Bounds on median eigenvalues of graphs of bounded degree

Bounds on median eigen v alues of graphs of b ounded degree Hric ha A chary a ∗ Zilin Jiang ∗ , † Shengtong Zhang ‡ Abstract W e pro ve that for ev ery integer d ≥ 3 , the median eigenv alues of an y graph of maximum degree d are b ounded abov e by √ d − 1 . W e also prov e that, in three separate cases, the median eigen v alues of a graph of maxim um degree d are b ounded b elow b y − √ d − 1 : when the graph is triangle-free, when d − 1 is a p erfect square, or when d ≥ 75 . These results resolve, for all but finitely man y v alues of d , an op en problem of Mohar on median eigen v alues of graphs of maxim um degree d . As a byproduct, we establish an upp er b ound on the av erage energy of graphs of maximum degree at most d , generalizing a previous result of v an Dam, Haemers, and K o olen for d -regular graphs. 1 In tro duction Sp ectral graph theory has traditionally fo cused on extremal p ortions of the sp ectrum, particularly the largest eigenv alue and, more generally , the leading eigen v alues. In con trast, eigen v alues near the cen ter of the sp ectrum ha ve received muc h less attention. Among these, the me dian eigenvalues form a natural and imp ortant ob ject of study , motiv ated both by intrinsic sp ectral questions and b y applications arising from Hüc kel molecular orbital theory in chemistry [ 5 ]. Let G b e a simple graph of order n , and let λ 1 ≥ λ 2 ≥ · · · ≥ λ n denote the eigen v alues of its adjacency matrix A G . The me dian eigenvalues of G are the eigenv alues λ h and λ ℓ , where h = ⌊ ( n + 1) / 2 ⌋ and ℓ = ⌈ ( n + 1) / 2 ⌉ . The systematic study of median eigenv alues was initiated by F owler and Pisanski [ 2 , 3 ], who conducted computational exp erimen ts on graphs of maximum degree at most three, also known as sub cubic gr aphs . They conjectured that, apart from finitely many exceptions, every sub cubic graph has its median eigenv alues in the in terv al [ − 1 , 1] . ∗ Sc ho ol of Mathematical and Statistical Sciences, Arizona State Universit y , T emp e, AZ 85281, USA. Email: {hachary3, zilinj}@asu.edu . † Sc ho ol of Computing and Augmented In telligence, Arizona State Universit y , T empe, AZ 85281, USA. Supp orted in part b y the Simons F oundation through its T ra v el Supp ort for Mathematicians program and b y U.S. taxpa yers through NSF gran t 2451581. ‡ Departmen t of Mathematics, Stanford Universit y , Stanford, CA 94305, USA. Email: stzh1555@stanford.edu 1 Subsequen t work of Mohar [ 7 , 9 ] confirmed the conjecture of F owler and Pisanski for bip artite sub cubic graphs, and identified the Hea woo d graph, namely the incidence graph of the F ano plane, as the unique exception, with median eigen v alues ± √ 2 . Recen tly A c harya, Jeter, and Jiang [ 1 ] completely resolved the conjecture, pro ving that every subcubic graph, except for the Hea woo d graph, has its median eigenv alues in [ − 1 , 1] . With the sub cubic case settled, w e turn to graphs of maximum degree d for a general in teger d . Mohar [ 8 ] prov ed that the median eigen v alues of any graph of maximum degree d are at most √ d in absolute v alue, and conjectured that √ d could b e impro ved to √ d − 1 . Mohar also p oin ted out that the median eigenv alues of the incidence graph of a pro jective plane of order d − 1 are equal to ± √ d − 1 ; hence the b ound √ d − 1 w ould b e optimal whenever d − 1 is a prime p o wer. In this pap er, w e confirm Mohar’s conjecture for all but finitely many v alues of d . W e b egin with the upp er b ound on the median eigenv alues. Theorem 1.1. F or every inte ger d ≥ 3 , the me dian eigenvalues of any gr aph of maximum de gr e e d ar e at most √ d − 1 . W e then prov e the lo wer b ound on the median eigenv alues in three separate cases. Theorem 1.2. F or every inte ger d ≥ 3 , the me dian eigenvalues of any triangle-fr e e gr aph of maximum de gr e e d ar e at le ast − √ d − 1 . Theorem 1.3. F or every inte ger d ≥ 2 such that d − 1 is a p erfe ct squar e, the me dian eigenvalues of any gr aph of aver age de gr e e at most d ar e at le ast − √ d − 1 . Theorem 1.4. F or every inte ger d ≥ 75 , the me dian eigenvalues of any gr aph of maximum de gr e e d ar e at le ast − √ d − 1 . R emark. Theorem 1.3 replaces the maximum-degree hypothesis with an a verage-degree hypothesis. A key ingredient in the pro of of Theorem 1.4 is an upp er b ound on the aver age ener gy of a graph G on n vertices, defined by ε ( G ) = 1 n n X i =1 | λ i | . This notion is closely related to the ener gy of a graph, namely P i | λ i | , in tro duced b y Gutman [ 4 ] in connection with Hück el molecular orbital theory . The first result on av erage energy dates back to McClelland [ 6 ], who pro ved that the av erage energy of an y graph of av erage degree d is at most √ d (see Theorem 3.1). F or ev ery d -regular graph G , v an Dam, Haemers, and Koolen [ 11 ] pro ved that its a v erage energy is at most √ d − 1 + 1 / ( d + √ d − 1 ) , with equality if and only if G is a v ertex-disjoint union of incidence graphs of pro jectiv e planes of order d − 1 , or, when d = 2 , a v ertex-disjoint union of 2 triangles and hexagons 1 . Here w e generalize their result to graphs of b ounded maxim um degree, from whic h we obtain a bound on the median eigen v alues. Theorem 1.5. F or every inte ger d ≥ 3 and every gr aph G of maximum de gr e e d , its aver age ener gy satisfies ε ( G ) ≤ √ d − 1 + 1 d + √ d − 1 . Corollary 1.6. F or every inte ger d ≥ 3 , the me dian eigenvalues of any gr aph of maximum de gr e e d ar e at most √ d − 1 + 1 / ( d + √ d − 1) in absolute value. The rest of the pap er is organized as follows. In Section 2 we prov e Theorems 1.1 and 1.2. In Section 3 w e prov e Theorem 1.3, and in Section 4 w e prov e Theorem 1.5. The pro of of Theorem 1.4 is giv en in Section 5. W e conclude with some further remarks in Section 6. 2 Bounds on median eigen v alues Lemma 2.1. F or every n -vertex gr aph G of aver age de gr e e d , its eigenvalues λ 1 , . . . , λ n satisfy X i λ i = 0 , X i λ 2 i = dn, X i λ 3 i ≥ 0 , X i λ 4 i ≥  2 d 2 − d  n, and mor e over P i λ 3 i = 0 if and only if G is triangle-fr e e. Pr o of. Notice that P i λ k i = tr ( A k G ) counts the n umber of closed walks of length k in G . Clearly , P i λ i = 0 , and P i λ 2 i = dn , since this quantit y coun ts each edge of G t wice. The sum P i λ 3 i coun ts six times the n umber of triangles in G , and is therefore non-negative; moreov er, it is equal to 0 if and only if G is triangle-free. Finally , b y considering only the closed w alks of the form ababa , abaca , or abcba , we obtain the low er b ound via the Cauc hy–Sc h warz inequalit y: X i λ 4 i ≥ X i d 2 i + d i ( d i − 1) = X i 2 d 2 i − d i ≥  2 d 2 − d  n, where ( d 1 , . . . , d n ) denotes the degree sequence of G . R emark. Our b ound on P i λ 4 i is inspired by v an Dam, Haemers, and K o olen [ 11 , page 125], where they pro ve the same b ound for d -regular graphs. Pr o of of The or em 1.1. Set ε 0 = √ d − 1 . Consider the magic p olynomial f defined by f ( x ) = ( α − x )( x + ε 0 ) 2 ( x + d ) , 1 In [ 11 , Theorem 1.1], the bound on the a verage energy for d -regular graphs is stated as ( d + ( d 2 − d ) √ d − 1 ) / ( d 2 − d + 1) , which simplifies to √ d − 1 + 1 / ( d + √ d − 1) . 3 − d − ε 0 ε 0 d α Figure 1: The graph of the magic p olynomial. where α ∈ R is chosen so that f ( ε 0 ) = f ( d ) . Solving this equation yields α = d 2 + 2 dε 0 + 2 ε 2 0 d + 2 ε 0 . Since f ( − d ) = f ( − ε 0 ) = 0 , f ( ε 0 ) = f ( d ) , and α > d , it follows that the graph of the quartic p olynomial f has the shape shown in Figure 1. More precisely , f ( x ) ≥ 0 for x ∈ [ − d, d ] and f ( x ) > f ( d ) for x ∈ ( ε 0 , d ) . W e denote the co efficien ts of f ( x ) b y f ( x ) = − x 4 + c 3 x 3 + c 2 x 2 + c 1 x + c 0 , and, b y Viète’s form ulas, c 3 = α − ( d + 2 ε 0 ) < 0 and c 2 = α ( d + 2 ε 0 ) − 2 dε 0 − ε 2 0 = d 2 + ε 2 0 = d 2 + d − 1 . Let G b e an n -v ertex graph of maximum degree d , and let λ 1 , . . . , λ n b e the eigenv alues of G . W e adopt a probabilistic point of view: let λ b e c hosen uniformly at random from { λ 1 , . . . , λ n } . Using Lemma 2.1 and the fact that c 3 < 0 , we estimate the exp ectation of f ( λ ) as follows: E f ( λ ) = − 1 n X i λ 4 i + c 3 n X i λ 3 i + c 2 n X i λ 2 i + c 1 n X i λ i + c 0 ≤ − d (2 d − 1) + c 2 d + c 0 , (1) whic h, as a quadratic function of d , is increasing on ( −∞ , (1 + c 2 ) / 4] . Since c 2 = d 2 + d − 1 ≥ 4 d − 1 for d ≥ 3 , we hav e d ≤ d ≤ (1 + c 2 ) / 4 , and therefore E f ( λ ) ≤ − d (2 d − 1) + c 2 d + c 0 . W e no w consider the random v ariable µ with the following distribution: Pr( µ = − d ) = Pr( µ = d ) = 1 2( d 2 − d + 1) and Pr( µ = − ε 0 ) = Pr( µ = ε 0 ) = d 2 − d 2( d 2 − d + 1) . 4 A direct computation shows that E µ = 0 , E µ 2 = d, E µ 3 = 0 , E µ 4 = d (2 d − 1) . (2) Using the facts that f ( − d ) = f ( − ε 0 ) = 0 and f ( ε 0 ) = f ( d ) , we hav e E f ( λ ) ≤ − d (2 d − 1) + c 2 d + c 0 (2) = E f ( µ ) = Pr( µ = ε 0 or µ = d ) f ( d ) = 1 2 f ( d ) . On the other hand, since | λ i | ≤ d for all i , f ( x ) ≥ 0 for x ∈ [ − d, d ] , and f ( x ) > f ( d ) for x ∈ ( ε 0 , d ) , w e hav e E f ( λ ) ≥ Pr( ε 0 < λ ≤ d ) f ( d ) . Therefore Pr( ε 0 < λ ≤ d ) ≤ 1 / 2 , and equalit y can hold only if Pr( λ = d ) = 1 / 2 . W e claim that Pr ( λ = d ) < 1 / 2 . Since d is the largest eigen v alue of a connected comp onen t of G if and only if that component is d -regular, the P erron–F rob enius theorem implies that the m ultiplicity of d as an eigenv alue of G is at most n/ ( d + 1) , and hence Pr( λ = d ) < 1 / 2 . Finally , w e must hav e the strict inequality Pr ( ε 0 < λ ≤ d ) < 1 / 2 , whic h implies that the median eigen v alues of G are at most ε 0 = √ d − 1 . R emark. The incidence graph H of a pro jectiv e plane of order d − 1 has eigenv alues ± d with m ultiplicity 1 each, and eigenv alues ± √ d − 1 with m ultiplicity d 2 − d eac h. The random v ariable µ can b e interpreted as b eing chosen uniformly at random from the eigenv alues of H . One migh t b e tempted to apply the same strategy to obtain a lo w er b ound on the median eigen v alues b y considering the exp ectation E f ( − λ ) instead of E f ( λ ) , as this w ould b ound the probabilit y Pr ( − d ≤ λ < − ε 0 ) . Ho wev er, the term ( c 3 /n ) P i λ 3 i in (1) w ould b ecome − ( c 3 /n ) P i λ 3 i , whic h is no longer guaran teed to be non-p ositiv e. Nev ertheless, the triangle-free assumption circum ven ts this difficulty . Pr o of of The or em 1.2. The pro of is nearly identical to that of Theorem 1.1, except that w e consider E f ( − λ ) instead of E f ( λ ) and modify the probabilistic argumen t accordingly . Since G is triangle-free, Lemma 2.1 gives P i λ 3 i = 0 , so (1) remains v alid. 3 Lo w er b ound when d − 1 is a p erfect square T o prov e Theorem 1.3, we use McClelland’s upp er b ound on av erage energy [ 6 ], together with a c haracterization of the extremal graphs. Theorem 3.1 (McClelland [ 6 ]) . F or every gr aph G with aver age de gr e e at most d , its aver age ener gy satisfies ε ( G ) ≤ √ d with e quality if and only if G is an empty gr aph or a matching, and d is e qual to 0 or 1 r esp e ctively. 5 Pr o of. Since P i λ 2 i ≤ dn by Lemma 2.1, using the Cauc hy–Sc h warz inequality , w e hav e ε ( G ) = 1 n X i | λ i | ≤ s 1 n X i λ 2 i ≤ √ d. Equalit y holds if and only if | λ i | = √ d for all i , and d is equal to the av erage degree of G . Supp ose that all eigenv alues of G ha ve the same absolute v alue. Let C b e an arbitrary connected comp onen t of G . By the Perron–F robenius theorem, the largest eigenv alue of C has multiplicit y 1 . If all eigenv alues of C ha ve the same absolute v alue and still sum to 0 , then C m ust hav e order at most 2 ; that is, C is either a single vertex or a single edge. Since this holds for ev ery connected comp onen t of G , the graph G is either an empty graph or a matc hing. The main idea in the pro ofs of Theorem 1.3 is to com bine av erage-energy estimates with an in tegrality argument. W e now present the pro of of Theorem 1.3. Pr o of of The or em 1.3. Fix an integer d ≥ 2 suc h that d − 1 is a p erfect square. Let G b e a graph of a verage degree at most d , and let ε = ε ( G ) b e the av erage energy of G . Set ε 0 = √ d − 1 ∈ N + and ε 1 = √ d. W e shall rep eatedly use the iden tity ( ε 1 − ε 0 )( ε 1 + ε 0 ) = ε 2 1 − ε 2 0 = 1 . (3) Assume for the sake of contradiction that the lo wer median eigen v alue λ ℓ of G is less than − ε 0 . Let I denote the set of indices i suc h that λ i  = − ε 0 . Since Q i ∈ I ( x − λ i ) ∈ Z [ x ] , w e deduce that Q i ∈ I ( ε 0 + λ i ) is a nonzero integer, and in particular, B := Y i ∈ I | ε 0 + λ i | ≥ 1 . (4) W e shall derive a contradiction by showing that the last inequality fails. W e partition I in to tw o subsets J and K as follows: J = { i : λ i < − ε 0 } and K = { i : λ i > − ε 0 } . Notice that J ⊇ { ℓ, . . . , n } , and in particular, | J | ≥ n/ 2 . (5) This partition of I = J ∪ K allo ws us to factor B as B J := Y i ∈ J | ε 0 + λ i | and B K := Y i ∈ K | ε 0 + λ i | . 6 W e first estimate B J . By the inequality of arithmetic and geometric means, we hav e B J ≤ 1 | J | X i ∈ J | ε 0 + λ i | ! | J | . (6) Since λ i < − ε 0 < 0 for all i ∈ J , and P i λ i = 0 by Lemma 2.1, we ha ve X i ∈ J | ε 0 + λ i | = − X i ∈ J ( λ i + ε 0 ) =      X i ∈ J λ i      − ε 0 | J | = 1 2      X i ∈ J λ i      +      X i / ∈ J λ i      ! − ε 0 | J | ≤ 1 2 X i | λ i | − ε 0 | J | = εn 2 − ε 0 | J | (5) ≤ ( ε − ε 0 ) | J | . Substituting this into (6), we obtain B J ≤ ( ε − ε 0 ) | J | < ( ε 1 − ε 0 ) | J | , (7) where the last inequality follows from Theorem 3.1. No w we turn to B K . By the inequality of quadratic and geometric means, we hav e B K ≤ 1 | K | X i ∈ K ( ε 0 + λ i ) 2 ! | K | / 2 . (8) W e estimate tw o differen t linear combinations of S J := X i ∈ J ( ε 0 + λ i ) 2 and S K := X i ∈ K ( ε 0 + λ i ) 2 . F or the first linear combination, since P i λ i = 0 and P i λ 2 i = dn = ε 2 1 n b y Lemma 2.1, w e ha ve S J + S K ≤ X i ( ε 0 + λ i ) 2 = ε 2 0 n + 2 ε 0 X i λ i + X i λ 2 i = ( ε 2 0 + ε 2 1 ) n (5) ≤ ( ε 2 0 + ε 2 1 ) · 2 | J | . (9) F or the second linear com bination, applying the inequality of arithmetic and geometric means, w e ha ve ( ε 1 + ε 0 ) 4 S J + S K ≥ | I | Y i ∈ J ( ε 1 + ε 0 ) 4 ( ε 0 + λ i ) 2 · Y i ∈ K ( ε 0 + λ i ) 2 ! 1 / | I | (4) ≥ | I |  ( ε 1 + ε 0 ) 4 | J |  1 / | I | = | I | 2 | J |  ( ε 1 + ε 0 ) 2  2 | J | | I | · 2 | J | . One c hecks that the function x 7→ a x /x is increasing on [1 / ln a, ∞ ) . As ( ε 1 + ε 0 ) 2 ≥  √ 2 + 1  2 > e and 2 | J | / | I | ≥ 1 by (5), we hav e ( ε 1 + ε 0 ) 4 S J + S K ≥ ( ε 1 + ε 0 ) 2 · 2 | J | . (10) 7 Com bining (9) and (10), we can estimate S K as follo ws: S K ≤ ( ε 1 + ε 0 ) 4  ε 2 1 + ε 2 0  − ( ε 1 + ε 0 ) 2 ( ε 1 + ε 0 ) 4 − 1 · 2 | J | (3) = ( ε 1 + ε 0 ) 2 ( ε 2 1 + ε 2 0 ) − ( ε 1 + ε 0 ) 2 ( ε 1 − ε 0 ) 2 ( ε 1 + ε 0 ) 2 − ( ε 1 − ε 0 ) 2 · 2 | J | = ( ε 1 + ε 0 ) 2 | J | . Substituting this into (8), we obtain B K ≤  ( ε 1 + ε 0 ) 2 | J | | K |  | K | / 2 =  ( ε 1 + ε 0 ) 2 | K | / | J |  | K | / | J | ! | J | / 2 One c hecks that the function x 7→ ( a/x ) x is increasing on (0 , a/e ) . As ( ε 1 + ε 0 ) 2 > e and | K | / | J | ≤ 1 b y (5), we hav e B K ≤  ( ε 1 + ε 0 ) 2  | J | / 2 = ( ε 1 + ε 0 ) | J | . (11) Com bining (7) and (11), we obtain B J B K < ( ε 1 − ε 0 ) | J | ( ε 1 + ε 0 ) | J | (3) = 1 , whic h contradicts (4). 4 Upp er b ound on a v erage energy T o pro ve Theorem 1.5, the key analytic ingredient is a carefully chosen p olynomial with no x 3 term. Lemma 4.1. L et d b e an inte ger at le ast 2 , and define f ( x ) = ( x − d )  x − √ d − 1  2  x + d + 2 √ d − 1  . Then, for every gr aph G of maximum de gr e e d , the sp e ctrum { λ i } and the aver age ener gy ε ( G ) of G satisfy 1 n X i f ( | λ i | ) ≥ 2 √ d − 1  d + √ d − 1  2  ε ( G ) − √ d − 1 − 1 d + √ d − 1  . (12) Pr o of. Expanding f ( x ) yields x 4 − α x 2 + β x − γ , where the co efficien ts are defined b y α = d 2 + 3 d − 3 + 2 d √ d − 1 , β = 2 √ d − 1  d + √ d − 1  2 , γ = d ( d − 1)( d + 2 √ d − 1) . The left hand side of Lemma 4.1 can be rewritten as 1 n X i λ 4 i − α · 1 n X i λ 2 i + β ε ( G ) − γ , whic h by Lemma 2.1 is at least 2 d 2 − d − αd + β ε ( G ) − γ , (13) 8 where d is the av erage degree of G . Notice that the quadratic function x 7→ 2 x 2 − (1 + α ) x is decreasing on the in terv al ( −∞ , (1 + α ) / 4) . One c hecks that (1 + α ) / 4 > d for d ≥ 3 . Since d ≤ d < (1 + α ) / 4 , (13) is at least 2 d 2 − d − αd + β ε ( G ) − γ , which is equal to the righ t-hand side of (12). Pr o of of The or em 1.5. Let f b e the p olynomial defined in Lemma 4.1. Since | λ i | ≤ d for every i , the definition of f gives f ( | λ i | ) ≤ 0 . Substituting this in to (12) completes the pro of. The b ound on the median eigen v alues in Corollary 1.6 follo ws from the follo wing result. Theorem 4.2 (Theorem 3.1 of Li, Li, Shi, and Gutman [ 5 ]) . F or every gr aph, the me dian eigenvalues ar e at most the aver age ener gy of the gr aph in absolute value. Pr o of of Cor ol lary 1.6. It follo ws immediately from Theorems 1.5 and 4.2. Although the original statemen t of Theorem 4.2 is for connected graphs, its pro of do es not require this assumption. F or completeness, we provide a streamlined pro of. Pr o of of The or em 4.2. Let G b e a graph on n v ertices, let λ 1 ≥ · · · ≥ λ n b e its eigenv alues, and let λ h and λ ℓ b e the median eigen v alues of G . Since P i λ i = 0 by Lemma 2.1, we ha ve ε ( G ) = 2 n X i : λ i ≥ 0 λ i = − 2 n X i : λ i ≤ 0 λ i . Recall that λ h and λ ℓ are the median eigen v alues of G , where h = ⌊ ( n + 1) / 2 ⌋ and ℓ = ⌈ ( n + 1) / 2 ⌉ . W e first prov e that λ h ≤ ε ( G ) . If λ h ≥ 0 , then h ≥ n/ 2 , and hence ε ( G ) = 2 n X i : λ i ≥ 0 λ i ≥ 2 hλ h n ≥ λ h . Otherwise, if λ h < 0 , then λ h ≤ ε ( G ) holds trivially . The pro of of λ ℓ ≥ − ε ( G ) is analogous. 5 Lo w er b ound when d is at least 75 Finally , we present the pro of of Theorem 1.4, which reuses a couple of ideas from the pro of of Theorem 1.3. The pro of of the following optimization problem is deferred to Appendix A. Prop osition 5.1. Fix an inte ger d ≥ 75 . Set ε 0 = √ d − 1 and ε 1 = √ d − 1 + 1 d + √ d − 1 . Then ther e exists δ ∈ ( ε 0 , d ) such that the maximum value of (2 ε 0 ) x + y + z  ε 0 + ε 1 2 ε 0  ( ε − ε 0 ) x  α ( δ ) · ( ε 1 − ε ) y  y / 2 ( d − ε 0 ) z 9 under the c onstr aint that ε 0 ≤ ε ≤ ε 1 , x ≥ 1 2 , 0 ≤ y ≤ 1 2 − z , 0 ≤ z ≤ 1 ( δ − ε 0 ) 2 , is strictly less than 1 , wher e α ( δ ) is define d by α ( δ ) = 2 ε 0 ( d + ε 0 ) 2 ( d − δ )( d + 2 ε 0 + δ ) . Pr o of of The or em 1.4. Fix an integer d ≥ 75 . Let G b e a graph of maxim um degree d , let λ 1 ≥ λ 2 ≥ · · · ≥ λ n b e the eigenv alues of its adjacency matrix A G , let λ h and λ ℓ b e the median eigen v alues of G , and let ε b e the a verage energy of G . W e define the constan ts ε 0 and ε 1 as in Prop osition 5.1. Assume for the sak e of contradiction that the lo w er median eigenv alue λ ℓ is less than − ε 0 . By Theorem 4.2, we are done as so on as ε ≤ ε 0 . Hereafter, in view of Theorem 1.5, w e assume that ε 0 ≤ ε ≤ ε 1 . (14) Let I denote the set of indices i suc h that | λ i |  = ε 0 . Since Q i ∈ I ( x − λ 2 i ) ∈ Z [ x ] , w e deduce that Q i ∈ I  ε 2 0 − λ 2 i  is a nonzero integer. Consequently , Y i ∈ I   ε 2 0 − λ 2 i   ≥ 1 . (15) W e shall deriv e a contradiction b y showing that the last inequality fails whenev er d ≥ 75 . W e factor the ab o v e pro duct as A := Y i ∈ I | ε 0 + | λ i || and B := Y i ∈ I | ε 0 − | λ i || . W e first estimate A . Applying the inequalit y of arithmetic and geometric means, we obtain A ≤ ε 0 + 1 | I | X i ∈ I | λ i | ! | I | . (16) Since λ ℓ < − ε 0 , w e hav e { ℓ, . . . , n } ⊆ I , and so | I | ≥ n/ 2 . Since ε = 1 n X i ∈ I | λ i | + X i / ∈ I | λ i | ! = 1 n X i ∈ I | λ i | + ( n − | I | ) ε 0 ! , w e obtain ε 0 + 1 | I | X i ∈ I | λ i | = ε 0 + 1 | I | ( nε − ( n − | I | ) ε 0 ) = 2 ε 0 + n | I | ( ε − ε 0 ) . One chec ks that the function x 7→ (1 + 1 /x ) x is increasing on [0 , ∞ ) . Since | I | ≤ n , w e hav e the estimate A ≤  2 ε 0 + n | I | ( ε − ε 0 )  | I | = (2 ε 0 ) | I |  1 + ε − ε 0 2 ε 0 · n | I |  | I | ≤ (2 ε 0 ) | I |  1 + ε − ε 0 2 ε 0  n (14) ≤ (2 ε 0 ) | I |  1 + ε 1 − ε 0 2 ε 0  n = (2 ε 0 ) | I |  ε 0 + ε 1 2 ε 0  n . (17) 10 W e no w turn to B . Let J = { i : λ i < − ε 0 } . Notice that J ⊆ { ℓ, . . . , n } , and in particular, J ⊆ I and | J | ≥ n/ 2 . (18) W e estimate part of B corresponding to J using the inequalit y of arithmetic and geometric means: B J := Y i ∈ J | ε 0 − | λ i || = Y i ∈ J ( | λ i | − ε 0 ) ≤ 1 | J | X i ∈ J | λ i | − ε 0 ! | J | . Since λ i with i ∈ J ha ve the same sign and P i λ i = 0 by Lemma 2.1, we ha ve X i ∈ J | λ i | =      X i ∈ J λ i      = 1 2      X i ∈ J λ i      +      X i / ∈ J λ i      ! ≤ 1 2 X i | λ i | = 1 2 nε ≤ | J | ε, whic h implies that B J ≤ ( ε − ε 0 ) | J | . (19) W e shall further divide I \ J into tw o subsets K and L as follows: K = { i ∈ I \ J : | λ i | ≤ δ } and L = { i ∈ I \ J : | λ i | > δ } , where δ ∈ ( ε 0 , d ) is a parameter to b e determined later, and it dep ends only on d . T o estimate the part of B corresp onding to K , we apply the inequalit y of arithmetic and geometric means: B K := Y i ∈ K | ε 0 − | λ i || = Y i ∈ K ( ε 0 − | λ i | ) 2 ! 1 / 2 ≤ 1 | K | X i ∈ K ( ε 0 − | λ i | ) 2 !! | K | / 2 . (20) Define g ( x ) as follows: g ( x ) = ( d − x )( ε 0 − x ) 2 ( x + d + 2 ε 0 ) . Clearly g ( x ) = − f ( x ) , where f ( x ) is defined in Lemma 4.1, and g ( x ) ≥ 0 for all x ∈ [0 , d ] . By Lemma 4.1, we hav e 1 n X i g ( | λ i | ) ≤ 2 ε 0 ( d + ε 0 ) 2 ( ε 1 − ε ) . (21) Observ e that the quadratic function x 7→ ( d − x )( x + d + 2 ε 0 ) is decreasing on (0 , d ) . F or eac h i ∈ K , since 0 ≤ | λ i | ≤ δ < d , we hav e ( ε 0 − | λ i | ) 2 = g ( | λ i | ) ( d − | λ i | )( | λ i | + d + 2 ε 0 ) ≤ g ( | λ i | ) ( d − δ )( δ + d + 2 ε 0 ) . Summing o ver i ∈ K , w e obtain X i ∈ K ( ε 0 − | λ i | ) 2 ≤ P i ∈ K g ( | λ i | ) ( d − δ )( δ + d + 2 ε 0 ) ≤ P i g ( | λ i | ) ( d − δ )( δ + d + 2 ε 0 ) (21) ≤ 2 ε 0 ( d + ε 0 ) 2 ( d − δ )( δ + d + 2 ε 0 ) ( ε 1 − ε ) n = α ( δ ) · ( ε 1 − ε ) · n, 11 where α ( δ ) is defined in Prop osition 5.1. Substituting the last estimate in to (20), w e obtain that B K ≤  α ( δ ) · ( ε 1 − ε ) n | K |  | K | / 2 . (22) T o estimate part of B corresponding to L , w e simply use | λ i | ≤ d to get that B L ≤ ( d − ε 0 ) | L | . (23) W e also estimate | L | here. Using the fact that P i λ 2 i equals the degree sum of G , whic h is b ounded b y nd , w e obtain that X i ∈ L ( ε 0 − | λ i | ) 2 ≤ X i ( ε 0 − | λ i | ) 2 ≤ nε 2 0 − 2 ε 0 X i | λ i | + X i λ 2 i ≤ n (2 d − 1 − 2 ε 0 ε ) (14) ≤ n (2 d − 1 − 2 ε 2 0 ) = n. Since | λ i | > δ > ε 0 for i ∈ L , the ab o v e inequality implies that | L | ≤ n ( δ − ε 0 ) 2 . (24) Set x = | J | /n , y = | K | /n , and z = | L | /n . Then | I | /n = x + y + z ≤ 1 . In view of (14) , (18) , and (24) , the parameters ε, x, y , z satisfy the constrain ts in Prop osition 5.1. Combining the b ounds (17), (19), (22), and (23), we obtain that Y i ∈ I   d − 1 − λ 2 i   ! 1 /n = ( AB J B K B L ) 1 /n ≤ (2 ε 0 ) x + y + z  ε 0 + ε 1 2 ε 0  ( ε − ε 0 ) x  α ( δ ) · ( ε 1 − ε ) y  y / 2 ( d − ε 0 ) z . A ccording to Prop osition 5.1, there exists δ ∈ ( ε 0 , d ) such that the right hand side of the last inequalit y is strictly less than 1 , which contradicts (15). 6 Concluding remarks Our results confirm Mohar’s conjecture for all but 64 v alues of d — the exceptions are d ∈ { 4 , . . . , 74 } \ { 5 , 10 , 17 , 26 , 37 , 50 , 65 } . In particular, it would b e interesting to settle his conjecture for d = 4 . Conjecture 6.1. The me dian eigenvalues of any gr aph of maximum de gr e e 4 ar e at le ast − √ 3 . Inspired by the result of [ 1 ] on sub cubic graphs, it is natural to conjecture that the b ound can b e strengthened by excluding the incidence graph of the pro jective plane of order 3 . Conjecture 6.2. The me dian eigenvalues of any gr aph of maximum de gr e e 4 , exc ept for a vertex- disjoint union of incidenc e gr aphs of pr oje ctive planes of or der 3 , ar e at most √ 2 in absolute value. 12 F or graphs of maxim um degree 5 , Theorems 1.1 and 1.3 already imply that the median eigen v alues are b ounded in absolute v alue by 2 . W e note that the incidence graph of the pro jective plane of order 4 is not the only connected 5 -regular graph attaining this b ound. Indeed, the Cayley graph of the group Z / 12 Z generated b y {± 3 , ± 4 , 6 } is a connected 5 -regular graph whose median eigenv alues are − 2 and +1 . A significant related result is due to Mohar and T ayfeh-Rezaie [ 10 ], who prov ed that for ev ery in teger d ≥ 3 , the median eigenv alues of any bip artite graph G of maximum degree d are at most √ d − 2 in absolute v alue, unless G is the vertex-disjoin t union of incidence graphs of pro jective planes of order d − 1 , in which case the median eigen v alues are ± √ d − 1 . It is natural to relax the condition on maxim um degree to one on a v erage degree. F rom Theorems 3.1 and 4.2, we obtain the follo wing b ound on the median eigenv alues. Corollary 6.3. The me dian eigenvalues of any gr aph of aver age de gr e e d ar e at most √ d in absolute value. Theorem 1.2 suggests a stronger conjecture that impro ves the b ound from √ d to √ d − 1 . Conjecture 6.4. F or every r e al numb er d ≥ 2 , the me dian eigenvalues of any gr aph of aver age de gr e e at most d ar e at most √ d − 1 in absolute value. W e p oin t out that the conjecture fails for d ∈ [1 , 2) . F or example, the vertex-disjoin t union of a triangles and b single edges has av erage degree (6 a + 2 b ) / (3 a + 2 b ) , but the median eigen v alues are 1 in absolute v alue. A c kno wledgemen ts The authors gratefully ac knowledge the Simons Laufer Mathematical Sciences Institute (SLMath) for supp orting tra vel and accommo dation during the program Algebr aic and Analytic Metho ds in Combinatorics . References [1] Hric ha Ac hary a, Benjamin Jeter, and Zilin Jiang. Median eigenv alues of sub cubic graphs, 2025. arXiv:2502.13139 [math.CO] . [2] P atrick W. F o wler and T omaž Pisanski. HOMO-LUMO maps for chemical graphs. MA TCH Commun. Math. Comput. Chem. , 64(2):373–390, 2010. [3] P atrick W. F owler and T omaž Pisanski. HOMO-LUMO maps for fullerenes. A cta Chim. Slov. , 57(3):513–517, 2010. 13 [4] Iv an Gutman. The energy of a graph. Ber. Math.-Statist. Sekt. F orschungsz. Gr az , 103:1–22, 1978. [5] Xueliang Li, Yiy ang Li, Y ongtang Shi, and Iv an Gutman. Note on the HOMO-LUMO index of graphs. MA TCH Commun. Math. Comput. Chem. , 70(1):85–96, 2013. [6] Bernard J McClelland. Properties of the latent ro ots of a matrix: the estimation of π -electron energies. J. Chem. Phys. , 54(2):640–643, 1971. [7] Bo jan Mohar. Median eigen v alues of bipartite planar graphs. MA TCH Commun. Math. Comput. Chem. , 70(1):79–84, 2013. [8] Bo jan Mohar. Median eigenv alues and the HOMO-LUMO index of graphs. J. Combin. The ory Ser. B , 112:78–92, 2015. arXiv:1401.1865 [math.CO] . [9] Bo jan Mohar. Median eigenv alues of bipartite sub cubic graphs. Combin. Pr ob ab. Comput. , 25(5):768–790, 2016. arXiv:1309.7395 [math.CO] . [10] Bo jan Mohar and Behruz T a yfeh-Rezaie. Median eigenv alues of bipartite graphs. J. A lgebr aic Combin. , 41(3):899–909, 2015. arXiv:1312.2613 [math.CO] . [11] Edwin R. v an Dam, Willem H. Haemers, and Jack H. K o olen. Regular graphs with maximal energy p er vertex. J. Combin. The ory Ser. B , 107:123–131, 2014. arXiv:1210.8273 [math.CO] . A Optimization Pr o of of Pr op osition 5.1. Fix an integer d ≥ 75 . W e shall choose δ ∈ ( ε 0 , d ) later so that α ( δ ) · ( ε 1 − ε 0 ) ≥ 1 / 5 and δ − ε 0 ≥ √ 2 . (25) First, when ε, y , z are fixed, the function (2 ε 0 ) x + y + z  ε 0 + ε 1 2 ε 0  ( ε − ε 0 ) x  α ( δ ) · ( ε 1 − ε ) y  y / 2 ( d − ε 0 ) z (26) is prop ortional to (2 ε 0 ) x ( ε − ε 0 ) x , whic h is a decreasing function of x b ecause 2 ε 0 ( ε − ε 0 ) ≤ 2 ε 0 ( ε 1 − ε 0 ) = 2 √ d − 1 d + √ d − 1 < 1 . Therefore, the maximum v alue of (26) is attained when x = 1 / 2 . Under this assumption, the function b ecomes (2 ε 0 ) 1 / 2+ y + z  ε 0 + ε 1 2 ε 0  ( ε − ε 0 ) 1 / 2  α ( δ ) · ( ε 1 − ε ) y  y / 2 ( d − ε 0 ) z . 14 Second, when y and z are fixed, the ab ov e function is prop ortional to ( ε − ε 0 ) 1 / 2 ( ε 1 − ε ) y / 2 , which attains its maximum when ε = ( y ε 0 + ε 1 ) / ( y + 1) . Under this assumption, the function becomes (2 ε 0 ) 1 / 2+ y + z  ε 0 + ε 1 2 ε 0  ε 1 − ε 0 y + 1  1 / 2  α ( δ ) · ( ε 1 − ε 0 ) y + 1  y / 2 ( d − ε 0 ) z . (27) Third, when z is fixed, the ab o v e function is prop ortional to (2 ε 0 ) y  α ( δ ) · ( ε 1 − ε 0 ) y + 1  ( y +1) / 2 One chec ks that if a 2 b > 3 e/ 2 , then the function y 7→ a y ( b/ ( y + 1)) ( y +1) / 2 is increasing on [0 , 1 / 2] . Since (2 ε 0 ) 2 · α ( δ ) · ( ε 1 − ε 0 ) ≥ 4( d − 1) / 5 ≥ 57 by (25) , the maximum v alue of (27) is attained when y = 1 / 2 − z . Under this assumption, the function b ecomes ( ε 0 + ε 1 )  ε 1 − ε 0 3 / 2 − z  1 / 2  α ( δ ) · ( ε 1 − ε 0 ) 3 / 2 − z  1 / 4 − z / 2 ( d − ε 0 ) z . (28) F ourth, the ab ov e function is prop ortional to ( d − ε 0 ) z (3 / 2 − z ) 3 / 4 − z / 2 . One chec ks that if a > e − 1 / 2 , then the function z 7→ a z / (3 / 2 − z ) 3 / 4 − z / 2 is increasing on [0 , 1 / 2] . Since d − ε 0 ≥ 60 for d ≥ 75 and 1 / ( δ − ε 0 ) 2 ≤ 1 / 2 b y (25) , the maxim um v alue of (28) is attained when z = 1 / ( δ − ε 0 ) 2 =: z ∗ . Under this assumption, the function b ecomes ( ε 0 + ε 1 )  ε 1 − ε 0 3 / 2 − z ∗  1 / 2  α ( δ ) · ( ε 1 − ε 0 ) 3 / 2 − z ∗  1 / 4 − z ∗ / 2 ( d − ε 0 ) z ∗ . (29) Supp ose that 75 ≤ d ≤ 139 . W e c ho ose δ ∈ ( ε 0 , d ) so that α ( δ ) · ( ε 1 − ε 0 ) = 1 / 4 , which is a quadratic equation in δ . Solving this equation gives δ = p ( d + ε 0 )( d − 7 ε 0 ) − ε 0 . One chec ks n umerically for ev ery d ∈ { 75 , . . . , 139 } that (25) is satisfied and (29) is strictly less than 1 . Supp ose that d ≥ 140 . W e c ho ose δ ∈ ( ε 0 , d ) so that α ( δ ) · ( ε 1 − ε 0 ) = 1 / 5 , whic h is a quadratic equation in δ . Solving this equation giv es δ = p ( d + ε 0 )( d − 9 ε 0 ) − ε 0 . One chec ks that δ − ε 0 ≥ d/ 3 for d ≥ 140 , (30) and so (25) is satisfied. Since α ( δ ) · ( ε 1 − ε 0 ) = 1 / 5 , the maximum v alue of (29) simplifies to ( ε 0 + ε 1 )  ε 1 − ε 0 3 / 2 − z ∗  1 / 2  1 15 / 2 − 5 z ∗  1 / 4 − z ∗ / 2 ( d − ε 0 ) z ∗ = ε 0 + ε 1 √ d + ε 0 · √ 5 (15 / 2 − 5 z ∗ ) 3 / 4 − z ∗ / 2 · exp  ln( d − ε 0 ) ( δ − ε 0 ) 2  . 15 W e estimate the three factors in the last expression separately . First, we hav e ε 0 + ε 1 √ d + ε 0 ≤ 2 √ d + 1 /d √ d = 2 + 1 /  d √ d  ≤ 2 + 1 / 444 . Second, since z ∗ = 1 / ( δ − ε 0 ) 2 ≤ 9 /d 2 ≤ 1 / 2000 via (30), we hav e √ 5 (15 / 2 − 5 z ∗ ) 3 / 4 − z ∗ / 2 ≤ √ 5 (15 / 2 − 1 / 400) 3 / 4 − 1 / 4000 ≤ 1 2 + 1 / 40 . Third, since d 7→ ln d/d 2 is a decreasing function on ( √ e, ∞ ) , we hav e that exp  ln( d − ε 0 ) ( δ − ε 0 ) 2  (30) ≤ exp  9 ln d d 2  ≤ exp  9 ln 140 140 2  ≤ 111 110 . Therefore, the pro duct of the three factors is at most 2 + 1 / 444 2 + 1 / 40 · 111 110 = 889 891 whic h is strictly less than 1 . 16