Spectral radius, toughness and $k$-factor of graphs

Sp ectral radius, toughness, binding n um b er and k -factor of graphs Y uan yuan Chen a,b , Huiqiu Lin a,b ∗ , Sh uc heng Li b a Scho ol of Mathematics, East China University of Scienc e and T e chnolo gy, Shanghai 200237, China b Col le ge of Mathematics and System Scienc e, Xinjiang University, Urumqi, Xinjiang 830017, China Abstract A k -regular spanning subgraph of G is called a k -factor. In this pap er, we pro vide sp ectral radius and edge conditions to ensure that a graph G with δ ( G ) ≥ k admits a k -factor. F an, Lin and Lu [Europ ean J. Combin. 110 (2023) 103701] presented a tigh t sufficient condition in terms of the sp ectral radius for a connected 1-tough graph to contain a connected 2-factor (Hamilton cycle). Then it is in teresting to consider the following problem: What is the sp ectral radius condition to guarantee the existence of a k -factor with k ≥ 3 in a connected 1-tough graph G with δ ( G ) ≥ k ? W e completely solv e this problem, and we further obtain a sufficient sp ectral radius condition for the existence of a k -factor in a connected 1-binding graph, whic h solv es an imp ortan t problem p osed b y F an and Lin [Electron. J. Com bin. 31 (2024) 1–30]. AMS Classification: 05C42, 05C50 Keyw ords: Spectral radius, F actor; T oughness, Binding num b er, Size 1 In tro duction Throughout this pap er, w e consider only finite, undirected and simple connected graphs. F or a v ertex v ∈ V ( G ), let N G ( v ) and d G ( v ) b e the neigh b orho od and degree of v in G , resp ectiv ely . The largest eigenv alue of A ( G ), denoted b y ρ ( G ), is called the sp e ctr al r adius of G . Given t wo graphs G 1 and G 2 , the disjoint union G 1 ∪ G 2 is the graph ∗ Corresp onding author; Email addresses: c henyy de@sina.com (Y. Chen), huiqiulin@126.com (H. Lin), sh uchengli666@163.com (S. Li). 1 with vertex set V ( G 1 ) ∪ V ( G 2 ) and edge set E ( G 1 ) ∪ E ( G 2 ), and the join G 1 ∨ G 2 is the graph obtained from G 1 ∪ G 2 b y adding all edges b et ween G 1 and G 2 . An [ a, b ]-factor of a graph G is a spanning subgraph H suc h that a ≤ d H ( v ) ≤ b for eac h v ∈ V ( G ). In particular, for a p ositiv e in teger k , a [ k , k ]-factor is called a k -factor. The initial study of factors w as due to Danish mathematician P etersen [35] in 1891. After that, man y researches ha ve b een conducted on this topic. There are abundan t achiev ements of studying the existence of factors in graphs from the sp ectral p erspec tiv e. Brou wer and Haemers [4] initiated the research of establishing sufficient conditions for a regular graph to hav e a p erfect matc hing in terms of the third largest adjacency eigenv alue. Their result w as improv ed in [7–9] and extended in [29, 30] to obtain a regular factor. Recently , the researc hers hav e also paid muc h attention to the existence of factors in a graph from the prosp ectiv e of the spectral radius, which can b e seen in [4, 6–11, 15, 18, 19, 27–31, 37] and the references therein, among others. Cho, Hyun, O and Park [10] p osed the following sp ectral v ersion conjecture for the existence of [ a, b ]-factors in graphs. Conjecture 1 (Cho, Hyun, O and Park [10]) . Let a, b b e tw o p ositiv e in tegers with a ≤ b , and G a graph of order n , where n ≥ a + 1 and na ≡ 0( mod 2). If ρ ( G ) > ρ ( K a − 1 ∨ ( K n − a ∪ K 1 )), then G contains an [ a, b ]-factor. F an and Lin [19] prov ed the conjecture for the case n ≥ 3 a + b − 1. Later, W ei and Zhang [37] completely confirmed this conjecture. As a corollary , F an and Lin also obtained the following result. Theorem 1 (F an and Lin [19]) . L et k n b e an even inte ger and k ≥ 1 . If G is a gr aph of or der n ≥ 4 k − 1 with ρ ( G ) > ρ ( K k − 1 ∨ ( K n − k ∪ K 1 )) , then G c ontains a k -factor. Let δ ( G ) b e the minim um degree of G . Note that δ ( G ) ≥ k is a trivial necessary condition for a graph to contain an k -factor. One ma y notice that the minim um degree of the extremal graph in Theorem 1 is k − 1. Based on Theorem 1, a natural and interesting problem arises: Problem 1. Giv e a sufficient sp ectral radius condition to ensure that a graph G with δ ( G ) ≥ k admits a k -factor. Let G 1 n,k b e the graph obtained from K k ∨ (( k + 1) K 1 ∪ K n − 2 k − 1 ) b y adding k − 1 edges b et w een ( k − 1) K 1 and a v ertex u k +1 whic h b elongs to K n − 2 k − 1 , see Fig. 1 (a). Concerning Problem 1, w e prov e the follo wing result. Theorem 2. L et G b e a c onne cte d gr aph of or der n ≥ 13 k 2 − 11 k − 1 with minimum de gr e e δ ( G ) ≥ k and k ≥ 1 . If ρ ( G ) ≥ ρ ( G 1 n,k ) , 2 Fig1 . Graphs G 1 n,k and G 2 n,k then G c ontains a k -factor, unless G ∼ = G 1 n,k . Let e ( G ) b e the n umber of edges in G . W e also pro vide the edge condition to guaran tee the existence of a k -factor in connected graphs. Theorem 3. L et G b e a c onne cte d gr aph of or der n ≥ 13 k 2 − 11 k − 1 with minimum de gr e e δ ( G ) ≥ k and k ≥ 1 . If e ( G ) >  n − k − 1 2  + k ( k + 1) + k − 1 , then G c ontains a k -factor. Ch v´ atal [14] defined the toughness of a non-complete graph G as t ( G ) = min { | S | c ( G − S ) : S ⊂ V ( G ) and c ( G − S ) > 1 } , where c ( G − S ) denotes the num b er of comp onen ts of G − S . A graph G is t -tough if | S | ≥ tc ( G − S ) for ev ery S ⊆ V ( G ) with c ( G − S ) > 1. Man y researchers also fo cused on the existence of factors in a graph from the prosp ectiv e of toughness, see [2, 3, 11–13, 20, 32]. A Hamiltonian cycle is a connected 2-factor of a graph. F an, Lin and Lu [20] determined the sp ectral radius condition to guarantee the existence of a Hamiltonian cycle among 1-tough graphs. Then it is in teresting to consider the following problem: Problem 2. What is the sp ectral radius condition to guarantee the existence of k ( ≥ 3)- factor in connected 1-tough graph G with δ ( G ) ≥ k ? Let G 2 n,k b e the graph obtained from K k ∨ (( k + 1) K 1 ∪ K n − 2 k − 1 ) by adding k − 2 edges b etw een ( k − 2) K 1 and a vertex u k +1 ∈ V ( K n − 2 k − 1 ) and an edge v 1 u k +2 , with v 1 ∈ V (( k − 2) K 1 ) and u k +2 ∈ V ( K n − 2 k − 1 ), as shown in Fig. 1 (b). Using some of the results established in Theorem 2, we can completely solve Problem 2. Theorem 4. L et G b e a 1-tough c onne cte d gr aph of or der n ≥ 13 k 2 − 11 k − 1 with minimum de gr e e δ ( G ) ≥ k and k ≥ 3 . If ρ ( G ) ≥ ρ ( G 2 n,k ) , 3 then G c ontains a k -factor, unless G ∼ = G 2 n,k . F or a subset S of G , we denote by N G ( S ) the neighborho o d of v ∈ S . The binding num- b er of G , denoted by b ( G ), w as in tro duced by W o odall [38] to measure how w ell the vertices of G are b ound together; is defined to b e min { | N G ( S ) | | S | : ∅  = S ⊂ V ( G ) and | N G ( S ) | < | V ( G ) |} . A non-complete graph G is r -binding if b ( G ) ≥ r . In the literature there are a n umber of results sho wing that v arious prop erties of G are consequences of assumptions on the v alue of b ( G ), including toughness [5, 16, 38], factors [1, 21, 23–26, 36, 38], etc. F an and Lin [21] prop osed the following problem. Problem 3. Whic h 1-binding graphs with δ ( G ) ≥ k hav e a k -factor? As for Problem 3, they ha ve carried out preliminary work from the viewp oin t of sp ectral radius for k = 1 , 2. They indicated that when k ≥ 3, Problem 3 seems more complicated. Theorem 5 (F an and Lin [21]) . L et G b e a 1-binding c onne cte d gr aph of or der n ≥ 21 with minimum de gr e e δ ( G ) ≥ 2 . If ρ ( G ) ≥ ρ ( G 1 n, 2 ) , then G c ontains a 2 -factor, unless G ∼ = G 1 n, 2 . In fact, b y Theorem 2 one sees that the binding n umber of the extremal graph G 1 n,k is at least 1 for k ≥ 2. Hence, the following corollary is a direct consequence of Theorem 2. This is also a generalization of Theorem 5. Corollary 1. L et G b e a 1-binding c onne cte d gr aph of or der n ≥ 13 k 2 − 11 k − 1 with minimum de gr e e δ ( G ) ≥ k ≥ 2 . If ρ ( G ) ≥ ρ ( G 1 n,k ) , then G c ontains a k -factor, unless G ∼ = G 1 n,k . 2 Preliminaries F or X, Y ⊆ V ( G ), w e denote b y e G ( X , Y ) the n um b er of edges in G with one endpoint in X and one endp oint in Y . Lemma 2.1 (See [34]) . L et k b e a p ositive inte ger, and let G b e a gr aph. Then G c ontains a k -factor if and only if δ G ( S, T ) = k ( | S | − | T | ) + X v ∈ T d G ( v ) − e G ( S, T ) − q G ( S, T ) ≥ 0 for al l disjoint subsets S, T ⊆ V ( G ) , wher e q G ( S, T ) is the numb er of the c omp onents C of G − ( S ∪ T ) such that e G ( V ( C ) , T ) + k | V ( C ) | ≡ 1 ( mod 2) . Mor e over, δ G ( S, T ) ≡ k | V ( G ) | ( mod 2) . 4 By the well-kno wn Perron-F rob enius theorem (cf. [17, Section 8.8]), we can easily deduce the follo wing result. Lemma 2.2. If H is a sp anning sub gr aph of a c onne cte d gr aph G , then ρ ( H ) ≤ ρ ( G ) , with e quality if and only if H ∼ = G . Lemma 2.3 (See [22, 33]) . L et G b e a gr aph on n vertic es and m e dges with minimum de gr e e δ ( G ) ≥ 1 . Then ρ ( G ) ≤ δ ( G ) − 1 2 + r 2 e ( G ) − nδ ( G ) + ( δ ( G ) + 1) 2 4 , with e quality if and only if G is either a δ ( G ) -r e gular gr aph or a bide gr e e d gr aph in which e ach vertex is of de gr e e either δ ( G ) or n − 1 . Lemma 2.4 (See [34]) . L et G b e a c onne cte d gr aph, and let u, v b e two vertic es of G . Supp ose that v 1 , v 2 , . . . , v s ∈ N G ( v ) \ N G ( u ) with s ≥ 1 , and G ∗ is the gr aph obtaine d fr om G by deleting the e dges v v i and adding the e dges uv i for 1 ≤ i ≤ s . L et x b e the Perr on ve ctor of A ( G ) . If x u ≥ x v , then ρ ( G ) < ρ ( G ∗ ) . Lemma 2.5 (See [21]) . L et a and b b e two p ositive inte gers. If a ≥ b ≥ 3 , then  a 2  +  b 2  <  a + 1 2  +  b − 1 2  . 3 Pro of of Theorems 2-4 F or an y S ⊆ G , let G [ S ] b e the subgraph of G induced by S and e ( S ) b e the num b er of edges in G [ S ]. Let G be the set of all connected graphs of order n ≥ 13 k 2 − 11 k − 1 with δ ( G ) ≥ k and no k -factor. Pro of of Theorem 2. Supp ose that G ∗ ∈ G with the maximum sp ectral radius. By Lemma 2.1, there exist tw o disjoin t subsets S, T ⊆ V ( G ∗ ), satisfying | S ∪ T | as large as p ossible suc h that δ G ∗ ( S, T ) = k ( | S | − | T | ) + X v ∈ T d G ∗ ( v ) − e G ∗ ( S, T ) − q G ∗ ( S, T ) ≤ − 2 , (1) where q G ∗ ( S, T ) is the num b er of the comp onen ts C of G ∗ − ( S ∪ T ) such that e G ∗ ( C, T ) + k | V ( C ) | ≡ 1 ( mod 2). Let C 1 , C 2 , . . . , C q b e the comp onen ts of G ∗ − ( S ∪ T ) such that e G ∗ ( C i , T ) + k | V ( C ) | ≡ 1 ( mod 2) and | V ( C i ) | = c i where 1 ≤ i ≤ q . Let | S | = s , | T | = t and q G ∗ ( S, T ) = q . By (1), we obtain X v ∈ T d G ∗ ( v ) ≤ k ( t − s ) + e G ∗ ( S, T ) + q G ∗ ( S, T ) − 2 . (2) 5 T ogether with Lemma 2.2 and the choice of G ∗ , w e get G ∗ [ S ∪ T ] = K s,t and G ∗ − T ∼ = K s ∨ ( K c 1 ∪ · · · ∪ K c q ∪ K c q +1 ), where c q +1 = n − t − s − P q i =1 c i . Now, w e divide the pro of in to the following eigh t claims. Claim 1. t ≥ s + 1. Pr o of . Otherwise, s ≥ t . If q = 0, then δ G ∗ ( S, T ) = k ( s − t ) + P v ∈ T d G ∗ − S ( v ) ≥ 0, which con tradicts (1). If q ≥ 1, then we ha v e δ G ∗ ( S, T ) = k ( s − t ) + X v ∈ T d G ∗ − S ( v ) − q ≥ k ( s − t ) + q X i =1 e G ∗ ( C i , T ) − q ≥ 0 , whic h also contradicts (1). 2 Claim 2. e ( G ∗ ) >  n − k − 1 2  + k + 2. Pr o of . Note that G 1 n,k ∈ G and contains a K n − k − 1 as a prop er subgraph. Then we hav e ρ ( G ∗ ) ≥ ρ ( G 1 n,k ) > ρ ( K n − k − 1 ) = n − k − 2 by Lemma 2.2. Combining this with Lemma 2.3, we obtain n − k − 2 < ρ ( G ∗ ) ≤ k − 1 2 + r 2 e ( G ∗ ) − nk + ( k + 1) 2 4 . The claim follo ws immediately . 2 Claim 3. If q ≥ 1, then c i ≥ k , where 1 ≤ i ≤ q . Pr o of . Otherwise, there exists some C j suc h that c j = k − a , where 1 ≤ a ≤ k − 1 and e G ∗ ( C j , T ) + k c j ≡ 1 ( mod 2) . (3) Then we divide the pro of into the following t wo cases. Case 1. k is even. Then w e hav e k ( k − a ) − 1 is o dd and e G ∗ ( C j , T ) is o dd b y (3). If e G ∗ ( C j , T ) ≤ k ( k − a ) − 1. Then let T ′ = T ∪ V ( C j ) and so | T ′ | = t ′ = t + c j = t + k − a . W e ha ve δ G ∗ ( S, T ′ ) = k ( s − t ′ ) + X v ∈ T d G ∗ ( v ) + X v ∈ V ( C j ) d G ∗ ( v ) − ( e G ∗ ( S, T ) + e G ∗ ( S, C j )) − ( q − 1) = k ( s − t ) + X v ∈ T d G ∗ ( v ) − e G ∗ ( S, T ) − q − k ( k − a ) + e G ∗ ( C j , T ) + 1 = δ G ∗ ( S, T ) − ( k ( k − a ) + 1 − e G ∗ ( C j , T )) ≤ δ G ∗ ( S, T ) (since e G ∗ ( C j , T ) ≤ k ( k − a ) − 1) , 6 whic h con tradicts the maximality of S ∪ T . If e G ∗ ( C j , T ) ≥ k ( k − a ) + 1. Then let S ′ = S ∪ V ( C j ) and so | S ′ | = s ′ = s + c j = s + k − a . W e ha ve δ G ∗ ( S ′ , T ) = k ( s ′ − t ) + X v ∈ T d G ∗ ( v ) − ( e G ∗ ( S, T ) + e G ∗ ( C j , T )) − ( q − 1) = k ( s − t ) + X v ∈ T d G ∗ ( v ) − e G ∗ ( S, T ) − q + k ( k − a ) − e G ∗ ( C j , T ) + 1 = δ G ∗ ( S, T ) − ( e G ∗ ( C j , T ) − k ( k − a ) − 1) ≤ δ G ∗ ( S, T ) (since e G ∗ ( C j , T ) ≥ k ( k − a ) + 1) , whic h also contradicts the maximality of S ∪ T . Case 2. k is o dd. If k − a is o dd, then k ( k − a ) − 1 is ev en, and hence e G ∗ ( C j , T )) is ev en by (3). If k − a is even, then k ( k − a ) − 1 is o dd, and hence e G ∗ ( C j , T )) is o dd b y (3). Therefore, w e may distinguish cases based on the v alues of e G ∗ ( C j , T )) and k ( k − a ) − 1. Using the same argument as in Case 1, we can find a larger set T ′ or S ′ suc h that δ G ∗ ( S, T ′ ) ≤ − 2 or δ G ∗ ( S ′ , T ) ≤ − 2, which con tradicts the maximality of S ∪ T . This completes the pro of of claim 3. 2 Claim 4. q ≤ 1. Pr o of . Otherwise, q ≥ 2. W e can obtain that c i ≥ k , 1 ≤ i ≤ q by Claim 3 and so n ≥ q k + s + t . Recall that G ∗ [ S ∪ T ] = K s,t and G ∗ − T ∼ = K s ∨ ( K c 1 ∪ · · · ∪ K c q ∪ K c q +1 ), where c q +1 = n − t − s − P q i =1 c i . Then b y Lemma 2.5 and inequality (2), w e ha ve e ( G ∗ ) ≤ X v ∈ T d G ∗ ( v ) + q − 1 X i =1  c i 2  +  c q +1 2  +  s 2  + s ( n − s − t ) ≤ k ( t − s ) + st + q − 2 + ( q − 1)  k 2  +  n − t − s − ( q − 1) k 2  +  s 2  + s ( n − s − t ) . (4) W e divide the pro of into the following four cases. Case 1. 1 ≤ t ≤ k and q ≥ 3. Then by Claims 1-2 and inequality (4), we ha ve 0 >  n − k − 1 2  + k + 2 − e ( G ∗ ) ≥ ( k q − 2 k + t − 1) n + (1 + q − q 2 ) k 2 2 − (2 q ( s + t ) − 4 s − 5) k 2 − (1 + t + 2 s ) t 2 + 5 − q ≥ t 2 + (2 k q − 4 k − 3) t 2 + k 2 q 2 − (3 k 2 + 2 k + 2) q 2 + k 2 + 5 k 2 + 5 − s (since n ≥ q k + s + t ) ≥ t 2 + (2 k q − 4 k − 5) t 2 + k 2 q 2 − (3 k 2 + 2 k + 2) q 2 + k 2 + 5 k 2 + 6 (since s ≤ t − 1) ≥ t 2 + (2 k − 5) t 2 + k 2 − 2 k 2 + 3 (since q ≥ 3) ≥ k 2 + k + 2 2 (since t ≥ 1) > 0 (since k ≥ 1) , a contradiction. 7 Case 2. 1 ≤ t ≤ k and q = 2. Recall that n ≥ 13 k 2 − 11 k − 1. If t ≥ 2. Then by Claims 1-2 and inequality (4), we ha ve 0 >  n − k − 1 2  + k + 2 − e ( G ∗ ) ≥ ( t − 1) n − k 2 + t 2 + 4 k t + t − 5 k 2 + 3 − st (Since inequality (4)) ≥ ( t − 1) n − k 2 + 3 t 2 + 4 k t − t − 5 k 2 + 3 (since s ≤ t − 1) ≥ n − 4 k 2 + 5 2 k + 4 (since 2 ≤ t ≤ k ) > 0 (since n ≥ 13 k 2 − 11 k − 1) , a contradiction. Let t = 1 and q = 2, then s = 0 b y Claim 1. W e assert that c i ≥ k + 1 for i = 1 , 2. Otherwise, there exists some C i suc h that c i = k b y Claim 3, where i = 1 , 2. Then w e ha ve e G ∗ ( C i , T ) = k as δ ( G ∗ ) ≥ k . Which contradicts of e G ∗ ( C i , T ) + k c i ≡ 1 ( mod 2). Therefore, by inequality (4), we can get e ( G ∗ ) ≤ k + k ( k +1) 2 +  n − k − 2 2  . Com bining this with Claim 2 and n ≥ 13 k 2 − 11 k − 1, we ha ve 0 >  n − k − 1 2  + k + 2 − e ( G ∗ ) ≥ n − k 2 2 − 3 k 2 > 0 , a contradiction. Case 3. t ≥ k + 1 and q ≥ 3. It is not hard to see that at least P q − 1 i =1 P q j = i +1 c i c j edges here are not in G . Without loss of generalit y , assume c q ≥ · · · ≥ c 1 ≥ k . Then we ha ve q − 1 X i =1 q X j = i +1 c i c j ≥ c 1 (( q − 1) c 1 + ( q − 2) c 1 + · · · + c 1 ) ≥ q ( q − 1) k 2 2 . Com bining this and inequality (2), we ha ve e ( G ∗ ) ≤ X v ∈ T d G ∗ ( v ) +  n − t 2  − q − 1 X i =1 q X j = i + 1 c i c j ≤ k ( t − s ) + st + q − 2 + ( n − t )( n − t − 1) 2 − k 2 ( q − 1) q 2 . Hence, by Claim 2, we obtain that 0 >  n − k − 1 2  + k + 2 − e ( G ∗ ) ≥ ( t − k − 1) n + k 2 ( q 2 − q + 1) − t ( t + 2 k − 1) + 5 k 2 + 5 − q − ( t − k ) s ≥ k 2 q 2 − (3 k 2 − 2 k t + 2 k + 2) q + ( t − 4 k − 5) t + k 2 + 5 k 2 + 6 (since n ≥ k q + s + t and s ≤ t − 1) ≥ k 2 + (2 t − 1) k + t 2 − 5 t + 6 2 (since q ≥ 3) ≥ 2 k 2 − k + 1 (since t ≥ k + 1) > 0 (since k ≥ 1) , a contradiction. 8 Case 4. t ≥ k + 1 and q = 2. If t ≥ k + 2, then by Claim 2 and inequality (4), we ha ve 0 >  n − k − 1 2  + k + 2 − e ( G ∗ ) ≥ ( t − 1) n − t 2 + k 2 + 4 k t + t − 5 k 2 + 3 − st (Since inequality (4)) ≥ t 2 − k 2 + k − 3 t 2 + 3 − s (since n ≥ 2 k + s + t ) ≥ t 2 − k 2 + k − 5 t 2 + 4 (since s ≤ t − 1) ≥ 1 (since t ≥ k + 2) , a contradiction. If t = k + 1, then b y Claim 2 inequality (4) we ha ve 0 >  n − k − 1 2  + k + 2 − e ( G ∗ ) ≥ nk − 3 k 2 − k + 2 − s ( k + 1) (Since inequality (4)) ≥ nk − 4 k 2 − 2 k + 2 (since s ≤ t − 1 = k ) > 0 (since n ≥ 13 k 2 − 11 k − 1) , a contradiction. Th us, it is completes the pro of of claim 4. 2 Claim 5. t ≥ k + 1. Pr o of . By Claim 4, inequalit y (2) and δ ( G ) ≥ k , w e hav e k t ≤ X v ∈ T d G ∗ ( v ) ≤ k ( t − s ) + st − 1 , and thus t ≥ k + 1 s . It is implies that t ≥ k + 1 b ecause t is a p ositiv e integer. 2 Claim 6. n ≥ s + t + k + 1. Pr o of . Otherwise, let n = s + t + b , where 1 ≤ b ≤ k . Then we hav e n ≤ 2 t − 1 + b as s ≤ t − 1. Recall that n ≥ 13 k 2 − 11 k − 1, so w e get t ≥ 13 2 k 2 − 11 2 k − b 2 ≥ 13 2 k 2 − 6 k . Note that e ( G ∗ ) ≤ X v ∈ T d G ∗ ( v ) +  n − t 2  ≤ k ( t − s ) + st − 1 + ( n − t )( n − t − 1) 2 . (5) If k ≥ 2 , then by Claim 2, w e ha ve 0 >  n − k − 1 2  + k + 2 − e ( G ∗ ) ≥ ( t − k − 1) n + k 2 + 5 k − t 2 − t 2 + 4 − ( t − k ) s − k t = k 2 + 5 k + t 2 − 3 t 2 + ( t − k − 1) b + 4 − 2 k t − s (since n = s + t + b ) ≥ ( t − 4 k − 5) t + k 2 + 5 k 2 + 5 (since s ≤ t − 1 , t ≥ k + 1) ≥ t + k 2 + 5 k 2 + 5  since t ≥ 13 2 k 2 − 6 k , k ≥ 2  > 0 , a contradiction. 9 Let k = 1, then t ≥ 2 and b = 1. W e hav e 0 >  n − k − 1 2  + k + 2 − e ( G ∗ ) ≥ ( t − 2) n − t 3 + 3 t 2 + 7 − ( t − 1) s (since k = 1) = t 2 − 5 t 2 + 5 − s (since n = s + t + 1) ≥ ( t − 3)( t − 4) 2 (since s ≤ t − 1) ≥ 0 , a contradiction. Th us, the claim hold. 2 Claim 7. t = k + 1. Pr o of . Otherwise, t ≥ k + 2. If t ≤ 3 k − 1, then by Claim 2, w e ha ve 0 >  n − k − 1 2  + k + 2 − e ( G ∗ ) ≥ ( t − k − 1) n + k 2 + 5 k − t 2 − t 2 + 4 − ( t − k ) s − k t (Since inequalit y (5)) ≥ ( t − k − 1) n + k 2 + 3 k − 3 t 2 + t 2 + 4 (since s ≤ t − 1) ≥ n + k 2 + 3 k − 3(3 k − 1) 2 + k + 2 2 + 4 (since k + 2 ≤ t ≤ 3 k − 1) = n − 13 k 2 + 11 k + 7 2 > 0 (since n ≥ 13 k 2 − 11 k − 1) , a contradiction. If t ≥ 3 k , then by Claims 2 and 7, w e ha ve 0 >  n − k − 1 2  + k + 2 − e ( G ∗ ) ≥ ( t − k − 1) n + k 2 + 5 k − t 2 − t 2 + 4 − ( t − k ) s − k t (Since inequalit y (5)) ≥ t 2 − t − k 2 + k 2 + 3 − s − k t (since n ≥ s + t + k + 1) ≥ t 2 − (2 k + 3) t − k 2 + k 2 + 4 (since s ≤ t − 1) ≥ k 2 − 4 k + 4 (since t ≥ 3 k ) ≥ 0 (since k ≥ 1) , a contradiction. Th us, the claim hold. 2 By inequalit y (2), Claim 2 ( s ≤ t − 1 = k ), Claim 4 ( q ≤ 1) and Claim 7 ( t = k + 1), w e ha ve X v ∈ T d G ∗ ( v ) ≤ k ( t − s ) + st + q − 2 ≤ k ( k + 1) + s − 1 ≤ k ( k + 1) + k − 1 . (6) 10 Equalit y holds if and only if equalit y holds simultaneously in each of the ab ov e inequalities. It is implies that G ∗ [ S ∪ T ] = K s,k +1 , s = k , P v ∈ T d G ∗ − S ( v ) = k − 1 and q = 1 (i.e., G ∗ − ( S ∪ T ) = C 1 ∼ = K n − 2 k − 1 and e G ∗ ( C 1 , T ) + k c 1 ≡ 1 ( mod 2) where c 1 = n − 2 k − 1). W e also obtain that G ∗ [ S ∪ C 1 ] = K k ∨ K n − 2 k − 1 = K n − k − 1 . Next, we consider separately the adjacency relations b et ween the set T and C 1 , as w ell as those within T itself, in G ∗ . Let T = { v 1 , . . . , v k +1 } , S = { u 1 , . . . , u k } and V ( C 1 ) = { u k +1 , . . . , u n − k − 1 } . Supp ose that x = ( x 1 , . . . , x n ) T is the P erron vector of A ( G ∗ ). Without loss of generalit y , assume that x u k +1 ≥ · · · ≥ x u n − k − 1 . By symmetry and eigen v alue equation ρ ( G ∗ ) x u = A ( G ∗ ) x u for any u ∈ V ( G ∗ ), we can easily get that x u 1 = · · · = x u k > x u k +1 ≥ · · · ≥ x u n − k − 1 . Let d C 1 ( v i ) = d i ( ≥ 0), for v i ∈ T . Supp ose that v i ∈ T with d i ≥ 1, that is N C 1 ( v i )  = ∅ . Then w e ha ve N C 1 ( v i ) = { u k +1 , . . . , u k + d i } . Otherwise, there exists a v ertex v j ∈ T ( d j ≥ 1) suc h that v j u t / ∈ E ( G ∗ ) and v j u t ′ ∈ E ( G ∗ ), where u t , u t ′ ∈ C 1 and k + 1 ≤ t ≤ k + d j < t ′ ≤ n − k − 1. Note that x u t ≥ x u t ′ , and let G 1 = G ∗ + v j u t − v j u t ′ . Then we can obtain that ρ ( G 1 ) > ρ ( G ∗ ) by Lemma 2.4, whic h con tradicts the maximality of ρ ( G ∗ ). Without loss of generality , assume that d 1 ≥ d 2 ≥ · · · ≥ d k +1 ≥ 0. Recall that N C 1 ( v i ) = { u k +1 , . . . , u k + d i } . Then N G ( u i +1 ) \{ u i } ⊆ N G ( u i ) \{ u i +1 } , where i ∈ [ k + 1 , k + d 1 ]. By symmetry and eigen v alue equation ρ ( G ∗ ) x u i = A ( G ∗ ) x u i for an y u i ∈ V ( C 1 ), we hav e x u k +1 ≥ · · · ≥ x u k + d 1 > x u k + d 1 +1 = · · · = x u n − k − 1 ≜ a ′ . Without loss of generality , we can assume that x v ∗ = max { x v i | v i ∈ T } and let d C 1 ( v ∗ ) = d ∗ . Clearly , d ∗ ≤ d 1 ≤ k − 1 and d T ( v ∗ ) ≤ ⌊ k − 1 2 ⌋ as P v ∈ T d G ∗ − S ( v ) = k − 1. Then we ha v e follo wing claim. Claim 8. a ′ ≥ x v ∗ . Pr o of . By eigenv alue equation ρ ( G ∗ ) x = A ( G ∗ ) x , we ha ve      ρ ( G ∗ ) x v ∗ = k x u 1 + P k + d ∗ i = k +1 x u i + P v i ∈ N T ( v ∗ ) x v i ρ ( G ∗ ) a ′ = k x u 1 + P k + d 1 i = k +1 x u i + ( n − 2 k − 2 − d 1 ) a ′ . F rom abov e equations, w e ha ve ρ ( G ∗ )( a ′ − x v ∗ ) = k + d 1 X i = k + d ∗ +1 x u i + ( n − 2 k − 2 − d 1 ) a ′ − X v i ∈ N T ( v ∗ ) x v i ≥ ( n − 2 k − 2 − d 1 ) a ′ − d T ( v ∗ ) x v ∗ (since x v ∗ = max { x v i | v i ∈ T } ) = ( n − 2 k − 2 − d 1 − d T ( v ∗ )) a ′ + d T ( v ∗ )( a ′ − x v ∗ ) ≥  n − 7 k + 1 2  a ′ + d T ( v ∗ )( a ′ − x v ∗ ) (since d 1 ≤ k − 1 , d T ( v ∗ ) ≤ ⌊ k − 1 2 ⌋ ) . That is ( ρ ( G ∗ ) − d T ( v ∗ ))( a ′ − x v ∗ ) ≥  n − 7 k + 1 2  a ′ ≥ 0 , 11 due to ρ ( G ∗ ) > n − k − 2 > d T ( v ∗ ), n ≥ 13 k 2 − 11 k − 1 > 7 k +1 2 when k ≥ 2 and n ≥ 4 when k = 1. It is implies that a ′ ≥ x v ∗ , as required. 2 F or an y v i ∈ T , by Claim 8, we ha ve x u k +1 ≥ x u k + d 1 > x u k + d 1 +1 ≥ x v i . Note that P v ∈ T d G ∗ − S ( v ) = k − 1, so 2 e G ∗ ( T ) + e G ∗ ( C 1 , T ) = k − 1. Therefore, w e assert G ∗ = G 1 n,k . Otherwise, let G 2 = G ∗ − X v i v j ∈ E G ∗ ( T ) v i v j − k +1 X i =1 k + d i X j = k +2 v i u j + k − 1 X i =1 v i u k +1 . Ob viously , G 2 ∼ = G 1 n,k . By Lemmas 2.2 and 2.4, w e hav e ρ ( G 2 ) > ρ ( G ∗ ), which con tradicts the maximality of ρ ( G ∗ ). Therefore G ∗ = G 1 n,k , which completes the pro of. Pro of of Theorem 3. Similarly , we obtain the conclusion using pro of b y contradiction. Let G ∈ G with maximum size and let S, T ⊆ V ( G ), satisfying | S ∪ T | as large as p ossible suc h that (1) hold. Let C 1 , C 2 , . . . , C q b e the comp onen ts of G − ( S ∪ T ) such that e G ( C i , T ) + k | V ( C i ) | ≡ 1 ( mod 2) and | V ( C i ) | = c i where 1 ≤ i ≤ q . Then w e also get G [ S ∪ T ] = K s,t and G − T ∼ = K s ∨ ( K c 1 ∪ · · · ∪ K c q ∪ K c q +1 ), where c q +1 = n − t − s − P q i =1 c i . Then b y using the same analysis as the pro of of Theorem 2, the Claims 1-7 are also hold. By Claim 4 ( q ≤ 1) and inequality (6), we hav e q = 1. Combining q = 1 and Claim 7 ( t = k + 1), w e can obtain that G − T ∼ = K n − k − 1 . Therefore, we hav e e ( G ) ≤ e ( G − T ) + X v ∈ T d G ( v ) ≤  n − k − 1 2  + k ( k + 1) + k − 1 . This completes the pro of. Pro of of Theorem 4. Assume to the contrary that ˜ G ( ∈ G ) is a connected 1-tough graph with maximum sp ectral radius. Let S, T ⊆ V ( ˜ G ), satisfying | S ∪ T | as large as possible such that (1) hold. Let | S | = s , | T | = t , q G ( S, T ) = q and let C 1 , C 2 , . . . , C q b e the comp onen ts of G − ( S ∪ T ) such that e ˜ G ( C i , T ) + k | V ( C i ) | ≡ 1 ( mod 2), where 1 ≤ i ≤ q . Then we ha ve s + t ≥ q due to ˜ G is a connected 1-tough graph. By using the same analysis as the proof of Theorem 2, the Claims 1-7 are also hold. Then w e can obtain that s = k , t = k + 1 and ˜ G [ S ∪ T ] = K k,k +1 , ˜ G [ S ∪ C 1 ] = K n − k − 1 . Let T = { v 1 , . . . , v k +1 } , S = { u 1 , . . . , u k } , V ( C 1 ) = { u k +1 , . . . , u n − k − 1 } and let d C 1 ( v i ) = d i ( ≥ 0), for v i ∈ T . Supp ose that x = ( x 1 , . . . , x n ) T is the Perron vector of A ( ˜ G ) and without loss of generality , assume that d 1 ≥ d 2 ≥ · · · ≥ d k +1 ≥ 0. Using the same analysis as the pro of of Claim 8 in Theorem 2. W e also obtain that N C 1 ( v i ) = { u k +1 , . . . , u k + d i } , x u k +1 ≥ · · · ≥ x u k + d 1 > x u k + d 1 +1 = · · · = x u n − k − 1 ≜ a ′ and a ′ ≥ x v ∗ , where x v ∗ = max { x v i | v i ∈ T } . Let U = S ∪ N C 1 ( T ). 12 Then | U | c ( G − U ) ≥ 1 b ecause of ˜ G is a connected 1-tough graph. In the following, we consider the adjacency relations b et ween T and C 1 , and b et ween T and T , resp ectiv ely , in ˜ G . Recall that P v ∈ T d T ∪ C 1 ( v ) = 2 e ˜ G ( T ) + e ˜ G ( C 1 , T ) = k − 1. Then we assert ˜ G = G 2 n,k . Otherwise, let G 2 = ˜ G − X v i v j ∈ E ˜ G ( T ) v i v j − k +1 X i =1 k + d i X j = k +2 v i u j + k − 2 X i =1 v i u k +1 + v 1 u k +2 , Ob viously , G 2 ∼ = G 2 n,k and G 2 n,k ∈ G is a connected 1-tough graph with k ≥ 3. Note that x u k +1 ≥ x u k +2 > x v i for any v i ∈ T . Then by Lemmas 2.2 and 2.4 we get ρ ( G 2 ) > ρ ( ˜ G ), whic h con tradicts the maximalit y of ρ ( ˜ G ). Thus ˜ G = G 2 n,k , whic h completes the pro of. Declaration of Comp eting In terest The authors declare that they hav e no known competing financial interests or p ersonal relationships that could hav e app eared to influence the work rep orted in this pap er. Ac kno wledgemen ts This research is supp orted by China Postdoctoral Science F oundation (No. 2025M783100), the National Natural Science F oundation of China (Nos. 12271162 and 12401468), Tian shan T alent T raining Program (No. 2024TSYCQNTJ0001), the Natural Science F oun- dation of Xinjiang Uygur Autonomous Region, (No. 2023D01C165), the Natural Sci- ence F oundation of Shanghai (No. 22ZR1416300) and the Program for Professor of Sp e- cial App oin tmen t (Eastern Scholar) at Shanghai Institutions of Higher Learning (No. TP2022031). References [1] J. Anderson, Perfect matchings in a graph, J. Combin. The ory Ser B. 10 (1971) 183–186. [2] N. Alon, T ough Ramsey graphs without short cycles, J. Algebr aic Combin. 4 (1995) 189–195. [3] D. Bauer, H. Bro ersma, H. V eldman, Not every 2-tough graph is Hamiltonian, Dis- cr ete Appl. Math. 99 (2000) 317–321. 13 [4] A. Brouw er, W. Haemers, Eigenv alues and p erfect matc hings, Line ar Algebr a Appl. 395 (2005) 155–162. [5] D. Bauer, N. Kahl, E. Sc hmeichel, D. W oo dall, M. Y atauro, T oughness and binding n umber. Discr ete Appl. Math. 165 (2014) 60–68. [6] S. Barik, S. Behera, On the smallest p ositiv e eigenv alue of bipartite unicyclic graphs with a unique p erfect matching, Discr ete Math. 346 (2) (2023) 113252. [7] S. Cioab˘ a, D. Gregory , W. Haemers, Matc hings in regular graphs from eigen v alues, J. Comb. The ory, Ser. B. 99 (2) (2009) 287–297. [8] S. Cioab˘ a, P erfect matchings, eigen v alues and expansion, C. R. Math. A c ad. Sci. So c. R. Can. 27 (4) (2005) 101–104. [9] S. Cioab˘ a, D. Gregory , Large match ings from eigenv alues, Line ar Algebr a Appl. 422 (1) (2007) 308–317. [10] E. Cho, J. Hyun, S. O, J. P ark, Sharp conditions for the existence of an even [ a, b ]- factor in a graph, Bul l. Kor e an Math. So c. 58 (1) (2021) 31–46. [11] Y. Cui, M. Kano, Some results on o dd factors of graphs, J. Gr aph The ory 12 (1988) 327–333. [12] Y. Chen, D. F an, H. Lin, F actors, sp ectral radius and toughness in bipartite graphs, Discr ete Appl. Math. 355 (2024) 223–231. [13] Y. Chen, X. Gu, H. Lin, Generalized toughness and sp ectral radius of graphs, Discr ete Math. 349 (2026) 114776. [14] V. Ch v´ atal, T ough graphs and Hamiltonian circuits, Discr ete Math. 5 (1973) 215–228. [15] M. Ellingham, Y. Nam, H. V oss, Connected ( g , f )-factors, J. Gr aph The ory 39 (2002) 62–75. [16] W. Go ddard, H. Swart. On the toughness of a graph. Quaestiones Math. 13 (2) (1990) 217–232. [17] C. Go dsil, G. Ro yle, Algebraic Graph Theory , Graduate T exts in Mathematices, 207, Springer-V erlag, New Y ork, 2001. [18] X. Gu, Regular factors and eigenv alues of regular graphs, Eur op e an J. Combin. 42 (2014) 15–25. 14 [19] D. F an, H. Lin, H. Lu, Sp ectral radius and [ a, b ]-factors in graphs, Discr ete Math. 345 (2022) 112892. [20] D. F an, H. Lin, H. Lu, T oughness, hamiltonicity and sp ectral radius in graphs, Eu- r op e an Journal of Combinatorics 110 (2023) 103701. [21] D. F an, H. Lin, Binding n umber, k -factor and sp ectral radius of graphs, Ele ctr on. J. Combin. 31 (1) (2024) 1–30. [22] Y. Hong, J. Shu, K. F ang, A sharp upp er b ound of the sp ectral radius of graphs, J. Combin. The ory Ser. B 81 (2001) 177–183. [23] Z. Hu, K. La w, W. Zang, An optimal binding num b er condition for bipancyclism. SIAM J. Discr ete Math. 27 (2013) 597–618. [24] Y. Hao, S. Li, Y. Y u, Bipartite binding n umber, k -factor and sp ectral radius of bipartite graphs, Discr ete Math. 348 (2025) 114511. [25] P . Katerinis, D. W o odall, Binding num b ers of graphs and the existence of k -factors. Quart. J. Math. Oxfor d Ser. 38 (2) (1987) 221–228. [26] M. Kano, N. T okushige, Binding n umbers and f -factors of graphs. J. Combin. The ory Ser. B. 54 (2) (1992) 213–221. [27] M. Kano, H. Lu, Q. Y u, F ractional factors, comp onen t factors and isolated vertex conditions in graphs, Ele ctr on. J. Comb. 26 (4) (2019) 4–33. [28] S. Kim, S. O, J. P ark, H. Ree, An o dd [1 , b ]-factor in regular graphs from eigen v alues, Discr ete Math. 343 (8) (2020) 111906. [29] H. Lu, Regular factors of regular graphs from eigenv alues, Ele ctr on. J. Comb. 17 (1) (2010) 2056–2065. [30] H. Lu, Regular graphs, eigen v alues and regular factors, J. Gr aph The ory 69 (4) (2012) 349–355. [31] H. Lu, Z. W u, X. Y ang, Eigenv alues and [1 , n ]-o dd factors, Line ar A lgebr a Appl. 433 (4) (2010) 750–757. [32] R. Liu, A. F an, J. Shu, Sp ectral extremal problems on factors in tough graphs, and b ey ond, Discr ete Math. 348 (2025) 114593. [33] V. Nikiforo v, Some inequalities for the largest eigenv alue of a graph, Combin. Pr ob ab. Comput. 11 (2) (2002) 179–189. 15 [34] E. Nosal, Eigen v alues of graphs, Master thesis, Universit y of Calgary , 1970. [35] J. Petersen, Die Theorie der regul ¨ a ren graphs, Acta Math. 15 (1) (1891) 193–220. [36] R. Shi, The binding num b er of a graph and its triangle, A cta Math. Appl. Sinic a (English Ser.) 3 (1985) 79–86. [37] J. W ei, S. Zhang, Pro of of a conjecture on the sp ectral radius condition for [ a, b ]- factors, Discr ete Math. 346 (2023) 113269. [38] D. W oo dall, The binding n umber of a graph and its Anderson n umber, J. Combin. The ory Ser. B. 15 (1973) 225–255. 16