Spectra and Laplacian spectra of arbitrary powers of lexicographic products of graphs
Nair Abreu, Domingos M. Cardoso, Paula Carvalho, Cybele T. M. Vinagre
aa r X i v : . [ m a t h . C O ] N ov Spectra and Laplacian spectra of arbitrarypowers of lexicographic products of graphs
Nair Abreu , Domingos M. Cardoso , Paula Carvalho , andCybele T. M. Vinagre PEP/COPPE, Universidade Federal do Rio de Janeiro, Rio de Janeiro, Brasil.Email: [email protected] Centro de Investiga¸c˜ao e Desenvolvimento em Matem´atica e Aplica¸c˜oes Departamento de Matem´atica, Universidade de Aveiro, 3810-193, Aveiro, Portugal.Email: (dcardoso,paula.carvalho)@ua.pt Instituto de Matem´atica e Estat´ıstica, Universidade Federal Fluminense, Niter´oi,Brasil. Email: [email protected]ff.br
August 13, 2018
Abstract
Consider two graphs G and H . Let H k [ G ] be the lexicographic productof H k and G , where H k is the lexicographic product of the graph H byitself k times. In this paper, we determine the spectrum of H k [ G ] and H k when G and H are regular and the Laplacian spectrum of H k [ G ] and H k for G and H arbitrary. Particular emphasis is given to the least eigenvalue ofthe adjacency matrix in the case of lexicographic powers of regular graphs,and to the algebraic connectivity and the largest Laplacian eigenvaluesin the case of lexicographic powers of arbitrary graphs. This approachallows the determination of the spectrum (in case of regular graphs) andLaplacian spectrum (for arbitrary graphs) of huge graphs. As an example,the spectrum of the lexicographic power of the Petersen graph with thegoogol number (that is, 10 ) of vertices is determined. The paper finishwith the extension of some well known spectral and combinatorial invariantproperties of graphs to its lexicographic powers. AMS Subject Classification : 05C50, 05C76, 15A18.
Keywords : Graph spectra, graph operations, lexicographic product of graphs.1
Introduction
The lexicographic product of a graph H by himself several times is a very specialgraph product, it is a kind of fractal graph which reproduces its copy in eachof the positions of its vertices and connects all the vertices of each copy withanother copy when they are placed in positions corresponding to adjacent ver-tices of H . This procedure can be repeated, reproducing a copy of the previousiterated graph in each of the positions of the vertices of H and so on. Despitethe spectrum and Laplacian spectrum of the lexicographic product of two graphs(with some restrictions regarding the spectrum) expressed in terms of the twofactors be well known (see [4], where a unified approach is given), it is not thecase of the spectra and Laplacian spectra of graphs obtained by iterated lexi-cographic products, herein called lexicographic powers, of regular and arbitrarygraphs, respectively. A lexicographic power H k of a graph H can produce a graphwith a huge number of vertices whose spectra and Laplacian spectra may not bedetermined using their adjacency and Laplacian matrices, respectively. The ex-pressions herein deduced for the spectra and Laplacian spectra of lexicographicpowers can be easily programmed, for example, in Mathematica, and the resultscan be obtained immediately. For instance, the spectrum of the 100-th lexico-graphic power of the Petersen graph, presented in Section 3, was obtained byMathematica and the computations lasted only a few seconds. Notice that suchlexicographic power has the googol number (that is, 10 ) of vertices.The paper is organized as follows. In the next section, the notation is intro-duced and some preliminary results are given. The main results are introducedin Section 3, where the spectra (Laplacian spectra) of H k [ G ] and H k , when G and H are regular (arbitary) graphs, are deduced. Particular attention is givento the Laplacian index and algebraic connectivity of the lexicographic powers ofarbitrary graphs. In Section 4, the obtained results are applied to extend somewell known properties and spectral relations of combinatorial invariants of graphs H to its lexicographic powers H k . In this work we deal with simple and undirected graphs. If G is such a graph oforder n , its vertex set is denoted by V ( G ) and its edge set by E ( G ). The elementsof E ( G ) are denoted by ij , where i and j are the extreme vertices of the edge ij .The degree of j ∈ V ( G ) is denoted by d G ( j ), the maximum and minimum degreeof the vertices in G are δ ( G ) and ∆( G ) and the set of the neighbors of a vertex j is N G ( j ). The adjacency matrix of G is the n × n matrix A G whose ( i, j )-entry isequal to 1 whether ij ∈ E ( G ) and is equal to 0 otherwise. The Laplacian matrix of G is the matrix L G = D − A G , where D is the diagonal matrix whose diagonal2lements are the degrees of the vertices of G . Since A G and L G are symmetricmatrices, theirs eigenvalues are real numbers. From Gerˇsgorin’s theorem, theeigenvalues of L G are nonnegative. The multiset (that is, the set with possiblerepetitions) of eigenvalues of a matrix M is called the spectrum of M and denoted σ ( M ). Throughout the paper, we write σ A ( G ) = { λ [ g ]1 , . . . , λ [ g s ] s } (respectively, σ L ( G ) = { µ [ l ]1 , . . . , µ [ l t ] t } ) when λ > . . . > λ s ( µ > . . . > µ t ) are the distincteigenvalues of A G ( L G ) indexed in decreasing order - in this case, γ [ r ] means thatthe eigenvalue γ has multiplicity r . If convenient, we write γ ( G ) in place of γ toindicate an eigenvalue of a matrix associated to G , and we denote the eigenvaluesof A G (respectively, L G ) indexed in non increasing order, as λ ( G ) ≥ · · · ≥ λ n ( G )( µ ( G ) ≥ · · · ≥ µ n ( G )).As usually, the adjacency matrix eigenvalues of a graph G are called the eigenvalues of G . We remember that µ n ( G ) = 0 (the all one vector is theassociated eigenvector) and its multiplicity is equal to the number of componentsof G . Besides, µ n − ( G ) is called the algebraic connectivity of G [9]. Furtherconcepts not defined in this paper can be found in [5, 7].The lexicographic product (also called the composition ) of the graphs H and G is the graph H [ G ] (also denoted by H ◦ G ) for which the vertex set is thecartesian product V ( H ) × V ( G ) and such that a vertex ( x , y ) is adjacent tothe vertex ( x , y ) whenever x is adjacent to x or x = x and y is adjacentto y (see [15] and [17] for notations and further details). This graph operationwas introduced by Harary in [13] and Sabidussi in [19]. It is immediate that thelexicographic product is associative but it is not commutative.The lexicographic product was generalized in [20] as follows: consider a graph H of order n and graphs G i , i = 1 , . . . , n , with vertex sets V ( G i )s two by twodisjoints. The generalized composition H [ G , . . . , G n ] is the graph such that V ( H [ G , . . . , G n ]) = n [ i =1 V ( G i ) and E ( H [ G , . . . , G n ]) = n [ i =1 E ( G i ) ∪ [ ij ∈ E ( H ) E ( G i ∨ G j ) , where G i ∨ G j denotes the join of the graphs G i and G j . This operation iscalled in [4] the H -join of graphs G , . . . , G n . In [20] and [4], the spectrum of H [ G , . . . , G n ] is provided, where H is an arbitrary graph and G , . . . , G n areregular graphs. Furthermore, in [10] and [4], using different approaches, it wascharacterized the spectrum of the Laplacian matrix of H [ G , . . . , G n ] for arbitrarygraphs.Now, let us focus on the spectrum of the adjacency and Laplacian matrix ofthe above generalized graph composition. Theorem 2.1. [4] Let H be a graph such that V ( H ) = { , . . . , n } and, for = 1 , . . . , n , let G j be a p j -regular graph with order m j . Then σ A ( H [ G , . . . , G n ]) = n [ j =1 ( σ A ( G j ) \ { p j } ) ! ∪ σ ( C ) , (1) where C = p c . . . c n − c n c p . . . c n − c n ... ... . . . ... ... c ( n − c ( n − . . . p n − c ( n − n c n c n . . . c n ( n − p n (2) and c ij = (cid:26) √ m i m j if ij ∈ E ( H ) , otherwise. (3)Let H be a graph of order n and G be an arbitrary graph. If, for 1 ≤ i ≤ n , G i is isomorphic to G , it follows immediately that H [ G , . . . , G n ] = H [ G ], a factalso noted in [2]. In particular case of a regular graph G , Theorem 2.1 impliesthe corollary below. Corollary 2.2.
Let H be a graph of order n with σ A ( H ) = { λ [ h ]1 ( H ) , . . . , λ [ h t ] t ( H ) } and let G be a p -regular graph of order m such that σ A ( G ) = { λ [ g ]1 ( G ) , . . . , λ [ g s ] s ( G ) } .Then σ A ( H [ G ]) = { p [ n ( g − , . . . , λ [ ng s ] s ( G ) } ∪ { ( mλ ( H ) + p ) [ h ] , . . . , ( mλ t ( H ) + p ) [ h t ] } . Now it is worth to recall the following result.
Theorem 2.3. [4] Let H be a graph such that V ( H ) = { , . . . , n } and, for each j ∈ { , . . . , n } , let G j be a graph of order m j with Laplacian spectrum σ L ( G j ) .Then the Laplacian spectrum of H [ G , . . . , G n ] is given by σ L ( H [ G , . . . , G n ]) = n [ j =1 ( s j + ( σ L ( G j ) \ { } )) ! ∪ σ ( C ) , where s j = (cid:26) P i ∈ N H ( j ) m i , if N H ( j ) = ∅ , , otherwiseand s j + ( σ L ( G j ) \ { } ) means that the number s j is added to each element of σ L ( G j ) \ { } , and C = s − c . . . − c n − − c n − c s . . . − c n − − c n ... ... . . . ... ... − c ( n − − c ( n − . . . s n − − c ( n − n − c n − c n . . . − c n ( n − s n with (4)4 ij = (cid:26) √ m i m j if ij ∈ E ( H ) , otherwise . (5)Assuming that G , . . . , G n are all isomorphic to a particular graph G we havethe next corollary, which was also proved in [2]. Corollary 2.4.
Let H be a graph of order n with σ L ( H ) = { µ ( H ) , . . . , µ n ( H ) } and let G be a graph of order m such that σ L ( G ) = { µ ( G ) , . . . , µ m ( G ) } . Then σ L ( H [ G ]) = n [ j =1 { md H ( j ) + µ i ( G ) : 1 ≤ i ≤ m − } ! ∪{ mµ ( H ) , . . . , mµ n ( H ) } . Let us consider the graphs obtained by an arbitrary number of iterations of thelexicographic product of a graph by another as follows: H [ G ] = G, H [ G ] = H [ G ] and H k [ G ] = H [ H k − [ G ]] , for all integer k ≥ . Example 3.1.
Let us consider the graph H = C (the cycle with four vertices)and G = K (the complete graph with two vertices). Then H [ G ] = K and H [ G ] = C [ K ] are depicted in Figure 1. Furthermore, the Figure 2 depicts thegraph H [ G ] = C [ K ] = C [ C [ K ]] . In what follows, we adopt the traditional notation of the union of sets fordenoting the union of multisets, where the repeated elements of the multisets A and B appear in A ∪ B as many times as we count them in A and B . bb bb bb b bbb Figure 1: The graphs H [ G ] = K and H [ G ] = C [ K ].5 b bb b bbb bb bb b bbbbb bb b bbb bb bb b bbb Figure 2: The graph H [ G ] = C [ K ]. p -regular graph G anda q -regular graph H The next theorem states the regularity degree, order and spectrum of H k [ G ], for k ≥
0, when G and H are both regular connected graphs. Theorem 3.2.
Let H be a q -regular connected graph of order n with σ A ( H ) = { q, γ [ h ]2 ( H ) , . . . , γ [ h t ] t ( H ) } and G be a p -regular connected graph of order m with σ A ( G ) = { p, γ [ g ]2 ( G ) , . . . , γ [ g s ] s ( G ) } . Then for each integer k ≥ , H k [ G ] is a r k -regular graph of order ν k with r k = mq n k − n − p,ν k = mn k ,σ A ( H k [ G ]) = n γ [ n k g ]2 ( G ) , . . . , γ [ n k g s ] s ( G ) o ∪ { r k } ∪ Λ k where Λ k = k − [ i =0 n ( mn i γ ( H ) + r i ) [ n k − − i h ] , . . . , ( mn i γ t ( H ) + r i ) [ n k − − i h t ] o , assuming that Λ = ∅ .Proof. Since H [ G ] = G , the case k = 0 follows. Furthermore, the case k = 1follows from Corollary 2.2, since H [ G ] = H [ G ] (notice that r = mq + p = mγ ( H ) + p ). Let us assume that the result holds for k − k ≥ r k = ν k − q + r k − = mq ( n k − + n k − + · · · + n + 1) + p = mq n k − n − p and ν k = ν ( H k [ G ]) = ν k − n = mn k . Additionally, replacing in the Corollary 2.2the graph G by H k − [ G ] it follows that σ A ( H k [ G ]) = { γ [ n k g ]2 ( G ) , . . . , γ [ n k g s ] s ( G ) } ∪ { r k } ∪ Λ k , where Λ k = k − [ i =0 n ( mn i γ ( H ) + r i ) [ n k − − i h ] , . . . , ( mn i γ t ( H ) + r i ) [ n k − − i h t ] o . Example 3.3.
For the graphs of Figure 1, we have m = 2 , n = 4 , p = 1 and q = 2 . From Theorem 3.2, we obtain the following degree, order and spectra for C k [ K ] for a given k ≥ integer: r k = 2 × k − − k +1 − ,ν k = ν ( H k [ G ]) = 2 × k ,σ A ( H k [ G ]) = n ( − [4 k ] o ∪ (cid:26) k +1 − (cid:27) ∪ Λ k , where Λ k = S k − i =0 (cid:26)(cid:16) × i × i +1 − (cid:17) [4 k − − i × , (cid:16) × i ( −
2) + i +1 − (cid:17) [4 k − − i ] (cid:27) = S k − i =0 n ( i +1 − ) [4 k − − i × , ( − i +1 + i +1 − ) [4 k − − i ] o . In particular, for k = 2 (the graph of Figure 2), it follows that r = 21 ,ν = ν ( H [ G ]) = 32 and σ A ( H [ G ]) = { , (5) [2] , (1) [8] , ( − [16] , ( − [4] , − } . We may consider the graph obtained by an arbitrary number of iterations ofthe lexicographic product of a graph for itself. In fact, for a given q -regular graph H of order n , we assume that H = K , H = H and that H k = H k − [ H ] for k ≥
2. Then, as immediate consequence of Theorem 3.2, we have the followingcorollary.
Corollary 3.4.
Let H be a connected q -regular graph of order n such that σ A ( H ) = { q, γ [ h ]2 ( H ) , . . . , γ [ h t ] t ( H ) } . Then, for each integer k ≥ , H k is a r k -regular graph f order ν k , such that r k = q n k − n − ,ν k = n k and σ A ( H k ) = k − [ i =0 { ( n i γ ( H ) + r i ) [ n k − − i h ] , . . . , ( n i γ t ( H ) + r i ) [ n k − − i h t ] } ! ∪ { r k } . Remark 3.5.
The least eigenvalue of H k is λ n k ( H k ) = n k − λ n ( H ) + q n k − − n − .Proof. In fact, based on the Corollary 3.4, we obtain λ n k ( H k ) = min ≤ i ≤ k − { n i γ t ( H ) + r i } = min ≤ i ≤ k − { n i γ t ( H ) + q n i − n − } = n k − γ t ( H ) + q n k − − n − n k − λ n ( H ) + q n k − − n − . The third equality above is obtained taking into account that for every i ∈{ , . . . , k − } , n k − γ t ( H ) + q n k − − n − ≤ n i γ t ( H ) + q n i − n − ⇔ ( n k − − n i ) γ t ( H ) ≤− q n k − − n i n − ⇔ γ t ( H ) ≤ − qn − and last inequality holds since the graph H has atleast one edge and then (see [5]) γ t ( H ) ≤ − ≤ − qn − . Remark 3.6.
Let H be a p -regular graph of order n . Then for all k ∈ N and forall nonnegative integer q , σ A ( H k ) \ { r k } ⊂ σ A ( H k + q ) , where r k is the regularity of H k and this inclusion means that all eigenvalues with the respective multiplicitiesof the multiset σ A ( H k ) \ { r k } belong to the multiset σ A ( H k + q ) .Proof. This is a direct consequence of Corollary 3.4.
Example 3.7.
Let us apply the Corollary 3.4 to the powers of the Pertersengraph P k , with k ∈ { , , } . k Spectrum of P k k = 1 3 , [5] , − [4] k = 2 33 , [5] , [50] , − [40] , − [4] k = 3 333 , [5] , [50] , [500] , − [400] , − [40] , − [4] k = 100 3 × X i =0 i , [5 × ] , − [4 × ] , m + 3 m − X i =0 i ! [5 × − m ] , m = 1 , . . . , , − m + 6 m − X i =1 i ! [4 × − m ] , m = 1 , . . . , .Notice that the graph P k has k vertices, in particular P has the googolnumber of vertices . All the computations were done by Mathematica andlasted just a few seconds. .2 The Laplacian spectra In this section we characterize the Laplacian spectrum of the iterated lexico-graphic product H k [ G ], where G and H are arbitrary graphs. The particularcases of the Laplacian spectra of these iterated lexicographic products, when H is regular and when H is arbitrary but equal to G are also presented. Theorem 3.8.
Let G be a graph of order m such that σ L ( G ) = { µ ( G ) , . . . ,µ m ( G ) } and let H be a graph such that V ( H ) = [ n ] and σ L ( H ) = { µ ( H ) , . . . ,µ n ( H ) } . Then, for each integer k ≥ , H k [ G ] is a graph of order ν k = mn k suchthat σ L ( H k [ G ]) = Ω kG ∪ Γ kH where Ω kG = [ ( j ,j ,...,j k ) ∈ [ n ] k ( µ l ( G ) + m k X i =1 n i − d H ( j i ) : 1 ≤ l ≤ m − ) and Γ kH = k [ i =2 [ ( j i ,...,j k ) ∈ [ n ] k − i +1 ( mn i − µ l ( H ) + m k X r = i n r − d H ( j r ) : 1 ≤ l ≤ n − ) ∪ (cid:8) mn k − µ j ( H ) : 1 ≤ j ≤ n (cid:9) . Proof.
Corollary 2.4 give us the assertion in case k = 1. Given an integer k ≥ σ L ( H k − [ G ]) = Ω k − G ∪ Γ k − H , whereΩ k − G = [ ( j ,...,j k − ) ∈ [ n ] k − ( µ l ( G ) + m k − X i =1 n i − d H ( j i ) : 1 ≤ l ≤ m − ) andΓ k − H = k − [ i =2 [ ( j i ,...,j k − ) ∈ [ n ] k − i ( mn i − µ l ( H ) + m k − X r = i n r − d H ( j r ) : 1 ≤ l ≤ n − ) ∪ (cid:8) mn k − µ j ( H ) : 1 ≤ j ≤ n (cid:9) . Then, by Corollary 2.4, σ L ( H k [ G ]) = σ L ( H [ H k − [ G ]]) == n [ j k =1 (cid:8) mn k − d H ( j k ) + x : x ∈ Ω k − G (cid:9)! ∪ n [ j k =1 (cid:8) mn k − d H ( j k ) + y : y ∈ Γ k − H (cid:9)! , where n [ j k =1 (cid:8) mn k − d H ( j k ) + x : x ∈ Ω k − G (cid:9) =9 n [ j k =1 [ ( j ,...,j k − ) ∈ [ n ] k − ( mn k − d H ( j k ) + µ l ( G ) + m k − X i =1 n i − d H ( j i ) : 1 ≤ l ≤ n ) = [ ( j ,...,j k − ,j k ) ∈ [ n ] k ( µ l ( G ) + m k X i =1 n i − d H ( j i ) : 1 ≤ l ≤ n ) = Ω kG and n [ j k =1 (cid:8) mn k − d H ( j k ) + y : y ∈ Γ k − H (cid:9) == n [ j k =1 k − [ i =2 [ ( j i ,...,j k − ) ∈ [ n ] k − i ( mn k − d H ( j k ) + mn i − µ l ( H ) + m k − X r = i n r − d H ( j r ) : 1 ≤ l ≤ n − ) ∪ (cid:8) mn k − d H ( j k ) + mn k − µ l ( H ) : 1 ≤ l ≤ n − (cid:9) ! ∪{ mn k − µ j ( H ) : 1 ≤ j ≤ n } == k [ i =2 [ ( j i ,...,j k ) ∈ [ n ] k − i +1 ( mn i − µ l ( H ) + m k X r = i n r − d H ( j r ) : 1 ≤ l ≤ n − ) ∪{ mn k − µ j ( H ); 1 ≤ j ≤ n } = Γ kH . As immediate consequence of the above theorem, for a regular graph H itfollows Corollary 3.9.
Let G and H as in Theorem 3.8, with H is q -regular. Then, foreach integer k ≥ , σ L ( H k [ G ]) = ((cid:18) µ l ( G ) + mq n k − n − (cid:19) [ n k ] : 1 ≤ l ≤ m − ) ∪ { } ∪ k +1 [ i =2 ((cid:18) mn i − µ l ( H ) + mqn i − n k − i +1 − n − (cid:19) [ n k − i +1 ] : 1 ≤ l ≤ n − ) . Proof.
From Theorem 3.8, for all integer k ≥
1, it follows that σ L ( H k [ G ]) = Ω kG ∪ Γ kH , where Ω kG = [ ( j ,...,j k ) ∈ [ n ] k ( µ l ( G ) + mq k X i =1 n i − : 1 ≤ l ≤ m − ) = ((cid:18) µ l ( G ) + mq n k − n − (cid:19) [ n k ] : 1 ≤ l ≤ m − ) and10 kH = k [ i =2 [ ( j i ,...,j k ) ∈ [ n ] k − i +1 ( mn i − µ l ( H ) + mq k X r = i n r − : 1 ≤ l ≤ n − ) ∪ (cid:8) mn k − µ j ( H ) : 1 ≤ j ≤ n (cid:9) = k +1 [ i =2 ((cid:18) mn i − µ l ( H ) + mqn i − n k − i +1 − n − (cid:19) [ n k − i +1 ] : 1 ≤ l ≤ n − ) ∪ { } . Now, let us consider the case G = H . Corollary 3.10.
Let H be a graph such that V ( H ) = [ n ] , with σ L ( H ) = { µ ( H ) ,. . . , µ n ( H ) } . Then H k is a graph of order ν k = n k such that σ L ( H k ) = k − [ i =1 [ ( j i ,...,j k − ) ∈ [ n ] k − i ( n i − µ l ( H ) + k − X r = i n r d H ( j r ) : 1 ≤ l ≤ n − ) ∪ (cid:8) n k − µ j ( H ) : 1 ≤ j ≤ n (cid:9) , for all k ≥ .Proof. The first statement is obvious. Regarding the second statement, applyingagain Theorem 3.8 for k ≥ σ L ( H k ) = σ L ( H k − [ H ]) == [ ( j ,j ,...,j k − ) ∈ [ n ] k − ( µ l ( H ) + n k − X i =1 n i − d H ( j i ) : 1 ≤ l ≤ n − ) ∪ k − [ i =2 [ ( j i ,...,j k − ) ∈ [ n ] k − i ( n i − µ l ( H ) + n k − X r = i n r − d H ( j r ) : 1 ≤ l ≤ n − ) ∪{ nn k − µ j ( H ) : 1 ≤ j ≤ n } == k − [ i =1 [ ( j i ,...,j k ) ∈ [ n ] k − i ( n i − µ l ( H ) + k − X r = i n r d H ( j r ) : 1 ≤ l ≤ n − ) ∪{ n k − µ j ( H ) : 1 ≤ j ≤ n } . Finally, the next proposition determines the algebraic connectivity and thelargest Laplacian eigenvalue of H k , for k ≥ roposition 3.11. If H is a connected graph of order n with σ L ( H ) = { µ ( H ) , . . . ,µ n ( H ) } and k ≥ , then µ n k − ( H k ) = n k − µ n − ( H ) and (6) µ ( H k ) = n k − µ ( H ) (7) Proof.
Let k ≥ H k is among the values n k − µ n − ( H ) and n i − µ n − ( H ) + P k − r = i n r d H ( j r ) for 1 ≤ i ≤ k −
1. We may recall that δ ( H ) ≥ µ n − ( H ); then, forall 1 ≤ i ≤ k −
1, it holds that n i − µ n − ( H ) + k − X r = i n r d H ( j r ) ≥ n i − µ n − ( H ) + k − X r = i n r δ ( H ) ≥ n i − µ n − ( H ) + µ n − ( H ) k − X r = i n r = µ n − ( H ) n i − + k − X r = i n r ! = µ n − ( H ) k − X r = i − n r = µ n − ( H ) k X r = i n r − ≥ µ n − ( H ) n k − . Thus the equality (6) is proved. Now, let us prove the equality (7).Applying again Corollary 3.10, it follows that the largest Laplacian eigenvalue of H k is among the values n k − µ ( H ) and n i − µ ( H )+ P k − r = i n r d H ( j r ), 1 ≤ i ≤ k − µ ( H ) ≥ ∆( H ) + 1, for 1 ≤ i ≤ k −
1, it follows that n i − µ ( H ) + k − X r = i n r d H ( j r ) ≤ n i − µ ( H ) + k − X r = i n r ( µ ( H ) − µ ( H ) k − X r = i − n r − k − X r = i n r = µ ( H ) n i − n k − i +1 − n − − n i n k − i − n − µ ( H ) n i − ( n k − i − n − n k − i ) − n i n k − i − n − µ ( H ) n k − + n k − i − n − n i − ( µ ( H ) − n ) ≤ n k − µ ( H )The last inequality is obtained taking into account that µ ( H ) − n ≤ Spectral and combinatorial invariant proper-ties of lexicographic powers of graphs
In this section, a few well known spectral and combinatorial invariant propertiesof a graph H are extended to the lexicographic powers of H . For instance,considering that H has order n ≥
2, for all k ≥
1, we may deduce that δ ( H k ) = δ ( H ) n k − n − (cid:18) ∆( H k ) = ∆( H ) n k − n − (cid:19) . (8)Notice that since H has order n , then H k has order n k . The equalities (8) canbe proved by induction on k , taking into account that they are obviously truefor k = 1. Assuming that the equalities (8) are true for k −
1, with k ≥
2, itis immediate that a vertex of H k with minimum (maximum) degree is a mini-mum (maximum) degree vertex of the copy of H k − located in the position of aminimum (maximum) degree vertex of H , and then its degree in H k is equal to δ ( H )( n k − − n − + n k − ) (cid:16) ∆( H )( n k − − n − + n k − ) (cid:17) .For an arbitrary graph G , let q ( G ) and q n ( G ) be the largest and the leasteigenvalue of the signless Laplacian matrix of G (that is, the matrix A G + D ),respectively. Taking into account the relations 2 δ ( G ) ≤ q ( G ) ≤ G ), whichwere proved in [6], and also the inequality q n ( G ) < δ ( G ) [8], for the lexicographicpower k of a graph H we obtain the inequalities2 δ ( H ) n k − n − ≤ q ( H k ) ≤ H ) n k − n − q n ( H k ) < δ ( H ) n k − n − . Denoting the distance between two vertices x and y in G by d G ( x, y ) andthe diameter of G by diam( G ), we may conclude the following interesting resultconcerning the diameter of the iterated lexicographic products of graphs. Proposition 4.1.
Let H be a connected not complete graph and let G be anarbitrary graph of order m . For very k ∈ N diam ( H k +1 ) = diam ( H k [ G ]) = diam ( H ) . Proof.
Consider V ( H ) = { , . . . , n } and x, y ∈ V ( H k [ G ]) ( x, y ∈ V ( H k +1 )).Then we have two cases (a) they are both in the same copy of H k − [ G ] ( H k )located in the position of the vertex i ∈ V ( H ) or (b) they are in different copiesof H k − [ G ] ( H k ) located in the positions of the vertices r, s ∈ V ( H ).(a) If x and y are adjacent, then d H k [ G ] ( x, y ) = 1 ( d H k +1 ( x, y ) = 1), otherwisesince there exists a vertex j ∈ V ( H ) such that ij ∈ E ( H ) and then there is13 path x, z, y , where z is a vertex of the copy of H k − [ G ] ( H k ) located in theposition of the vertex j ∈ V ( H ). Therefore, d H k [ G ] ( x, y ) = 2 ( d H k +1 ( x, y ) =2).(b) In this case, assuming that r, j , . . . , j t , s is a shortest path in H connectingthe vertices r and s , there are vertices z , . . . , z t in the copies of H k − [ G ]( H k ) located in the positions of the vertices j , . . . , j t , respectively, suchthat x, z , . . . , z t , y is a path of length d H ( r, s ). Regarding the stability number α ( G ) (the maximum cardinality of a vertex subsetof an arbitrary graph G with pairwise nonadjacent vertices), according to [11], α ( H [ G ]) = α ( H ) α ( G ) for an arbitrary graph H . Thus we may conclude that α ( H k ) = α ( H ) k (and, denoting the complement of graph F by F and the cliquenumber by ω ( F ), since H [ G ] = H [ G ], ω ( H k ) = ω ( H ) k ). Furthermore, from thespectral upper bound α ( G ) ≤ n µ ( G ) − δ ( G )∆( G ) , independently deduced in [18] and [12]for an arbitrary graph G , and taking into account (8) and (7), considering the k -th lexicographic power of a graph H of order n we obtain α ( H k ) ≤ n k µ ( H k ) − δ ( H k )∆( H k ) ≤ n k n − n k − n k − µ ( H ) − δ ( H )∆( H ) . Considering a graph G of order m and a graph H of order n , it is well known thatthe lexicographic product H [ G ] is connected if and only if H is a connected graph[14]. On the other hand, according to [11], if both G and H are not complete,then υ ( H [ G ]) = mυ ( H ), where υ ( H ) denotes the vertex connectivity of H (thatis, the minimum number of vertices whose removal yields a disconnected graph).Therefore, υ ( H k ) = n k − υ ( H ) . Furthermore, we may conclude that when H isconnected not complete (and then H k is also connected not complete), n k − µ n − ( H ) ≤ υ ( H k ) ≤ δ ( H ) n k − n − . In fact, it should be noted that υ ( G ) ≤ δ ( G ) and, when G is not complete, µ n − ( G ) ≤ υ ( G ), see [9]. Therefore, taking into account (6) and (8) we obtain n k − µ n − ( H ) = µ n − ( H k ) ≤ υ ( H k ) ≤ δ ( H k ) = δ ( H ) n k − n − .14 .3 The chromatic number Concerning the relations of the chromatic number of a graph G of order n withits spectrum, the following lower bound due to Hoffman in [16] is well known. χ ( G ) ≥ − λ ( G ) λ n ( G ) . As direct consequence, if a graph H is q -regular of order n , taking into accountthe Remark (3.5), we may conclude the following lower bound on the chromaticnumber of H k : χ ( H k ) ≥ − r k λ n k ( H k ) = 1 − q n k − n − (cid:16) n k − λ n ( H ) + q n k − − n − (cid:17) = 1 − n k − n k − (cid:16) ( n − λ n ( H ) q + 1 (cid:17) − . Acknowledgement
The research of Nair Abreu is partially supported by Project Universal CNPq442241/2014 and Bolsa PQ 1A CNPq, 304177/2013-0. The research of Domin-gos M. Cardoso and Paula Carvalho is supported by the Portuguese Foundationfor Science and Technology (“FCT-Funda¸c˜ao para a Ciˆencia e a Tecnologia”),through the CIDMA - Center for Research and Development in Mathematics andApplications, within project UID/MAT/04106/2013. Cybele Vinagre thanks thesupport of FAPERJ, through APQ5 210.373/2015 and the hospitality of Depart-ment of Mathematics of University of Aveiro, Portugal, where this paper wasfinished.
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