Anti-adjacency eigenvalues of mixed extension of star graph
aa r X i v : . [ m a t h . C O ] F e b Anti-adjacency eigenvalues of mixed extension of star graph ∗ Jianfeng Wang a, † , Xingyu Lei a , Mei Lu b a School of Mathematics and Statistics, Shandong University of Technology, Zibo 255049, China b Department of Mathematical Sciences, TsingHua University, Beijing 100084, China
Abstract
The anti-adjacency matrix of graphs is constructed from the distance matrix of a graphby keeping each row and each column only the largest distances. This matrix can be in-terpreted as the opposite of the adjacency matrix, which is instead constructed from thedistance matrix of a graph by keeping each row and each column only the distances equal to1. The anti-adjacency eigenvalues of a graph are those of its anti-adjacency matrix. Employ-ing a novel technique introduced by Haemers [Spectral characterization of mixed extensionsof small graphs, Discrete Math. 342 (2019) 2760–2764], we characterize all connected graphswith exactly one positive anti-adjacency eigenvalues. On this basis, we identify the connectedgraphs with all but at most two anti-adjacency eigenvalues equal to − AMS classification:
Keywords : Mixed extension; Anti-adjacency matrix; Eccentricity matrix; Eigenvalues. Introduction
To study the graphs determined by the spectrum of adjacency matrix, Haemers [6] introduceda useful operation on a graph named as mixed extension . Consider a graph G with vertex set { , . . . , n } . Let V , . . . , V n be mutually disjoint nonempty finite sets. We define a graph H withvertex set the union of V , . . . , V n as follows. For each i , the vertices of V i are either all mutuallyadjacent ( V i is a clique), or all mutually nonadjacent ( V i is a coclique). When i = j , a vertexof V i is adjacent to a vertex of V j if and only if i and j are adjacent in G . We call H a mixedextension of G . We represent a mixed extension by an n -tuple ( t , . . . , t n ) of nonzero integers,where t i > V i is a clique of order t i , and t i < V i is a cocliqueof order − t i . Haemers’s definition is more convenient and powerful, which is motivated by aconcrete question for which the pineapple graphs are determined by the adjacency spectrum[20]. We refer to [6, 7] for basic results on mixed extensions and to [1] for graph spectra.Along with other techniques, Haemers [6] determined all graphs with at most three eigenval-ues unequal to 0 and −
1, consisting of all mixed extensions of graphs on at most three verticestogether with some particular mixed extensions of the paths P and P . Subsequently, Haemerset al. [7] investigated the mixed extension of P on being determined by the adjacency spectrumand presented several cospectral families. Comparatively, Cioabˇa et al. [2] provided another ∗ The first two authors are supported by National Natural Science Foundation of China (No. 11971274) andthe third author is supported by National Natural Science Foundation of China (No. 11771247). † Corresponding author.
Email addresses: [email protected] (J.F. Wang), [email protected] (X. Lei), [email protected](M. Lu). ±
1. Moreover, Cioabˇa et al. [3] identi-fied the graphs with all but two eigenvalues equal to − − − − − anti-adjacency matrix of graphs. We only consider finite, simple and connected graphs.Let G = ( V ( G ) , E ( G )) be a graph with order | V ( G ) | = n . The distance d G ( v, w ) between twovertices v and w is the minimum length of the paths joining them. The diameter of G , denotedby dim( G ), is the greatest distance between any two vertices in G . The eccentricity ε G ( u ) of thevertex u ∈ V ( G ) is given by ε G ( u ) = max { d ( u, v ) | v ∈ V ( G ) } . Then the anti-adjacency matrix (or eccentricity matrix ) A ( G ) = ( ǫ uv ) of G are defined as follows [24]: ǫ uv = (cid:26) d G ( u, v ) if d G ( u, v ) = min { ε G ( u ) , ε G ( v ) } , . By comparing the definitions, it turns out that A ( G ) is equal to the D MAX -matrix introducedby Randi´c in [16] as a tool for Chemical Graph Theory. Anyway, since the importance of vertex-eccentricity is not limited to applications to chemistry, the author asserted that such matrixmight open new directions of exploration in other branches of graph theory as well.Remark, the matrix A ( G ) is constructed from the distance matrix by only keeping the largestdistances for each row and each column, whereas the remaining entries become null. That is why A ( G ) can be interpreted as the opposite of the adjacency matrix, which is instead constructedfrom the distance matrix by keeping only distances equal to 1 on each row and each column.From this point of view, A ( G ) and A ( G ) are extremal among all possible distance-like matrices.As a contrast with A ( G ), the anti-adjacency matrix has some fantastic properties, one of whichis that A ( G ) of a connected graph is not necessarily irreducible. See the published papers[11, 22, 24, 26, 27] and the arXiv preprints [8, 12, 21] for more results about this newer matrix.We next introduce some notations borrowed from spectral graph theory. The A -polynomial of G is defined as φ ( G, λ ) = det( λI − A ( G )), where I is the identity matrix. The roots ofthe A -polynomial are the A -eigenvalues and the A -spectrum , denoted also by Spec A ( G ), of G is a multiset consisting of the A -eigenvalues. Since A ( G ) is symmetric, the A -eigenvalues arereal. Let ξ ≥ ξ ≥ · · · ≥ ξ n be the anti-adjacency eigenvalues of a graph with order n . If ξ ′ > ξ ′ > · · · > ξ ′ k are all distinct A -eigenvalues, then the A -spectrum can be written asSpec A ( G ) = { ξ , ξ , . . . , ξ n } = (cid:26) ξ ′ ξ ′ · · · ξ ′ k m m · · · m k (cid:27) , where m i is the algebraic multiplicity of the eigenvalue ξ ′ i (1 ≤ i ≤ k ). For any graphs matrix M , the M -cospectral graphs are non-isomorphic ones with the same M -spectrum. We say that G is determined by its M -spectrum if no M -cospectral graphs of G exist. Usually, M is theadjacency, or the Laplacian, or the anti-adjacency matrices and so on.The most important reason why the authors [25] tended to build a spectral theory basedon the anti-adjacency matrix is that they tried to detect the proportion of cospectral graphsrelating to two famous conjectures: One is that almost all graphs are adjacency cospectralposed by Schwenk [17]; the other is that almost all graphs are determined by the adjacency (orLaplacian) spectrum formally proposed by Haemers [5]. In their paper, the authors [25] showedthat, when n → ∞ , the fractions of non-isomorphic cospectral graphs with respect the adjacencyand the anti-adjacency matrix behave like those only concerning the self-centered graphs withdiameter two. Moreover, they also obtained that the connected graphs have exactly two distinct2 -eigenvalues iff G is the r -antipodal graphs, which could be used to construct much more A -cospectral graphs.Recall, a classical resut, due to Smith [4, Theorem 6.7], is that a connected graph hasexactly one positive adjacency eigenvalue if and only if G is a complete multipartite graph K n ,n , ··· ,n t , which is just the mixed extension of complete graph K t of type ( − n , − n , . . . , − n t ).Contrastively, we will determine the graphs with exactly one positive anti-adjacency eigenvalue,which is the mixed extension of star S ,n − in Theorem 1.1. Based on this result, we classifythe graphs with all but two anti-adjacency eigenvalues equal to − − Theorem 1.1.
A connected graph G has exactly one positive A -eigenvalue if and only if G isthe mixed extension of star S ,q +1 of type ( t , − p, t , . . . , t q ) with p, q ≥ and t j ≥ ( ≤ j ≤ q ),where (i) t = 1 , p + q ≥ ; (ii) t = 2 , p, q ≥ ; (iii) t = 3 , ≤ q ≤ − p (0 ≤ p ≤ ; (vi) t = 4 , ≤ q ≤ − p (0 ≤ p ≤ ; (v) t ≥ , ≤ q ≤ − p (0 ≤ p ≤ . Theorem 1.2.
Let G be a connected graph. Then (i) None graph has exactly one A -eigenvalue different from and − . (ii) The graph with all but two A -eigenvalues equal to − and if and only if G is the mixedextension of star S , of type ( − t , − t ) , where t , t ≥ . Here is the remainder of the paper. In Section 2 we mainly give the proof of Theorem1.1 which are decomposed into a series of conclusions. Especially, we determine the spectraldistribution in the graphs with exactly one positive A -eigenvalue. In Section 3 we provide aproof for Theorem 1.2 based on the results in previous section. In Section 4 we give someremarks and put forward several problems for further study. Graphs with exactly one positive A -eigenvalue Throughout the paper, let G be the set of graphs with exactly one positive A -eigenvalue. Fortwo graphs G and H , let G ∪ H be their disjoint union , and H ⊆ G (or H * G ) denote that H is (or not) an induced subgraph of G . We denote by G ∨ H the join obtained from G ∪ H byjoining each vertex of G to each one of H . Lemma 2.1 (Cauchy Interlace Theorem) . Let R be a real symmetric n × n matrix and let S bea principal submatrix of R with order m × m . Then, for i = 1 , , · · · , m , λ n − m + i ( R ) ≤ λ i ( S ) ≤ λ i ( R ) , where λ ( R ) ≥ λ ( R ) ≥ · · · ≥ λ n ( R ) and λ ( S ) ≥ λ ( S ) ≥ · · · ≥ λ m ( S ) are respectively theeigenvalues of R and S . H ⊆ G , the principal submatrix indexed by V ( H ) may not be the anti-adjacency matrixof H . However, for any vertex u ∈ V ( H ), if there exists a vertex v ∈ V ( H ) such that ε H ( u ) = ε G ( u ) = d G ( u, v ), then A ( H ) is a principal submatrix of A ( G ) which together with Lemma 2.1induces the following lemma. Lemma 2.2.
Let G be a connected graph with order n and H be an induced subgraph with order n ′ . If diam( G ) ≤ and diam( H ) ≤ , then A ( H ) is a principal submatrix of A ( G ) . Thus, for i = 1 , , · · · , n ′ , ξ n − n ′ + i ( G ) ≤ ξ i ( H ) ≤ ξ i ( G ) , where ξ i ( G ) ( i = 1 , , . . . , n ) is the A -eigenvalue of G . Label Graph ξ F S (5 , −
3) 4 − √ F S (5 , − ,
2) 0 . F S (5 , − , ,
2) 0 . F S (4 , − √ − √ F S (4 , − ,
2) 0 . F S (4 , − , ,
2) 0 . F S (4 , − , , ,
2) 0 . F S (3 , −
5) 5 − √ F S (3 , − ,
2) 0 . F S (3 , − , ,
2) 0 . F S (3 , − , , ,
2) 0 . F S (3 , − , , , ,
2) 0 . A -eigenvalues of F – F .For convince, let S ( t , t , . . . , t k ) be the mixed extension of star S ,k of the type ( t , t , . . . , t k ).Then the subsequent lemma follows from Lemma 2.2 and Table 1. Lemma 2.3.
Let G = S ( t , − p, t , . . . , t q ) be a mixed extension with p, q ≥ , t j ≥ ( ≤ j ≤ q ).If G ∈ G , then the graphs F – F in Table1 are not the induced subgraphs of G . As usual, let C n , P n , K n denote the cycle , path and complete graph of order n , respectively. Lemma 2.4.
For any G ∈ G , diam( G ) ≤ . Proof.
Assume that diam( G ) ≥
3. Let P d +1 = v v . . . v d − v d be a path with length diam( G ) = d of G . Then ε G ( v ) = ε G ( v d ) = d and d − ≤ ε G ( v ) , ε G ( v d − ) ≤ d . We next distinct thefollowing cases to be discussed. Case 1. ε G ( v ) = d and ε G ( v d − ) ≤ d , or ε G ( v ) ≤ d and ε G ( v d − ) = d . For the symmetry,we show the former situation. Without loss of generality, for some u ∈ V ( G ) set ε G ( v ) = d G ( v , u ) = d = ε G ( u ). Then the principal submatrix of A ( G ) indexed by { v , v d , v , u } is W = d A v u d A v d u d A uv A uv d d . d G ( v i , u ) ≤ d we have A v i u ∈ { , d } (0 ≤ i ≤ d ). Thereby, the principal submatrix of A ( G )indexed by { v , v d , v , u } is one of the following matrices: W = d d d d , W = d d d d d d , W = d dd dd d , W = d dd d dd d d . A direct calculation shows that the second largest eigenvalues of the first three matrices aboveare respectively d > − √ d > − √ d >
0, and that the characteristic polynomial of W is φ W ( λ ) = ( λ + d ) f ( λ ) , where f ( λ ) = ( λ − dλ − d λ + d ) . For d ≥ f ( − d ) = d + 3 d − d − d < f (0) = d > f ( d ) = − d < f ( d ) = d − d − d + d >
0. Hence, the second eigenvalue ξ ( W ) of W is greater than 0.By Lemma 2.1, we get ξ ( G ) ≥ min { ξ ( W i ) | i = 2 , , , } >
0, a contradiction.
Case 2. ε G ( v ) = d − ε G ( v d − ). Then, the the principal submatrix of A ( G ) indexed by { v , v d , v , v d − } is W = d d − d d − d − d − with the second largest eigenvalue ξ ( W ) = ( − d + √ − d + 5 d ) >
0, and so ξ ( G ) ≥ ξ ( W ) > G ) ≤ G ∈ G . P ① v ① v ① v ① v C ① v ① v ① v ① v P ∪ K ① v ① v ① v ① v Fig. 1 : Graphs in Lemma 2.5.
Lemma 2.5.
None graph in G contains one in { P , C , P ∪ K } as an induced subgraph. Proof.
By Lemma 2.4 it suffices to consider diam( G ) = 2. Assume by the contradiction that P = v v v v ⊆ G . Due to ε G ( v i ) ≤ ≤ i ≤ d G ( v i , v j ) ≤ i, j ∈ { , , , } .Hence, we get d G ( v , v ) = 2 , d G ( v , v ) = 2 , d G ( v , v ) = 2 and ε G ( v i ) = 2 ( i = 1 , , , A ( G ) indexed by these four vertices are W = with the second largest eigenvalue ξ ( W ) = √ −
1. By Lemma 2.1 we get ξ ( G ) ≥ ξ ( W ) > P * G . 5et C ⊆ G . Clearly, the principal submatrix of A ( G ) indexed by V ( C ) is W = whose the second largest eigenvalue is ξ ( W ) = 2. Thus, ξ ( G ) ≥ >
0, a contradiction.For P ∪ K ⊆ G , the principal submatrix of A ( G ) indexed by those vertices is W = with the second largest eigenvalue ξ ( W ) ≈ . ξ ( G ) >
0, a contradiction.This completes the proof.Note that the vertex set of S ( t , t , . . . , t k ) is V ∪ V ∪ · · · ∪ V k , where the corresponding setof t i is V i = { v i , v i , . . . , v in i } (0 ≤ i ≤ k ). Lemma 2.6.
Let G ′ = S ( t , t , . . . , t k ) with k ≥ and t i ≥ ( ≤ i ≤ k ) . If G ′ ∈ G and G ∈ G is the connected graph obtained from G ′ by adding a new vertex w , then w must be adjacent toall vertices of V . Proof.
For the mixed extension of star S ,k , we get that any vertex in V is adjacent to thoseones in other V ′ i s (1 ≤ i ≤ k ), and that the vertices in V i and the vertices in V j are not mutuallyadjacent (1 ≤ i = j ≤ k ).Assume by way of contradiction that there exists some vertex v ∈ V is not adjacent to w .For v i ∈ V i and v j ∈ V j ( i = j ), by the above statement we get v , v i and v , v j are adjacent.Due to P ∪ P * G (see Lemma 2.5), w is adjacent to at least one vertex of v i and v j . Then G [ v , v i , w, v j ] ∼ = C if w is adjacent to both v i and v j , and otherwise G [ v , v i , w, v j ] ∼ = P .This contradicts Lemma 2.5.As shown above, w must be adjacent to all vertices of V . Proposition 2.7.
Let G ∈ G with order n . Then G is the mixed extension S ( t , t , . . . , t k ) ,where k ≥ and t i ≥ ( ≤ i ≤ k ). Proof.
By Lemma 2.4, if diam( G ) = 1 we get G ∼ = K n ∼ = S ( t , n − t ). We next set diam( G ) = 2.If G is a tree, by diam( G ) = 2 we get G ∼ = S ,n − ∼ = S (1 , , . . . ,
1) ( n ≥ G containing at least one cycle. By C , P * G (Lemma 2.5), then G only contains the triangle C . If n = 3, then G ∼ = C ∼ = S (1 , n = 4, clearly we get G = S (1 , ,
2) or S (2 , , n ≥ n −
1. Due to the connectedness of G , thereexists a vertex w such that G ′ = G − w is connected. If diam( G ′ ) = 1, then G ′ ∼ = K n − ∼ = S ( t , n − − t ) ∈ G . If diam( G ′ ) ≥
3, then P ⊆ G ′ ⊆ G contradicting to Lemma 2.5. Hence,diam( G ′ ) = 2. By Lemma 2.2 we get ξ ( G ) ≥ ξ ( H ) ≥ ξ ( G ) ≥ ξ ( H ) ≥ · · · ≥ ξ n − ( H ) ≥ ξ n ( G )which, along with G ∈ G , leads to G ′ ∈ G . By inductive hypothesis, we can set G ′ ∼ = S ( t ′ , t ′ , . . . , t ′ k ), where P ki =0 t ′ i = n − V ′ i = { v i ′ , · · · , v it ′ i } , k ≥ t ′ i ≥ ≤ i ≤ k ). ByLemma 2.6, we get that w is adjacent to all vertex of V ′ .6f w is not adjacent to any vertex of V ′ ∪ · · · ∪ V ′ k , then G ∼ = S ( t ′ , t ′ , . . . , t ′ k , w isadjacent to a vertex (say, v s ′ ) of one set in { V ′ , . . . , V ′ k } (say, V ′ s ), then w must be adjacent toall vertices of V ′ s . Otherwise, there exits a vertex v st ′ a ∈ V ′ s such that w and v st ′ a is not adjacentwhich implies G [ w, v s ′ , v st ′ a , v j ′ ] ∼ = P ∪ K ⊆ G , contradicting to Lemma 2.5. Consequently, G ∼ = S ( t ′ , t ′ , . . . , t ′ i + 1 , . . . , t ′ k ). If w is adjacent to the vertices of at least two sets in { V ′ , . . . , V ′ k } (say, V ′ i and V ′ j (0 ≤ i = j ≤ k )), for v i ′ ∈ V ′ i and v j ′ ∈ V ′ j we conclude that w must be adjacentto all vertices of V ′ ∪· · ·∪ V ′ k . Otherwise, there is a vertex v ∈ V ′ ∪· · ·∪ V ′ k satisfying that v and w is not adjacent. In this case, G [ w, v, v i ′ , v j ′ ] ∼ = P if v ∈ V ′ i ∪ V ′ j , or G [ w, v, v i ′ , v j ′ ] ∼ = P ∪ K if v / ∈ V ′ i ∪ V ′ j , a contradiction. Therefore, w is adjacent to all vertices of V ′ ∪ · · · ∪ V ′ k , and so G ∼ = S ( t ′ + 1 , t ′ , . . . , t ′ k ).This finishes the proof.Let S ( t , t , . . . , t k ) be a mixed extension with k ≥
1. If t ≥ t ≥ · · · ≥ t q ≥ > t q +1 = · · · = t k = 1, then S ( t , t , . . . , t k ) ∼ = S ( t , − p, t , . . . , t q ), where 0 ≤ p, q ≤ k ( p = k − q ) and t j ≥ ≤ j ≤ q ). Proposition 2.8.
Let G = S ( t , − p, t , . . . , t q ) be a mixed extension with p, q ≥ and t j ≥ ( ≤ j ≤ q ). If G ∈ G , then (i) t = 1 , p + q ≥ ; (ii) t = 2 , p, q ≥ ; (iii) t = 3 , ≤ q ≤ − p (0 ≤ p ≤ ; (vi) t = 4 , ≤ q ≤ − p (0 ≤ p ≤ ; (v) t ≥ , ≤ q ≤ − p (0 ≤ p ≤ . Proof.
Let t ≥
5. If p ≥
3, then F ⊆ G contradicting to Lemma 2.3. Hence, p ≤
2. If p = 2,then q = 0 by F * G (see Lemma 2.3). If p = 0 ,
1, then q ≤ − p by F * G . Set t = 4. Since F * G , then p ≤
3. Due to F , F , F * G , we get q ≤ − p . Let t = 3. By F * G we obtain p ≤
4. In view of F i * G ( i = 9 , , , q ≤ − p ( p ≤ G isat least 2, we get p, q ≥ t = 2 and p + q ≥ t = 1.So far, we have shown the sufficiency of Theorem 1.1. We next show its necessary. Write { t , t , · · · , t q } as the multiset { k · t , · · · , k h · t h } , where k i is the number of t i (1 ≤ i ≤ h ).Clearly, S ( t , − p, t , . . . , t q ) = S ( t , − p, k · t , · · · , k h · t h ) and h P i =1 k i = q . Proposition 2.9.
Let G = S ( t , − p, k · t , · · · , k h · t h ) be a mixed extension with p, q ≥ and t j ≥ ( ≤ j ≤ h ). Under the conditions (i)–(v) in Proposition 2.8, then G has exactly onepositive eigenvalue. Furthermore, (i) p + q ≤ . For t ≥ , then G ∼ = K n and Spec A ( G ) = (cid:26) n − − n − (cid:27) . (ii) p + q ≥ and p ≥ . For t = 1 , q = 0 or t = 3 , q = 4 − p or t = 4 , q = 3 − p , we get Spec A ( G ) = (cid:26) ξ − − ξ − t h ξ · · · − t ξ h +3 − t n − t − p − q + 1 t − p − k h − · · · k − k − (cid:27) , here ξ ∈ ( − t h , − , ξ i ∈ ( − t h − i , − t h − i +1 ) (1 ≤ i ≤ h − .Otherwise, for the other cases in conditions (i)–(v) of Proposition 2.8, Spec A ( G ) = (cid:26) ξ ξ − − ξ − t h ξ · · · − t ξ h +4 − t n − t − p − q t − p − k h − · · · k − k − (cid:27) , where ξ ∈ ( − t h , − , ξ i ∈ ( − t h − i , − t h − i +1 ) (1 ≤ i ≤ h − . (iii) p + q ≥ and p = 0 . For t = 3 , q = 4 or t = 4 , q = 3 , then Spec A ( G ) = (cid:26) ξ − − t h ξ − t h − · · · − t ξ h +1 − t n − t − q + 1 t − k h − k h − − · · · k − k − (cid:27) , where ξ i ∈ ( − t h − i , − t h − i +1 ) (1 ≤ i ≤ h − .Otherwise, for the other cases in conditions (i)–(v) of Proposition 2.8, Spec A ( G ) = (cid:26) ξ ξ − − t h ξ − t h − · · · − t ξ h +2 − t n − t − q t − k h − k h − − · · · k − k − (cid:27) , where ξ i ∈ ( − t h − i , − t h − i +1 ) and ≤ i ≤ h − . Proof. If p + q ≤
1, then G = K n is the mixed extension S ( t , n − t ) withSpec A ( G ) = (cid:26) n − − n − (cid:27) . Let p + q ≥
2. Labelling the vertices of G properly, we get A ( G ) = J − I J J J . . . JJ J − I ) 2 J J . . . JJ J J . . . J ... ... ... ... . . . ... J J J J . . . , (1)with the A -polynomial φ ( G, λ ) = λ n − t − p − q ( λ + 1) t − ( λ + 2) p − g t , − p, ··· ,t q ( λ ) , (2)where g t , − p, ··· ,t q ( λ ) = h Y i =1 ( λ + 2 t i ) k i − h ( λ ) (3)and h ( λ ) = ( λ − pλ − t λ +3 λ + t p − t − p +2) h Y i =1 ( λ +2 t i ) − ( λ +2)(2 λ − t +2) h X j =1 ( k j t j h Y i =1 i = j ( λ +2 t i )) . Obviously, the polynomial (3) has q + 2 real roots denoted by λ ≥ · · · ≥ λ q +2 . When k i ≥ − t i is a root of (3) with multiplicity k i − ≤ i ≤ h ). Thus, the number of these roots is q − h , and hence the remaining h + 2 roots of (3) are the roots of h ( λ ). For 2 ≤ i ≤ h , we have h ( − t i − ) h ( − t i ) = k i − t i − k i t i (2 t i − − t i − t i − + t − t i + t − × [ i − Y j =1 (2 t j − t i − )(2 t j − t i )]( t i − t i − )( t i − − t i ) × [ h Y j = i +1 (2 t j − t i − )(2 t j − t i )] . k i − t i − k i t i (2 t i − − t i − t i − + t − t i + t − > i − Q j =1 (2 t j − t i − )(2 t j − t i ) > t i − t i − )( t i − − t i ) < h Q j = i +1 (2 t j − t i − )(2 t j − t i ) >
0, then h ( − t i − ) h ( − t i ) < h ( λ ) that is in ( − t i − , − t i ) (2 ≤ i ≤ h ). At present, we have got q − G ∈ G we get λ >
0, and so we need find the other two roots of h ( λ ).If p ≥
1, then h ( − t h ) = − k h t h (2 t h + 2)( t + 4 t h − h − Q i =1 ( − t h + 2 t i ) < h ( −
2) = p ( t + 2) h Q i =1 (2 t i − >
0. Hence, λ ∈ ( − t h , − h ( −
1) = t ( p − h Q i =1 (2 t i −
1) + t h P j =1 ( k j t j h Q i =1 i = j (2 t i − h ( − > p + q ≥
2. As well, h (0) = ( t p − t − p + 2) h Y i =1 t i − − t + 2) h X j =1 (2 k j t j h Y i =1 i = j t i ) = 2 h ( qt + pt − t − q − p + 2) h Y i =1 t i . Under the conditions (i)–(v) in Proposition 2.8, if t = 1, then q ≥ h (0) = 2 h ( − q ) h Q i =1 t i < q ≥ q = 0). If t = 2, then q ≥ h (0) = 2 h ( − h Q i =1 t i <
0. If t = 3, then0 ≤ q ≤ − p and h (0) = 2 h ( p + q − h Q i =1 t i < q < − p (or q = 4 − p ). If t = 4,then 0 ≤ q ≤ − p and h (0) = 2 h (2 p + 2 q − h Q i =1 t i < q < − p (or q = 3 − p ).If t ≥
5, then 0 ≤ q ≤ − p and h (0) = 2 h (3 p + 3 q − h Q i =1 t i <
0. Consequently, for the cases t = 1 , q = 0, or t = 3 , q = 4 − p , or t = 4 , q = 3 − p we get λ = 0, and hence Spec A ( G ) = (cid:26) ξ − − ξ − t h ξ · · · − t ξ h +3 − t n − t − p − q + 1 t − p − k h − · · · t − k − (cid:27) , where ξ ∈ ( − t h , −
2) and ξ i ∈ ( − t h − i , − t h − i +1 ) (1 ≤ i ≤ h − λ ∈ ( − , Spec A ( G ) = (cid:26) ξ ξ − − ξ − t h ξ · · · − t ξ h +4 − t n − t − p − q t − p − k h − · · · t − k − (cid:27) , where ξ ∈ ( − t h , −
2) and ξ i ∈ ( − t h − i , − t h − i +1 ) (1 ≤ i ≤ h − p = 0, then q ≥ φ ( G, λ ) = λ n − t − q ( λ + 1) t − Q hi =1 ( λ + 2 t i ) k i − l ( λ )with l ( λ ) = ( λ − t + 1) h Y i =1 ( λ + 2 t i ) − (2 λ − t + 2) h X j =1 ( k j t j h Y i =1 i = j ( λ + 2 t i )) . A straightforward calculation shows that l ( −
1) = − t h Y i =1 (2 t i −
1) + t h X j =1 ( k j t j h Y i =1 i = j (2 t i − t [ X j =1 (( k j − t j h Y i =1 i = j (2 t i − t − t ) h Y i =3 (2 t i −
1) + h X j =3 ( k j t j h Y i =1 i = j (2 t i − h Y i =2 (2 t i − > l (0) = ( − t + 1) h Q i =1 t i − ( − t + 2) h P j =1 (2 k j t j h Q i =1 i = j t i ) = 2 h − ( qt − t − q + 2) h Q i =1 t i . Underthe conditions (i)–(v) in Proposition 2.8, if t = 1, then q ≥ l (0) = − h − q h Q i =1 t i <
0; if t = 2, then l (0) = − h h Q i =1 t i <
0; if t = 3, then 2 ≤ q ≤ l (0) = 2 h − ( q − h Q i =1 t i < q < q = 4); if t = 4, then 2 ≤ q ≤ l (0) = 2 h − (2 q − h Q i =1 t i < q < q = 3); if t ≥
5, then l (0) = − h h Q i =1 t i <
0. Consequently, for ( t , q ) = (3 ,
4) or( t , q ) = (4 ,
3) we have λ = 0, andSpec A ( G ) = (cid:26) ξ − − t h ξ − t h − · · · ξ h +1 − t n − t − q + 1 t − k h − k h − − · · · k − (cid:27) where ξ i ∈ ( − t h − i , − t h − i +1 ) (1 ≤ i ≤ h − λ ∈ ( − ,
0) andSpec A ( G ) = (cid:26) ξ ξ − − t h ξ − t h − · · · ξ h +2 − t n − t − q t − k h − − t h − · · · k − (cid:27) with ξ i ∈ ( − t h − i , − t h − i +1 ) (1 ≤ i ≤ h − Proof of Theorem 1.1 . This theorem follows from Propositions 2.7, 2.8 and 2.9.
Remark 2.10.
Recently, Sorgun and K¨u¸c¨uk [19] independently studied the graphs with exactlyone positive A -eigenvalue; however, their proof is incomplete. Graphs with all but at most two A -eigenvalues equal to − and Let K n ,n ,...,n l be the complete multipartite graph with n ≥ n ≥ · · · ≥ n l . Lemma 3.1. [22]
Let G ∼ = K n ∨ K n ,...,n l with n ≥ , n r ≥ and l ≥ ≤ r ≤ l ) . Then φ ε ( G, λ ) = ( λ + 1) n − ( λ + 2) n − n − l [( λ − n + 1) l Y r =1 ( λ − n r + 2) − n l X r =1 l Y s =1 r = s n r ( λ − n s + 2)] . Lemma 3.2. [22]
Let G be a graph with order n and least A -eigenvalue ξ n ( G ) . Then ξ n ( G ) = − if and only if (i) G ∼ = K n ,n ,...,n l , where l ≥ and n r ≥ ≤ r ≤ l ) ; (ii) G ∼ = K n ∨ K n ,...,n l , where n r ≥ ≤ r ≤ l ) and ≤ l ≤ if n = 1 , ≤ l ≤ if n = 2 or l = 2 if n ≥ . Note that the connected graphs with order n ≥ A -eigenvalue. Let S denote the set of connected graphs with all but at most two A -eigenvalues equal to − G ∈ S , G might have exactly one positive A -eigenvalue, or two positive A -eigenvalues,or one positive and one negative A -eigenvalue different from − Lemma 3.3.
Let G ∈ S with order n . Then None graph in S has exactly one A -eigenvalue different from and − . (ii) G has two positive A -eigenvalues if and only if G ∼ = K n ,n with n , n ≥ . Proof. If G ∈ S has exactly one or two positive A -eigenvalue different from 0 and −
2, thenthe least A -eigenvalue of G is −
2. By Lemma 3.2, we discuss the following two cases.
Case 1. G ∼ = K n ,n ,...,n l , where l ≥ n r ≥ ≤ r ≤ l ). Since Spec A ( K n ,n ,...,n l ) = { n − , n − , . . . , n l − , − ( n − l ) } , then G has at least two positive A -eigenvalues.Thereby, G ∼ = K n ,n with n , n ≥ Case 2. G ∼ = K n ∨ K n ,...,n l , where n ≥ l ≥ n ≥ · · · ≥ n l ≥ ≤ r ≤ l ). By Lemma3.1, we get φ ε ( G, λ ) = ( λ + 1) n − ( λ + 2) n − n − l f n ,n ,...,n l ( λ ) , (4)where f n ,n ,...,n l ( λ ) = ( λ − n + 1) l Q r =1 ( λ − n r + 2) − n l P r =1 l Q s =1 r = s n r ( λ − n s + 2)]. If n ≥
3, then G has at least three A -eigenvalues (i.e., one positive A -eigenvalues and at least two A -eigenvalues −
1) different from − n ≤
2. Due to Lemma 3.2, we have2 ≤ l ≤ Subcase 2.1. l = 2. From Lemma 3.2(ii) it follows that n ≥
1. By (4) we get f n ,n ,n ( λ ) = ( λ − n + 1)( λ − n + 2)( λ − n + 2) − n n ( λ − n + 2) − n n ( λ − n + 2) . By calculations we get f n ,n ,n ( −
2) = − n n < f n ,n ,n ( −
1) = ( n + n − n > f n ,n ,n (2 n −
2) = 2 n n ( n − n ) ≥ f n ,n ,n (2 n −
2) = 2 n n ( n − n ) ≤ f n ,n ,n (2( n + n + n ) −
2) = 2 n (4 n + 3 n ) + 2 n [3 n + 2( n − n + n (5 n − n [4 n + n (5 n −
2) + 2 n (5 n − >
0. Hence, G has more than two A -eigenvalue differentfrom − ξ ∈ [2 n − , n + n + n ) − , ξ ∈ [2 n − , n − , ξ n +2 ∈ ( − , − Subcase 2.2. l = 3. By Lemma 3.2(ii) we get n = 1 or 2. If n = 1, then G = K ∨ K n ,n ,n and f ,n ,n ,n ( λ ) = λ ( λ − n + 2)( λ − n + 2)( λ − n + 2) − n n ( λ − n + 2)( λ − n + 2) − n n ( λ − n + 2)( λ − n + 2) − n n ( λ − n + 2)( λ − n + 2) . We get f ,n ,n ,n ( −
2) = 4 n n n > f ,n ,n ,n (2 n −
2) = 4 n ( n − n )( n − n ) ≤ f ,n ,n ,n (2 n −
2) = 4 n ( n − n )( n − n ) ≥ f ,n ,n ,n (2 n −
2) = − n ( n − n )( n − n ) ≤ f ,n ,n ,n (2( n + n + n ) −
2) = 4( n ( − n + 4 n ) + n ( n + n )(8 n + 8 n −
5) + ( n + n )(4 n ( n − n − n + n (4 n − n (4 n + n (4 n − n n (16 n − n (16 n − > G has more than two A -eigenvalues different from − ξ ∈ [2 n − , n + n + n ) − , ξ ∈ [2 n − , n − , ξ ∈ [2 n − , n − , ξ n +3 ∈ ( − , n − n = 2, then G = K ∨ K n ,n ,n and A ( K ∨ K n ,n ,n ) is the principle submatrix of A ( K ∨ K n ,n ,n ). From Lemma 2.1, by the above discussion we get ξ ≥ n − , ξ ≥ n − ξ ≥ n −
2, a contradiction.
Case 2.3. l = 4. In view of Lemma 3.2(ii) we get n = 1 and G = K ∨ K n ,n ,n ,n . Note that A ( K ∨ K n ,n ,n ) is the principle submatrix of A ( K ∨ K n ,n ,n ,n ). From Lemma 2.1, by Case2.2 we obtain ξ ≥ n − , ξ ≥ n − ξ ≥ n −
2, a contradiction.As proved above, we get that there is no graph in S having exactly one positive A -eigenvalue,and that G ∼ = K n ,n ( n , n ≥
2) if G ∈ S has two positive A -eigenvalues.11e finally identify the graphs G ∈ S which have one positive and one negative A -eigenvalue.At this moment, G has exactly one positive A -eigenvalue, and so G ∈ G defined in Section 2. Lemma 3.4. G ∈ S ∩ G if and only if G is the star S ,p = S (1 , − p ) with p ≥ . Proof.
For G ∈ G , by Theorem 1.1 we get G ∼ = S ( t , − p, t , . . . , t q ) with the those restrictedconditions. Recall, t ≥ t ≥ · · · ≥ t q ≥
2. If t ≥
3, then by (3) we deduce that G at least two A -eigenvalues −
1, a contradiction. Hence, t ≤ Case 1. t = 1. By Theorem 1.1(i) we get p + q ≥ p, q ≥ Subcase 1.1. p = 0. Then q ≥
1, and thus G ∼ = S (1 , t , t , . . . , t q ) with t j ≥ ≤ j ≤ q ). If q = 1, then G ∼ = K n / ∈ S . So, set q ≥
2. Let g t , − p, ··· ,t q ( λ ) be defined in (3).If q = 2, then G ∼ = S (1 , t , t ). From Proposition 2.9(iii) it follows that G has three A -eigenvalues different from − ξ > , ξ n − ∈ ( − , , ξ n ∈ [ − t , − t )), a contradic-tion.If q = 3, then G ∼ = S (1 , t , t , t ). By Proposition 2.9(iii) we get that G has four A -eigenvaluesdifferent from − ξ > , ξ n − ∈ ( − , , ξ n − ∈ [ − t , − t ) , ξ n ∈ [ − t , − t )), acontradiction.If q ≥
4, then G = S (1 , t , · · · , t q ) and A ( S (1 , t , t , t )) is the principal submatrix of A ( S (1 , t , · · · , t q )). By Lemma 2.1 and the above conclusion, we get ξ > , ξ n − ≤ − t and ξ n ≤ − t , a contradiction. Subcase 1.2. p = 1. Then q ≥
0. If q = 0, then G = S (1 , − ∈ S ∩ G .If q = 1, then G ∼ = S (1 , − , t ) and G has three A -different from − ξ > , ξ n − ∈ ( − , , ξ n ∈ ( − t , − q = 2, then G ∼ = S (1 , − , t , t ). From Proposition 2.9(ii) it follows that G has four A -eigenvalues different from − ξ > , ξ n − ∈ ( − , , ξ n − ∈ [ − t , − , ξ n ∈ [ − t , − t )), a contradiction.If q ≥
3, then G ∼ = A ( S (1 , − , t , · · · , t q )) and A ( S (1 , − , t , t )) is the principal submatrixof A ( S (1 , − , t , · · · , t q )). By Lemma 2.1 and the above case, we get ξ > , ξ n − < − ξ n ≤ − t , a contradiction. Subcase 1.3. p ≥
2. Then q ≥
0. If q = 0, then G ∼ = S (1 , − p ) andSpec A ( S (1 , − p )) = { p p − p + 1 + p − , ( n − t − p − q ) , − p p − p + 1 + p − , − ( p − } . Thus, the star S (1 , − p ) ∈ S ∩ G .If q = 1, then G ∼ = S (1 , − p, t ). By Proposition 2.9 we get G has three A -eigenvalues differentfrom − ξ > , ξ n − p ∈ ( − , , ξ n ∈ ( − t , − q ≥
2, then G = S (1 , − p, t , · · · , t q ) and A ( S (1 , − , t , t )) is the principal submatrix of A ( S (1 , − p, t , · · · , t q )). Similarly to the last case of Subcase 1.2 we get a conflict. Case 2. t = 2. By Theorem 1.1 we get p, q ≥ Subcase 2.1. p = 0. Then q ≥ G ∼ = S (2 , t , t , . . . , t q ). If q = 1, then G ∼ = K n / ∈ S .If q = 2, then G ∼ = S (2 , t , t ) and G has three A -eigenvalues different from − ξ > , ξ n − ∈ ( − , , ξ n ∈ [ − t , − q ≥
3, then G ∼ = S (2 , t , · · · , t q ) and A ( S (1 , t , t , t )) is the principal submatrix of A ( S (2 , t , · · · , t q )). Similarly to Subcase 1.1 (when q = 3), we get a contradiction. Subcase 2.2. p = 1. So q ≥
0. If q = 0, then G ∼ = S (2 , − ∼ = C S . If q = 1, then G ∼ = S (2 , − , t ). By Proposition 2.9(ii) we get that G has three A -eigenvalues different from12 ξ > , ξ n − ∈ ( − , , ξ n ∈ ( − t , − q ≥
2, then G = S (2 , − , t , · · · , t q ) and A ( S (1 , − , t , t )) is the principal submatrix of A ( S (2 , − , t , · · · , t q )).Similarly to Subcase 1.2 (when q = 2), we get G / ∈ S . Subcase 2.3. p ≥
2. If q = 0, then G ∼ = S (2 , − p ) and we can get A -eigenvalues of G fromProposition 2.9(ii). G has three A -eigenvalues different from − ξ > , ξ n − p − ∈ ( − , , ξ n − p = − q = 1, then G ∼ = S (2 , − p, t ). Hence, G has three A -eigenvalues different from − ξ > , ξ n − p − ∈ ( − , , ξ n ∈ ( − t , − q ≥
2, then G ∼ = S (2 , − p, t , · · · , t q ) and A ( S (1 , − , t , t )) is the principal submatrix of A ( S (2 , − p, t , · · · , t q )). Similarly, G S .As discussed above, G ∈ S ∩ G if and only if G ∼ = S (1 , − p ) with p ≥ S ,p = S (1 , − p ) is a special kind of complete bipartite graphs K n ,n ∼ = S ( − n , − n ). Proof of Theorem 1.2.
This theorem follows from Lemmas 3.3 and 3.4.
As Haemers pointed out [7], the mixed extension of graphs is a special case of the so called generalized composition (see [18] for details). However, Haemers’s definition is more convenientand powerful. It is quite helpful for the classifications of graphs and further for identifying whichgraphs are determined by the spectra. On reflection, we propose the following problems, thefirst one of which is a natural step.
Problem 1.
Which graphs with exactly one positive A -eigenvalues are determined by their A -spectra? Problem 2.
Determine the graphs with all but two A -eigenvalues equal to − and − . Fowler and Pisanski [16,17] introduced the notion of the HL-index of a graph w.r.t. toadjacency matrix. It is related to the HOMO-LUMO separation studied in theoretical chemistry.Similarly, the HL -index R A ( G ) w.r.t. anti-adjacency matrix of a (molecular) graph G of order n is defined as R A ( G ) = max {| ξ H | , | ξ L |} , where H = ⌊ n +12 ⌋ , L = ⌈ n +12 ⌉ . See [13, 14, 15, eg.] for more details about the HL-index w.r.t.adjacency matrices of graphs.Actually, by cumbersome calculations we can completely determine the HL-index w.r.t. anti-adjacency matrix of graphs in Theorem 1.1. But we expect more general results about this topic. Problem 3.
Investigate the HL-index w.r.t. anti-adjacency matrix of graphs.
Another interesting problem is the nullity η A ( G ) of a graph G , which is defined to be themultiplicity of zero as an eigenvalue of the adjacency matrix. Similarly, we consider the nullity η A ( G ) of the anti-adjacency matrix. The nullity of a graph is important in mathematics, sinceit is related to the singularity of (anti-)adjacency matrix. Note that the anti-adjacency matricesof graphs seems to be usually more sparse than adjacency matrix. Hence, the nullity of anti-adjacency matrix may be larger than that of adjacency matrix. For example, η A ( P k +1 ) = 1and η A ( P k ) = 0 for k ≥
1; while η A ( P k +1 ) = 2 k − η A ( P k ) = 2 k − k ≥ roblem 4. For the anti-adjacency matrix, give lower and upper bounds of nullity involvinggraph parameters, and characterize the extreme graphs.
On the other hand, it is well-know that the HL-index and the nullity of graphs are of greatinterest in chemistry. As a comparison in [22], it seems that the spectral radius of adjacencymatrix of graphs is closely related to the chemical properties of octane isomers, while the spectralradius of anti-adjacency matrix of graphs may be more efficient for the benzenoid hydrocarbons.In the end, we pose the last one problem to finish this paper.
Problem 5.
With respect to the adjacency and anti-adjacency matrices of graphs, compare theHL-index and nullity of graphs and study their applications in the chemistry.
References [1] A.E. Brouwer, W.H. Haemers, Spectra of Graphs, Springer, 2012.[2] S.M. Cioab˘a, W.H. Haemers, J.R. Vermette, W. Wong, The graphs with all but two eigenvaluesequal to ±
1, J. Algebr. Comb. 41 (2015) 887–897.[3] S.M. Cioab˘a, W.H. Haemers, J.R. Vermette, The graphs with all but two eigenvalues equal to − P , Elect. J. Combin. 26(3)(2019) − −
2, Graphs Combin. 34 (2018) 395–414.[10] L. Lu, Q.X. Huang, X.Y. Huang, The graphs with exactly two distance eigenvalues different from − −
3, J Algebr Comb. 45 (2017) 629–647.[11] I. Mahato, R. Gurusamy, M.R. Kannan, S. Arockiaraj, Spectra of eccentricity matrices of graphs,Discrete Appl. Math. 285 (2020) 252–260.[12] I. Mahato, R. Gurusamy, M.R. Kannan, S. Arockiaraj, On the spectral radius and the energy ofeccentricity matrix of a graph, arXiv:1909.05609v1.[13] B. Mohar, Median eigenvalues and the HOMO-LUMO index of graphs, J. Combin. Theory, Ser. B112 (2015) 78–92.[14] B. Mohar, Median Eigenvalues of Bipartite Subcubic Graphs, Combin. Probab. Comput. 25 (2016)768–790.[15] Mohar, Tayfeh-Rezaie, Median eigenvalues of bipartite graphs, J. Algebr. Comb. 41 (2015) 899–909.[16] M. Randi´c, D MAX -Matrix of dominant distances in a graph, MATCH Commun. Math. Comput.Chem. 70 (2013) 221–238.[17] A.J. Schwenk, Almost all trees are cospectral, in: F. Harary (Ed.), New Directions in the Theory ofGraphs, Academic Press, New York, 1973, pp. 275–307.
18] A.J. Schwenk, Computing the characteristic polynoimial of a graph, in: R. Bary, F. Harary(Eds.),Graphs Combinatorics, in: Lecture Notes in Mathematics, vol. 406, Springer, Berlin, 1974, pp.153–172.[19] S. Sorgun, H. K¨u¸c¨uk, On the graphs having exactly one positive eccentricity eigenvalue,arXiv:2012.10933v1.[20] H. Topcu, S. Sorgun, W. H. Haemers, The graphs cospectral with the pineapple graph, DiscreteAppl. Math. 269 (2019) 52–59.[21] F. Tura, On the eccentricity energy of complete multipartite graph, arXiv:2002.07140v1.[22] J.F. Wang, X.Y. Lei, S.C. Li, W. Wei, On the eccentricity matrix of graphs and its applications tothe boiling point of hydrocarbons, Chem. Intel. Lab. Sys. 207 (2020) 104173.[23] J.F. Wang, L. Lu, M. Randi´c, G.Z. Li, Graph energy based on the eccentricity matrix, DiscreteMath. 342 (2019) 2636–2646.[24] J.F. Wang, M. Lu, F. Belardo, M. Randi´c, The anti-adjacency matrix of a graph: eccentricity matrix,Discrete Appl. Math. 251 (2018) 299–309.[25] J.F. Wang, M. Lu, M. Brunetti, L. Lu, X.Y. Huang, Spectral determinations and eccentricity matrixof graphs, manuscript and submitted.[26] J. Wang, M. Lu, L. Lu, F. Belardo, Spectral properties of the eccentricity matrix of graphs, DiscreteAppl. Math. 279 (2020) 168–177.[27] W. Wei, X.C. He, S.C. Li, Solutions for two conjectures on the eigenvalues of the eccentricity matrix,and beyond, Discrete Math. 343 (2020) 111925.18] A.J. Schwenk, Computing the characteristic polynoimial of a graph, in: R. Bary, F. Harary(Eds.),Graphs Combinatorics, in: Lecture Notes in Mathematics, vol. 406, Springer, Berlin, 1974, pp.153–172.[19] S. Sorgun, H. K¨u¸c¨uk, On the graphs having exactly one positive eccentricity eigenvalue,arXiv:2012.10933v1.[20] H. Topcu, S. Sorgun, W. H. Haemers, The graphs cospectral with the pineapple graph, DiscreteAppl. Math. 269 (2019) 52–59.[21] F. Tura, On the eccentricity energy of complete multipartite graph, arXiv:2002.07140v1.[22] J.F. Wang, X.Y. Lei, S.C. Li, W. Wei, On the eccentricity matrix of graphs and its applications tothe boiling point of hydrocarbons, Chem. Intel. Lab. Sys. 207 (2020) 104173.[23] J.F. Wang, L. Lu, M. Randi´c, G.Z. Li, Graph energy based on the eccentricity matrix, DiscreteMath. 342 (2019) 2636–2646.[24] J.F. Wang, M. Lu, F. Belardo, M. Randi´c, The anti-adjacency matrix of a graph: eccentricity matrix,Discrete Appl. Math. 251 (2018) 299–309.[25] J.F. Wang, M. Lu, M. Brunetti, L. Lu, X.Y. Huang, Spectral determinations and eccentricity matrixof graphs, manuscript and submitted.[26] J. Wang, M. Lu, L. Lu, F. Belardo, Spectral properties of the eccentricity matrix of graphs, DiscreteAppl. Math. 279 (2020) 168–177.[27] W. Wei, X.C. He, S.C. Li, Solutions for two conjectures on the eigenvalues of the eccentricity matrix,and beyond, Discrete Math. 343 (2020) 111925.