6) and (8), the residual of yty_t in the core equation is O(n−1)O(n^-1). Expanding yty_t in the common eigenbasis of QnQ_n and Q, the spectral separation by γ implies that every component orthogonal to wjw_j is O(n−1)O(n^-1), uniformly in t. Since (8) and unit normalization give ‖yt‖=1+O(n−1)\|y_t\|=1+O(n^-1), we obtain yt=σj,t,nwj+O(n−1),|σj,t,n|=1+O(n−1),y_t= _j,t,nw_j+O(n^-1), | _j,t,n|=1+O(n^-1), (9) uniformly in t. No choice of the sign of the eigenvector is needed below, because only the product of the two endpoint coordinates occurs. Multiplying (5)–(7) by n n and using (8) gives nxt=(yt)1μj,t+O(n−1),nzt=(yt)kμj,t+O(n−1). n\,x_t= (y_t)_1 _j,t+O(n^-1), n\,z_t= (y_t)_k _j,t+O(n^-1). Together with (4) and (9), this proves the uniform limit supt∈[0,1]|nxtzt−(wj)1(wj)kθj2|⟶0. _t∈[0,1] |nx_tz_t- (w_j)_1(w_j)_k _j^2 | 0. (10) Return to the unscaled eigenvalue λj,t(n)=nμj,t(n) _j,t(n)=n _j,t(n) of Bt(n,dn)B_t(n,d_n). Since it is simple, the Hellmann–Feynman formula applied to the uvuv entry gives dtλj,t(n)=2xtzt. ddt _j,t(n)=2x_tz_t. If θj>0 _j>0, then λj,t(n)>0 _j,t(n)>0 for all t and all sufficiently large n, and (4)–(10) yield, uniformly in t, 1np−2dtλj,t(n)p 1n^p-2 ddt _j,t(n)^p =2p(λj,t(n)n)p−1(nxtzt) =2p ( _j,t(n)n )^p-1 (nx_tz_t ) ⟶2pθjp−3(wj)1(wj)k. 2p\, _j^p-3(w_j)_1(w_j)_k. (11) If θj<0 _j<0, the corresponding large eigenvalue remains negative and contributes nothing to positive p-energy. Because the convergence in (11) is uniform on the compact interval [0,1][0,1], integration in t and summation over the positive eigenvalues of Q give 2p∑θj>0θjp−3(wj)1(wj)k=2p(Q+p−3)1k2p _ _j>0 _j^p-3(w_j)_1(w_j)_k=2p\,(Q_+^p-3)_1k from the k eigenvalues of order n. It remains to justify that the two eigenvalues near the two zero eigenvalues of SnS_n are negligible. Let λ=nμλ=nμ be either of these eigenvalues and let (x,y,z)(x,y,z) be a corresponding unit eigenvector. Weyl’s bound and (3) imply supt∈[0,1]|μ|⟶0. _t∈[0,1]|μ| 0. (12) Put δ=minj|θj|>0δ= _j| _j|>0. For large n, uniformly in t and for either stray eigenvalue, ‖(Qn−μI)−1‖≤2δ.\|(Q_n-μ I)^-1\|≤ 2δ. The unscaled core equation therefore gives the explicit estimate ‖y‖≤2δnx2+z2.\|y\|≤ 2δ n x^2+z^2. (13) Writing q=x2+z2q= x^2+z^2, normalization and (13) give q≥1/2q≥ 1/2 for all sufficiently large n. The two endpoint equations, viewed as a two-dimensional vector equation, now imply |λ|q≤ny12+yk2+tq≤(2δ+1)q.|λ|q≤ n y_1^2+y_k^2+tq≤ ( 2δ+1 )q. Consequently |λ|≤2δ+1|λ|≤ 2δ+1 (14) for both stray eigenvalues, uniformly for t∈[0,1]t∈[0,1] and all sufficiently large n. Their total positive p-energy at t=0t=0 and at t=1t=1 is therefore O(1)O(1), which is o(np−2)o(n^p-2) because p>2p>2. This completes the proof of (2). ∎ 3 Noninteger exponents We first prove Theorem 1 when p>2p>2 is not an integer. Put k=⌊p⌋≥2,r=p−3.k= p ≥ 2, r=p-3. Then k−3<r<k−2.k-3<r<k-2. (15) Let A=A(Pk)A=A(P_k) and choose a rational number 0<c<1ρ(A)=12cos(π/(k+1)).0<c< 1ρ(A)= 12 (π/(k+1)). (16) Then both Q=I+cA,M=I−cAQ=I+cA, M=I-cA are positive definite. Let D=diag(1,−1,1,−1,…).D=diag(1,-1,1,-1,…). Since DAD=−ADAD=-A, one has Q=DMDQ=DMD and therefore, by functional calculus, Qr=DMrD.Q^r=DM^rD. (17) Because cρ(A)<1cρ(A)<1, the generalized binomial series converges absolutely in operator norm: Mr=(I−cA)r=∑j=0∞(rj)(−cA)j.M^r=(I-cA)^r= _j=0^∞ rj(-cA)^j. (18) The distance between the endpoints of PkP_k is k−1k-1, so (Aj)1k=0(j<k−1),(A^j)_1k=0 (j<k-1), while (Ak−1)1k=1(A^k-1)_1k=1 and (Aj)1k≥0(A^j)_1k≥ 0 for all j. For every integer j≥k−1j≥ k-1, condition (15) implies that exactly j−k+2j-k+2 factors in (rj)=r(r−1)⋯(r−j+1)j! rj= r(r-1)·s(r-j+1)j! are negative. Hence sgn((rj)(−1)j)=(−1)j−k+2+j=(−1)k.sgn\! ( rj(-1)^j )=(-1)^j-k+2+j=(-1)^k. Thus every nonzero summand in the (1,k)(1,k) entry of (18) has the same sign, and the first one is nonzero. Therefore sgn(M1kr)=(−1)k.sgn(M^r_1k)=(-1)^k. (19) Since D11Dkk=(−1)k−1D_11D_k=(-1)^k-1, equations (17) and (19) give the strict sign (Qr)1k<0.(Q^r)_1k<0. (20) As Q is positive definite, Q+r=QrQ_+^r=Q^r. Write c=a/bc=a/b in lowest terms and take n through multiples of b, with d=cnd=cn. Lemma 1 and (20) give limn→∞ℰp+(Hn,d,k(1))−ℰp+(Hn,d,k(0))np−2=2p(Qr)1k<0. _n→∞ E_p^+(H_n,d,k^(1))-E_p^+(H_n,d,k^(0))n^p-2=2p(Q^r)_1k<0. Hence every sufficiently large such n is a counterexample, and there are infinitely many. 4 Integer exponents Now let p=m≥3p=m≥ 3 be an integer, and put ϑ=πm+1. = πm+1. Choose a rational c∈(0,1)c∈(0,1) such that 12cosϑ<c,1−2ccos(2ϑ)>0. 12 <c, 1-2c (2 )>0. (21) Such a choice always exists. The first lower bound is below 11; for m=3m=3 the second inequality is automatic, while for m≥4m≥ 4 the strict inequality cosϑ>cos(2ϑ) > (2 ) leaves a nonempty interval before the second spectral threshold. The eigenvalues of Q=Im+cA(Pm)Q=I_m+cA(P_m) are θj=1+2ccosjπm+1,1≤j≤m. _j=1+2c jπm+1, 1≤ j≤ m. By (21), θm<0<θm−1 _m<0< _m-1. Thus Q has exactly one negative eigenvalue and no zero eigenvalue. Set r=m−3r=m-3. Since the endpoints of PmP_m have distance m−1m-1 and r<m−1r<m-1, (Qr)1m=0.(Q^r)_1m=0. (22) Let wmw_m be the normalized sine eigenvector corresponding to θm _m. Then wm(1)wm(m)=2m+1(−1)m+1sin2ϑ.w_m(1)w_m(m)= 2m+1(-1)^m+1 ^2 . Since θm<0 _m<0 and r=m−3r=m-3, θmrwm(1)wm(m)>0. _m^rw_m(1)w_m(m)>0. Using the spectral decomposition in (22), and recalling that θm _m is the only negative eigenvalue, gives 0=(Q+r)1m+θmrwm(1)wm(m).0=(Q_+^r)_1m+ _m^rw_m(1)w_m(m). Therefore (Q+m−3)1m<0.(Q_+^m-3)_1m<0. (23) As in the noninteger case, take n through multiples of the denominator of c and put d=cnd=cn. Lemma 1 yields limn→∞ℰm+(Hn,d,m(1))−ℰm+(Hn,d,m(0))nm−2=2m(Q+m−3)1m<0. _n→∞ E_m^+(H_n,d,m^(1))-E_m^+(H_n,d,m^(0))n^m-2=2m(Q_+^m-3)_1m<0. This proves Theorem 1 at every integer m≥3m≥ 3, and together with the previous section completes its proof for every real p>2p>2. 5 An explicit p=4p=4 certificate Although the asymptotic theorem already settles p=4p=4, a concrete exact example is useful. Take four cliques C1,C2,C3,C4≅K9C_1,C_2,C_3,C_4 K_9, join consecutive cliques completely, add vertices u,vu,v, join u to all of C1C_1 and v to all of C4C_4, and omit uvuv. Call the resulting 3838-vertex graph J0J_0, and put J1=J0+uvJ_1=J_0+uv. An equitable-partition calculation gives χJ0(x) _J_0(x) =(x+1)32(x3−25x2+46x+153)(x3−7x2−98x−9), =(x+1)^32(x^3-25x^2+46x+153)(x^3-7x^2-98x-9), (24) χJ1(x) _J_1(x) =(x+1)34(x+7)(x−14)(x2−27x+98). =(x+1)^34(x+7)(x-14)(x^2-27x+98). (25) Let −a-a be the negative root of the first cubic in (24), and let −b,−c-b,-c be the two negative roots of the second, with b>c>0b>c>0. Their relevant negative spectra are J0:−1(32 times),−a,−b,−c,J1:−1(34 times),−7.J_0: -1\ (32 times),\ -a,-b,-c, J_1: -1\ (34 times),\ -7. Direct rational evaluations give a<421250,b<69391000,c<931000.a< 421250, b< 69391000, c< 931000. (26) Indeed a is the unique positive root of a3+25a2+46a−153a^3+25a^2+46a-153, while b,cb,c are the two positive roots of z3+7z2−98z+9z^3+7z^2-98z+9; the derivatives have fixed signs on the stated isolating intervals. Adding uvuv creates no new 44-cycle and both endpoint degrees are 99. From trA4=2|E|+4∑w(d(w)2)+8C4trA^4=2|E|+4 _w d(w)2+8C_4 we obtain trA(J1)4−trA(J0)4=2+4(9+9)=74.trA(J_1)^4-trA(J_0)^4=2+4(9+9)=74. Since total fourth energy is the fourth spectral moment, ℰ4+(J1)−ℰ4+(J0)=a4+b4+c4−2329.E_4^+(J_1)-E_4^+(J_0)=a^4+b^4+c^4-2329. Using (26), ℰ4+(J1)−ℰ4+(J0)<−1281106782111500000000000<0.E_4^+(J_1)-E_4^+(J_0)<- 1281106782111500000000000<0. (27) Numerically the difference is approximately −3.8556020556-3.8556020556. 6 A 1717-vertex exact counterexample and its full failure interval Take n=m=5n=m=5. Write G0=G5,5(0)G_0=G_5,5^(0) and G1=G5,5(1)G_1=G_5,5^(1). In this case det(xI−P0) (xI-P_0) =x3−8x2−39x+20=:f0(x), =x^3-8x^2-39x+20=:f_0(x), det(xI−P1) (xI-P_1) =x3−9x2−31x+54=:f1(x), =x^3-9x^2-31x+54=:f_1(x), det(xI−M0) (xI-M_0) =(x−5)(x+1), =(x-5)(x+1), det(xI−M1) (xI-M_1) =x2−3x−9. =x^2-3x-9. Hence χG0(x) _G_0(x) =(x+1)13(x−5)f0(x), =(x+1)^13(x-5)f_0(x), (28) χG1(x) _G_1(x) =(x+1)12(x2−3x−9)f1(x). =(x+1)^12(x^2-3x-9)f_1(x). (29) Because P0P_0 and P1P_1 are real symmetric, all roots of the two cubics are real. Their determinants are negative (the cubic constant terms are positive) while their traces are 88 and 99, respectively; hence each cubic has exactly one negative root. Write those roots as −a-a and −b-b, respectively. Then a3+8a2−39a−20 a^3+8a^2-39a-20 =0, =0, (30) b3+9b2−31b−54 b^3+9b^2-31b-54 =0. =0. (31) The negative eigenvalue of M1M_1 has absolute value c=3(5−1)2.c= 3( 5-1)2. Because NG0(u)∩NG0(v)=∅N_G_0(u)∩ N_G_0(v)= , adding uvuv creates no triangle. Therefore trA(G1)3=trA(G0)3,trA(G_1)^3=trA(G_0)^3, and Δ(3):=ℰ3+(G1)−ℰ3+(G0)=−55+275+b3−a3. (3):=E_3^+(G_1)-E_3^+(G_0)=-55+27 5+b^3-a^3. (32) On [3,∞)[3,∞) the polynomials in (30)–(31) are strictly increasing, and direct evaluation at 33 and 44 places both a and b in (3,4)(3,4). Exact evaluations then give a>753200,b<454125,5<22371000.a> 753200, b< 454125, 5< 22371000. Consequently Δ(3)<−59470157109<0. (3)<- 5947015710^9<0. (33) This alone gives a short exact 1717-vertex counterexample at the endpoint p=3p=3 of the previously open range. We now characterize all p for which this same graph pair is a counterexample. Lemma 2 (Generalized Descartes rule). Let F(s)=∑i=1rciais,0<a1<a2<⋯<ar,ci≠0.F(s)= _i=1^rc_ia_i^s, 0<a_1<a_2<·s<a_r, c_i≠ 0. Then the number of distinct real zeros of F is at most the number of sign changes in the sequence (c1,…,cr)(c_1,…,c_r). Proof. Set x=esx=e^s and ri=logair_i= a_i. Multiplying by the positive factor x−r1x^-r_1 reduces the claim to a generalized polynomial f(x)=c1+∑i=2rcixρi,0<ρ2<⋯<ρr,x>0.f(x)=c_1+ _i=2^rc_ix _i, 0< _2<·s< _r, x>0. We induct on r. The derivative has coefficient signs (c2,…,cr)(c_2,…,c_r), since all ρi _i are positive. If c1c_1 and c2c_2 have opposite signs, Rolle’s theorem gives Z(f)≤Z(f′)+1,Z(f)≤ Z(f )+1, matching the additional sign change at the front. If c1c_1 and c2c_2 have the same sign, then for sufficiently small x>0x>0 both f(x)f(x) and f′(x)f (x) have that sign. If f has a first positive zero, f′f must therefore have a zero at or before it; together with the Rolle zeros between consecutive zeros of f, this gives Z(f)≤Z(f′).Z(f)≤ Z(f ). The induction hypothesis completes the proof. ∎ Let x0<X0x_0<X_0 be the two positive roots of f0f_0, and x1<X1x_1<X_1 the two positive roots of f1f_1. The positive eigenvalues are therefore x0,5,X0for G0,x1,3+352,X1for G1.\x_0,5,X_0\ G_0, \x_1, 3+3 52,X_1 \ G_1. They satisfy the strict ordering 0<x0<x1<3+352<5<X0<X1.0<x_0<x_1< 3+3 52<5<X_0<X_1. (34) Indeed, f0(0)>0>f0(1)f_0(0)>0>f_0(1), while f1(1)>0>f1(2)f_1(1)>0>f_1(2); also (3+35)/2<5(3+3 5)/2<5. Next, f0(565)=−1924125<0,f_0 ( 565 )=- 1924125<0, so X0>56/5X_0>56/5. Since f1(x)−f0(x)=−x2+8x+34f_1(x)-f_0(x)=-x^2+8x+34 is decreasing for x>4x>4 and is already negative at 56/556/5, we have f1(X0)<0f_1(X_0)<0, hence X1>X0X_1>X_0. Thus Δ(p)=−x0p+x1p+(3+352)p−5p−X0p+X1p. (p)=-x_0^p+x_1^p+ ( 3+3 52 )^p-5^p-X_0^p+X_1^p. (35) The coefficient signs in increasing order of the bases are −,+,+,−,−,+,-,+,+,-,-,+, which have exactly three sign changes. Lemma 2 therefore implies that Δ has at most three distinct real zeros. One is the trivial zero Δ(0)=0. (0)=0. We next certify signs at p=2,3,4p=2,3,4. Since one edge is added, trA(G1)2−trA(G0)2=2.trA(G_1)^2-trA(G_0)^2=2. Using the negative spectra from (28)–(29), Δ(2)=a2−b2+95−212. (2)=a^2-b^2+ 9 5-212. (36) The additional exact bound 5>559250 5> 559250 gives Δ(2)>545801106>0. (2)> 54580110^6>0. (37) Equation (33) gives Δ(3)<0 (3)<0. Finally, the standard closed-walk identity trA4=2|E|+4∑w(d(w)2)+8C4trA^4=2|E|+4 _w d(w)2+8C_4 shows that trA(G1)4−trA(G0)4=42:trA(G_1)^4-trA(G_0)^4=42: the new edge contributes 22, the two endpoint degrees are both 55, and no new 44-cycle is created. Hence Δ(4)=43+a4−b4−c4. (4)=43+a^4-b^4-c^4. (38) Using the same rational bounds as above yields Δ(4)>92911859579214316000000000000>0. (4)> 92911859579214316000000000000>0. (39) We can now state the complete characterization. Theorem 2. For the 1717-vertex pair (G0,G1)(G_0,G_1) there exist unique numbers α∈(2,3),β∈(3,4)α∈(2,3), β∈(3,4) such that, for every p≥1p≥ 1, ℰp+(G1)<ℰp+(G0)⟺α<p<β.E_p^+(G_1)<E_p^+(G_0) α<p<β. Numerically, α≈2.4322519338217608578,β≈3.0408359954191677907.α≈ 2.4322519338217608578, β≈ 3.0408359954191677907. Proof. Continuity together with (37) and (33) produces a zero α∈(2,3)α∈(2,3); continuity together with (33) and (39) produces a zero β∈(3,4)β∈(3,4). Along with p=0p=0, these are three distinct real zeros. By Lemma 2, there can be no others. The signs at 2,3,42,3,4 then force exactly the stated sign pattern on p≥1p≥ 1. ∎ 7 Discussion Corollary 1 gives a complete negative resolution of the spectral-radius-style monotonicity program above. In particular, it disproves Conjecture 3 of Tang–Liu–Wang, arXiv v3 [2]: there is no exponent p≥1p≥ 1 for which positive p-energy is universally monotone under adding a missing edge. Thus, although the spectral radius itself satisfies ρ(G+uv)≥ρ(G)ρ(G+uv)≥ρ(G), none of the positive p-energies inherits a universal analogue. The mechanism is different at integer and noninteger exponents. Away from the integers, a fractional-power binomial series has a rigid endpoint sign dictated by path distance. At an integer, that leading term vanishes by locality, and moving the quotient across its first spectral threshold creates one negative mode whose removal from the positive spectral power restores a strict negative sign. The 3838-vertex example shows that the failure at p=4p=4 can be certified with elementary rational arithmetic. The 1717-vertex example gives a different perspective: for one fixed graph pair the failure occurs only on a bounded interval of exponents, even though the family construction produces counterexamples at every exponent by changing the graph. 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