On the Maximum Number of Spanning Trees in $C_4$-Free Graphs

ON THE MAXIMUM NUMBER OF SP ANNING TREES IN C 4 -FREE GRAPHS ANDR ´ AS LONDON Abstract. W e introduce a “Kirchhoff–T ur´ an” v arian t of the extremal C 4 problem: among all simple connected n -v ertex C 4 -free graphs G , maximize the num b er of spanning trees τ ( G ). F or the pro jective-plane orders n = q 2 + q + 1 w e compute an exact form ula for the Erd˝ os–R´ enyi orthogonal p olarit y graph E R q , namely τ ( E R q ) = n ( n − 3) / 2 , via a p olarit y sp ectral identit y and Kirc hhoff ’s matrix–tree theorem. W e also give an explicit general upper b ound on st ( n, C 4 ) at these n using a sharp degree-sequence inequality for τ ( G ) and a degree-balancing argument; this matc hes the low er b ound in the leading exp onential term. 1. The sp anning-tree Tur ´ an problem for C 4 F or a connected graph G , let τ ( G ) denote the num b er of spanning trees of G . Definition 1. F or n ∈ N define st( n, C 4 ) := max { τ ( G ) : | V ( G ) | = n, G is simple, connected, and C 4 -free } . Remark 1. Adding an edge strictly increases τ ( G ), so ev ery maximizer in Definition 1 is edge-maximal C 4 -free. (Connectivit y may b e imp osed since disconnected graphs hav e τ ( G ) = 0.) W arm-up: the triangle-free case. Define analogously st( n, C 3 ) := max { τ ( G ) : | V ( G ) | = n, G is simple, connected, and C 3 -free } . Man tel’s theorem identifies the edge-extremal C 3 -free graph as the balanced complete bipartite graph K ⌊ n/ 2 ⌋ , ⌈ n/ 2 ⌉ . Moreo v er, for complete bipartite graphs one has τ ( K a,b ) = a b − 1 b a − 1 . Th us a natural guess is that K ⌊ n/ 2 ⌋ , ⌈ n/ 2 ⌉ also maximizes τ ( G ) among triangle-free graphs. In con trast, for C 4 -freeness the edge-extremal constructions at the pro jectiv e-plane orders n = q 2 + q + 1 come from finite geometry (p olarit y graphs), suggesting that the corresp onding Kirc hhoff–T ur´ an problem may single out these algebraic constructions. 2. Two ingredients: a degree bound and degree balancing Theorem 1 (Klee–Naray anan–Sauermann [1]) . F or every simple gr aph G on n vertic es, τ ( G ) ≤ 1 n 2 Y v ∈ V ( G ) (deg( v ) + 1) . Remark 2. Inequalities upp er-b ounding τ ( G ) b y expressions inv olving degree data go back further; for background and related references see, e.g., [6]. Lemma 1 (Balancing) . Fix inte gers n ≥ 1 and S ≥ 0 . Among al l inte ger se quenc es ( d 1 , . . . , d n ) with P i d i = S , the pr o duct Q i ( d i + 1) is maximize d when the de gr e es differ by at most 1 , i.e. d i ∈ { a, a + 1 } for some inte ger a . Date : F ebruary 26, 2026. 2020 Mathematics Subje ct Classific ation. 05C30, 05C35, 05C50. Key wor ds and phr ases. spanning trees; matrix–tree theorem; extremal graph theory; C 4 -free graphs; orthogonal p olarit y graphs. 1 2 ANDR ´ AS LONDON Pr o of. If x ≥ y + 2, replacing ( x, y ) b y ( x − 1 , y + 1) preserv es the sum and changes the factor ( x + 1)( y + 1) to x ( y + 2). Since x ( y + 2) − ( x + 1)( y + 1) = x − y − 1 ≥ 1, the pro duct increases. Iterating remo ves all gaps ≥ 2. □ Theorem 2 (F ¨ uredi; see e.g. [5, 4]) . If q ≥ 14 is an inte ger and n = q 2 + q + 1 , then ex( n, C 4 ) ≤ 1 2 q ( q + 1) 2 . 3. Or thogonal polarity graphs and an exact sp anning-tree count W e follow the standard pro jective-plane definition (see [2, § 1]). Definition 2 (Orthogonal p olarit y graph E R q ) . Let q b e a prime p ow er and let PG (2 , q ) b e the Desarguesian pro jective plane: its p oints are the 1-dimensional subspaces of F 3 q . Fix an ortho gonal p olarity φ , i.e. a p olarity with exactly q + 1 absolute p oints [ 2 , § 1]. The (simple) orthogonal p olarit y graph E R q has v ertex set the p oin ts of PG (2 , q ), and distinct vertices p, p ′ are adjacen t if and only if p ∈ φ ( p ′ ). Remark 3 (Degrees) . Let n = q 2 + q + 1. In the lo op ed p olarity-graph mo del (lo ops at absolute p oin ts), the p olarit y graph is ( q + 1)-regular. Deleting the q + 1 lo ops pro duces the simple graph E R q , in which the q + 1 absolute p oints hav e degree q and the remaining q 2 v ertices hav e degree q + 1; see [2, § 1]. Lo op con ven tion. W e briefly use the lo op e d p olarit y-graph model: a lo op at a v ertex con tributes +1 to the degree and +1 to the diagonal entry of the adjacency matrix. Hence lo ops cancel in the Laplacian L = D − A , so the Laplacian (and therefore τ ) is unchanged if one deletes lo ops. Lemma 2 (Sp ectrum of p olarit y graphs; multiplicities from absolute p oin ts [ 3 , Lem. 2.1]) . L et G b e a p olarity gr aph of a pr oje ctive plane of or der q in the lo op e d mo del, with adjac ency matrix A and n = q 2 + q + 1 . Then A 2 = J + q I . Conse quently, the adjac ency eigenvalues ar e q + 1 and ± √ q . Mor e over, q + 1 has multiplicity 1 , and the multiplicities of ± √ q ar e uniquely determine d by tr ( A ) , i.e. by the numb er of absolute p oints (lo ops). Prop osition 1 (Exact formula for τ ( E R q )) . L et q b e a prime p ower and n = q 2 + q + 1 . Then τ ( E R q ) = n n − 3 2 . Pr o of. W ork in the lo op ed mo del (lo ops do not affect the Laplacian). By Lemma 2, the adjacency eigen v alues are q + 1 and ± √ q . F or an orthogonal p olarit y , there are exactly q + 1 absolute p oints [ 2 , § 1], so tr ( A ) = q + 1. Lemma 2 then forces the multiplicities of + √ q and − √ q to b e equal, i.e. eac h equals ( n − 1) / 2. Since the lo op ed p olarity graph is ( q + 1)-regular, L = ( q + 1) I − A and the nonzero Laplacian eigen v alues are ( q + 1) ± √ q , each with multiplicit y ( n − 1) / 2. Kirc hhoff ’s matrix–tree theorem in sp ectral form yields τ ( E R q ) = 1 n  ( q + 1) − √ q  n − 1 2  ( q + 1) + √ q  n − 1 2 = 1 n  ( q + 1) 2 − q  n − 1 2 = 1 n n n − 1 2 = n n − 3 2 . □ In particular, for n = q 2 + q + 1 we hav e the low er b ound st( n, C 4 ) ≥ τ ( E R q ) = n n − 3 2 . Remark 4 (Connectivity of E R q ) . In the lo op ed mo del the graph is ( q + 1)-regular and has adjacency eigenv alue q + 1 with multiplicit y 1 (Lemma 2). F or a regular graph, the multiplicit y of the eigenv alue equal to the degree equals the num b er of connected comp onents. Hence the lo op ed p olarit y graph is connected, and therefore the simple graph E R q is connected as well. ON THE MAXIMUM NUMBER OF SP ANNING TREES IN C 4 -FREE GRAPHS 3 4. An explicit upper bound a t n = q 2 + q + 1 Prop osition 2. L et n = q 2 + q + 1 with q ≥ 14 . F or every C 4 -fr e e gr aph G on n vertic es, τ ( G ) ≤ 1 n 2 ( q + 1) q +1 ( q + 2) n − ( q +1) . Conse quently, st( n, C 4 ) ≤ 1 n 2 ( q + 1) q +1 ( q + 2) n − ( q +1) . Pr o of. By Theorem 2, e ( G ) ≤ 1 2 q ( q + 1) 2 , hence X v ∈ V ( G ) deg( v ) = 2 e ( G ) ≤ q ( q + 1) 2 =: S max . The righ t-hand side of Theorem 1 is monotone in each degree, so under the single constrain t P v deg ( v ) ≤ S max it is maximized when P v deg ( v ) = S max . Lemma 1 then shows that among all in teger sequences of sum S max , the pro duct Q v ( deg ( v ) + 1) is maximized by degrees in { q , q + 1 } with exactly q + 1 copies of q and n − ( q + 1) copies of q + 1, since S max = q ( q 2 + q + 1) + q 2 = q n + ( n − ( q + 1)) . Th us Y v (deg( v ) + 1) ≤ ( q + 1) q +1 ( q + 2) n − ( q +1) , and Theorem 1 gives the b ound on τ ( G ). □ Remark 5 (Matching the leading exponential term) . Com bining Prop osition 1 and Prop osition 2 giv es, for n = q 2 + q + 1, log st( n, C 4 ) = n 2 log n + O ( √ n log n ) , so the low er and upp er b ounds agree in the leading n 2 log n term. 5. A quantit a tive deficit bound in a near-extremal regime F or n = q 2 + q + 1 define the balanced-pro duct en velope P ( S ) := max n n Y i =1 ( d i + 1) : d i ∈ Z ≥ 0 , n X i =1 d i = S o . By Lemma 1, writing S = na + r with a = ⌊ S/n ⌋ and 0 ≤ r < n , w e ha ve (1) P ( S ) = ( a + 2) r ( a + 1) n − r . Lemma 3 (One-step ratio) . F or every S ≥ 1 , writing S = na + r with a = ⌊ S/n ⌋ and 0 ≤ r < n , P ( S − 1) P ( S ) ≤ a + 1 a + 2 ≤ exp  − 1 a + 2  . Pr o of. F rom (1) , if r ≥ 1 then P ( S ) /P ( S − 1) = ( a +2) / ( a +1), hence P ( S − 1) /P ( S ) = ( a +1) / ( a + 2). If r = 0, then S = na and S − 1 = n ( a − 1) + ( n − 1), giving P ( S ) /P ( S − 1) = ( a + 1) /a and so P ( S − 1) /P ( S ) = a/ ( a + 1) ≤ ( a + 1) / ( a + 2). Finally , ( a + 1) / ( a + 2) = 1 − 1 a +2 ≤ e − 1 / ( a +2) . □ Prop osition 3 (Edge deficit suppresses τ (for t ≤ q 2 / 2)) . L et q ≥ 14 , n = q 2 + q + 1 , and let G b e C 4 -fr e e on n vertic es with e ( G ) = 1 2 q ( q + 1) 2 − t (0 ≤ t ≤ q 2 / 2) . Then τ ( G ) ≤ 1 n 2 ( q + 1) q +1 ( q + 2) n − ( q +1) · exp  − 2 t q + 2  . 4 ANDR ´ AS LONDON Pr o of. W rite S := P v deg ( v ) = 2 e ( G ) = S max − 2 t , where S max := q ( q + 1) 2 . By Theorem 1, τ ( G ) ≤ 1 n 2 P ( S ). Iterating Lemma 3 for S max − S = 2 t steps gives P ( S ) ≤ P ( S max ) exp  − 2 t X j =1 1 a j + 2  , a j := j S max − j n k . Since S max = q ( q + 1) 2 = q n + q 2 , so for every 1 ≤ j ≤ 2 t we hav e S max − j n ≤ S max − 1 n = q + q 2 − 1 q 2 + q + 1 < q + 1 , hence a j ≤ q . Therefore 1 a j +2 ≥ 1 q +2 for all 1 ≤ j ≤ 2 t , which yields P ( S ) ≤ P ( S max ) exp  − 2 t q + 2  . Finally , P ( S max ) = ( q + 1) q +1 ( q + 2) n − ( q +1) as in Prop osition 2. □ Conjecture 1. L et q b e a prime p ower and n = q 2 + q + 1 . Then st( n, C 4 ) = τ ( E R q ) = n n − 3 2 , and the maximizers ar e pr e cisely the ortho gonal p olarity gr aphs. Remark 6. A plausible route to Conjecture 1 is to connect near-maximalit y of τ ( G ) to near- extremalit y of e ( G ), and then inv oke known stability results for C 4 -free graphs to deduce that G is close to a p olarit y construction. Monotonicit y of τ under edge addition (Remark 1) would then force equality . 6. Future directions 6.1. The case of general n . While the p olarit y graphs E R q pro vide a natural candidate when n = q 2 + q + 1, the b ehavior of st( n, C 4 ) for general n remains op en. Problem 1. Determine the asymptotic b eha vior of log st ( n, C 4 ) for n far from pro jective-plane orders. Problem 2. Establish a “sp ectral stability” result: if a C 4 -free graph G satisfies τ ( G ) ≥ (1 − ε ) τ ( E R q ), then G is close to E R q in edit distance. 6.2. F orbidding K 2 ,t and C 2 k . The definition extends to any forbidden subgraph H by st( n, H ) := max { τ ( G ) : | V ( G ) | = n, G is simple, connected, and H -free } . The inequalit y of Klee–Naray anan–Sauermann (Theorem 1) applies verbatim for all H . Hence, once one has an upper bound on the degree sum (via ex ( n, H )), Lemma 1 yields explicit (en velope) upp er b ounds on st( n, H ). 6.2.1. The family K 2 ,t . Fix t ≥ 2. If G is K 2 ,t -free then any t wo vertices hav e at most t − 1 common neigh b ors, which implies the standard 2-path counting inequality: X v ∈ V ( G )  deg( v ) 2  ≤ ( t − 1)  n 2  . Moreo ver, ex ( n, K 2 ,t ) = O ( √ t n 3 / 2 ) by the K˝ ov´ ari–S´ os–T ur´ an theorem [ 7 ], and a sharp asymptotic upp er b ound due to F ¨ uredi [8] gives (2) ex( n, K 2 ,t ) ≤ 1 2 √ t − 1 n 3 / 2 + O ( n ) ( t fixed) . ON THE MAXIMUM NUMBER OF SP ANNING TREES IN C 4 -FREE GRAPHS 5 Prop osition 4. Fix t ≥ 2 . Ther e exists a c onstant C t > 0 such that for al l n and al l K 2 ,t -fr e e gr aphs G on n vertic es, τ ( G ) ≤ 1 n 2  1 + C t √ n  n . Conse quently, log st( n, K 2 ,t ) ≤ n 2 log n + O t ( n ) . Pr o of. F rom (2) , P v deg ( v ) = 2 e ( G ) ≤ C t n 3 / 2 for some C t . By Theorem 1 it suffices to upp er b ound Q v ( deg ( v ) + 1) under the single constrain t P v deg ( v ) ≤ C t n 3 / 2 . Lemma 1 sho ws the pro duct is maximized when the degrees are as equal as possible, i.e. av erage degree O ( √ n ), yielding Q v (deg( v ) + 1) ≤ (1 + C t √ n ) n . □ 6.2.2. Even cycles C 2 k . Fix k ≥ 2. The theorem of Bondy–Simonovits [ 9 ] implies ex ( n, C 2 k ) = O k ( n 1+1 /k ). Sharper b ounds are kno wn; for instance Bukh–Jiang [ 10 ] and He [ 11 ] prov e b ounds of the form (3) ex( n, C 2 k ) ≤ C k n 1+1 /k + O k ( n ) . Prop osition 5. Fix k ≥ 2 . Ther e exists a c onstant D k > 0 such that for al l n and al l C 2 k -fr e e gr aphs G on n vertic es, τ ( G ) ≤ 1 n 2  1 + D k n 1 /k  n . Conse quently, log st( n, C 2 k ) ≤ n k log n + O k ( n ) . Pr o of. F rom (3) , P v deg ( v ) = 2 e ( G ) ≤ 2 C k n 1+1 /k + O k ( n ). As b efore, Lemma 1 yields the en velope b ound with a verage degree O k ( n 1 /k ). □ AI tools disclosure. AI to ols w ere used only for language polishing and minor editorial suggestions; all mathematical conten t, results, and references were verified by the author. References [1] S. Klee, B. Nara yanan, and L. Sauermann. Sharp estimates for spanning trees. , 2021. Av ail- able at the authors’ webpages: https://sites.math.rutgers.edu/ ~ narayanan/pdf/counting_spanning_ trees.pdf . [2] X. Peng, M. T ait, and C. Timmons. On the c hromatic num ber of the Erd˝ os–R´ en yi orthogonal p olarity graph. Ele ctr onic Journal of Combinatorics 22(2), P2.21 (2015). [3] M. T ait and C. Timmons. 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Jiang. A b ound on the num b er of edges in graphs without an even cycle. Combinatorics, Pr ob ability and Computing 26(1) (2017), 1–15. [11] Z. He. A new upp er b ound on extremal n umber of even cycles. Ele ctr onic Journal of Combinatorics 28(2) (2021), P2.41. Institute of Informa tics, University of Szeged, Szeged, Hungar y Email addr ess : london@inf.u-szeged.hu, ORCID: 0000-0003-1957-5368