Discrete Mathematics. The signless Laplacian spectral radius of bicyclic graphs with prescribed degree sequences


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1 Discrete Mathematics 311 (2011) Contents lists available at ScienceDirect Discrete Mathematics journal homepage: wwwelseviercom/locate/disc The signless Laplacian spectral radius of bicyclic graphs with prescribed degree sequences Yufei Huang, Bolian Liu, Yingluan Liu School of Mathematical Science, South China Normal University, Guangzhou, , PR China a r t i c l e i n f o a b s t r a c t Article history: Received 24 December 2009 Received in revised form 7 September 2010 Accepted 13 December 2010 Available online 5 January 2011 Keywords: Signless Laplacian spectral radius Degree sequence Connected bicyclic graph Majorization In this paper, we characterize all extremal connected bicyclic graphs with the largest signless Laplacian spectral radius in the set of all connected bicyclic graphs with prescribed degree sequences Moreover, the signless Laplacian majorization theorem is proved to be true for connected bicyclic graphs As corollaries, all extremal connected bicyclic graphs having the largest signless Laplacian spectral radius are obtained in the set of all connected bicyclic graphs of order n (resp all connected bicyclic graphs with a specified number of pendant vertices, and all connected bicyclic graphs with given maximum degree) 2011 Elsevier BV All rights reserved 1 Introduction In this paper, only connected simple graphs are considered Let G = (V, E) be a connected simple graph with vertex set V = {v 1, v 2,, v n } and edge set E If E = V + c 1, then G = (V, E) is called a ccyclic graph, where c is a nonnegative integer Especially, if c = 0 (resp c = 1 and 2), then G is called a tree (resp unicyclic and bicyclic graphs) Let u, v V(G) If u is adjacent to v in G, then v is called the neighbor of u Let N(u) denote the neighbor set of u Then d(u) = d G (u) = N(u) is called the degree of u in G A nonincreasing sequence of nonnegative integers π = (d 1, d 2,, d n ) is called graphic if there exists a graph G having π as its vertex degree sequence (π is also called the degree sequence of G) For all other definitions, notations and terminologies on the spectral graph theory, not given here, see eg [1,4,6] Denote by D(G) = diag(d(u), u V) the diagonal matrix of vertex degrees of G and A(G) the (0, 1)adjacency matrix of G If G is connected, then the matrix Q (G) = D(G)+A(G) is called the signless Laplacian matrix of G (also called the unoriented Laplacian matrix of G in [12,18]), which may be first introduced in book [4] without giving a name The signless Laplacian matrices of graphs have received much attention in recent years (see eg [3,5,7 10,12,13,18]) The largest eigenvalue of Q (G) is called the signless Laplacian spectral radius of G and is denoted by µ(g) Note that Q (G) is an irreducible nonnegative matrix, then there exists only one unit positive eigenvector f = (f (u), v V(G)) with f = 1 such that Q (G)f = µ(g)f Such an eigenvector f is called the Perron vector of Q (G), and f (u) is called the µweight of vertex u [15] For a prescribed graphic degree sequence π, let G π = {G G is simple, connected and with π as its degree sequence} Motivated by the Brualdi Solheid problem [2], Zhang put forward the determination of graphs maximizing (or minimizing) the signless Laplacian spectral radius in the set G π [21] It has been proved in [20,21] that for a given degree sequence π of trees (resp unicyclic graphs), there exists a unique tree (resp unicyclic graph) that has the largest signless Laplacian spectral radius in G π This work is supported by NNSF of China (NO ) Corresponding author address: (B Liu) X/$ see front matter 2011 Elsevier BV All rights reserved doi:101016/jdisc
2 Y Huang et al / Discrete Mathematics 311 (2011) Let B π denote the class of all bicyclic graphs with bicyclic degree sequence π In this paper, we determine the unique maximal graph in B π with respect to the signless Laplacian spectral radius of G B π Moreover, recall the notion of majorization Let π = (d 1,, d n ) and π = (d,, 1 d n ) be two nonincreasing k sequences If i=1 d i k i=1 d i for k = 1, 2,, n 1 and n i=1 d i = n i=1 d i, then the sequence π is said to be majorized by the sequence π, which is denoted by π π Let π and π be two different tree (resp unicyclic) degree sequences with the same order Let G and G be the trees (resp unicyclic graphs) with the largest signless Laplacian spectral radius in G π and G π, respectively In [20] (resp [21]), the author proved that if π π, then µ(g) < µ(g ) Such a theorem is called the signless Laplacian majorization theorem for trees (resp unicyclic graphs) In Section 3, the signless Laplacian majorization theorem is proved to be true for bicyclic graphs As corollaries, all extremal bicyclic graphs having the largest signless Laplacian spectral radius are obtained in the set of all bicyclic graphs of order n (resp all bicyclic graphs with a specified number of pendant vertices, all bicyclic graphs with given maximum degree) 2 Preliminaries In this section, we introduce some definitions and lemmas which are useful in the presentations and proofs of our main results Let G be a graph with a root vertex v 1 V(G) Denote by dist(v, v 1 ) the distance between v V(G) and v 1 Besides, the distance dist(v, v 1 ) is called the height of vertex v in G, denoted by h(v) = dist(v, v 1 ) Definition 21 ([21]) Let G = (V, E) be a graph with a root v 1 V(G) A wellordering of the vertices is called a breadthfirstsearch ordering (BFSordering for short) if the following hold for all vertices u, v V : (1) u v implies h(u) h(v); (2) u v implies d(u) d(v); (3) suppose uv, xy E, uy, xv E with h(u) = h(x) = h(v) 1 = h(y) 1 If u x, then v y We call a graph that has a BFSordering of its vertices a BFSgraph Lemma 22 ([21]) Let G = (V, E) be a simple connected graph having the largest signless Laplacian spectral radius in G π and f be the Perron vector of Q (G) If V = {v 1,, v n } satisfies f (v i ) f (v j ) for i < j (ie, the vertices of V(G) are denoted with respect to f (v) in nonincreasing order), then d(v i ) d(v j ) for i < j Moreover, if f (v i ) = f (v j ), then d(v i ) = d(v j ) Corollary 23 Let G = (V, E) be a simple connected graph having the largest signless Laplacian spectral radius in G π and f be the Perron vector of Q (G) If d(u) > d(v), then f (u) > f (v), where u, v V Proof Suppose f (v) f (u) By Lemma 22, d(v) d(u), a contradiction Lemma 24 ([21]) Let G = (V, E) be a simple connected graph having the largest signless Laplacian spectral radius in G π Then G has a BFSordering consistent with the Perron vector f of Q (G) in such a way that u v implies f (u) f (v) From the Perron Frobenius Theorem of nonnegative matrices [15], we have Lemma 25 ([15]) If H is a proper connected subgraph of a connected simple graph G, then µ(h) < µ(g) Suppose that uv E(G) (resp uv E(G)) Denote by G uv (resp G + uv) the graph obtained from G by deleting (resp adding) the edge uv Lemma 26 ([20]) Let G = (V, E) G π and f be a Perron vector of Q (G) (1) Suppose uw i E and vw i E for i = 1,, k Let G = G + vw vw k uw 1 uw k If G is connected and f (v) f (u), then µ(g ) > µ(g) (2) If d(u) > d(v) and f (u) f (v), then there exists a connected graph G G π such that µ(g ) > µ(g) Lemma 27 ([20]) Let G = (V, E) G π and f be a Perron vector of Q (G) Assume that v 1 u 1, v 2 u 2 E, and v 1 v 2, u 1 u 2 E Let G = G + v 1 v 2 + u 1 u 2 v 1 u 1 v 2 u 2 If G is connected and (f (v 1 ) f (u 2 ))(f (v 2 ) f (u 1 )) 0, then µ(g ) µ(g) Moreover, the last equality holds if and only if f (v 1 ) = f (u 2 ) and f (v 2 ) = f (u 1 ) Corollary 28 Let G = (V, E) be a simple connected graph that has the largest signless Laplacian spectral radius in G π and f be a Perron vector of Q (G) Assume that v 1 u 1, v 2 u 2 E, and v 1 v 2, u 1 u 2 E Let G = G + v 1 v 2 + u 1 u 2 v 1 u 1 v 2 u 2 Then the degree sequence of G is π Moreover, (1) if G is connected and f (v 1 ) > f (u 2 ), then f (v 2 ) < f (u 1 ); (2) if G is connected and f (v 1 ) = f (u 2 ), then f (v 2 ) = f (u 1 )
3 506 Y Huang et al / Discrete Mathematics 311 (2011) Proof Obviously, the degree sequence of G is π Now suppose that f (v 2 ) f (u 1 ) in (1), and f (v 2 ) f (u 1 ) in (2) Since (f (v 1 ) f (u 2 ))(f (v 2 ) f (u 1 )) 0, by Lemma 27, we both have µ(g ) > µ(g), a contradiction An internal path of G is a path (or cycle) P with vertices v 1,, v k (or v 1 d(v 2 ) = = d(v k 1 ) = 2, where k > 1 = v k ) such that d(v 1 ), d(v k ) 3 and Lemma 29 ([14]) Let G be a simple connected graph and uv be an edge on an internal path of G If G uv is obtained from G by the subdivision of the edge uv into the edges uw and wv, then µ(g uv ) < µ(g) Let G be a simple connected nontrivial graph and u V(G) Let G(k, l) be the graph obtained from G by attaching two paths uu 1 u 2 u k and uv 1 v 2 v l at vertex u respectively, where k, l 1 Lemma 210 ([13]) If k l 2, then µ(g(k, l)) > µ(g(k + 1, l 1)) Lemma 211 ([11,19]) Let π = (d 1,, d n ) and π = (d 1,, d n ) be two nonincreasing graphic sequences If π π, then there exist a series of graphic degree sequences π 1,, π k such that (π =)π 0 π 1 π k π k+1 (=π ), and only two components of π i and π i+1 are different from 1, where i = 0, 1,, k Lemma 212 ([16]) Let m(v) = u N(v) d(u)/d(v) Then d(u)(d(u) + m(u)) + d(v)(d(v) + m(v)) µ(g) max uv E(G) d(u) + d(v) Lemma 213 ([17]) Let G be a connected graph with at least one edge Then µ(g) λ(g) + 1, where λ(g) is the Laplacian spectral radius of G, and is the maximum degree of G The first equality holds if and only if G is bipartite, and the second equality holds if and only if = n 1 3 Main results To begin with, we give a characterization of the degree sequences of connected bicyclic graphs as follows Proposition 31 ([1,11]) Let π = (d 1, d 2,, d n ) be a nonincreasing sequence Then π is graphic if and only if n i=1 d i is even and k d i k(k 1) + i=1 n min{k, d i }, where 1 k n (1) i=k+1 Proposition 32 Let π = (d 1, d 2,, d n ) be a positive nonincreasing integer sequence with even sum and n 4 If π is connected bicyclic graphic (also called a bicyclic degree sequence), then n i=1 d i = 2n + 2 and (1) holds Let θ(p, q, r) denote the θgraph, which is obtained from three vertexdisjoint paths (of orders p + 1, q + 1 and r + 1 respectively, where p, q, r > 0 and at most one of p, q, r equals 1) by identifying the three initial (resp terminal) vertices of them Let C(n 1, n 2 ) denote the graph obtained from two cycles C n1 and C n2 (of orders n 1 and n 2 respectively, where n 1, n 1 3) by identifying a vertex of C n1 with a vertex of C n2 For a prescribed nonincreasing bicyclic degree sequence π = (d 1, d 2,, d n ) with n 4, by Proposition 32, π should be one of the following four cases Moreover, we construct a special bicyclic graph B π with degree sequence π as follows (1) d n = 2 and d 1 = 4 Then d i = 2 (2 i n 1) Let B π = C(3, n 2) (2) d n = 2 and d 1 = 3 Then d 2 = 3 and d i = 2 (3 i n 1) by Proposition 32 Let B π = θ(1, 2, n 2) (3) d n = 1 and d 2 = 2 Then we have d 1 > 4, d i = 2 (i = 3, 4, 5), and d j 2 (6 j n 1) Denote by B π the connected graph of order n obtained from C(3, 3) and d 1 4 paths of almost equal lengths (ie, their lengths are different at most by one) by identifying the maximum degree vertex of C(3, 3) with one end vertex of each path of the d 1 4 paths (4) d n = 1 and d 2 3 Then d 1 d 2 3, and we use the breadthfirstsearch method to define a special bicyclic graph B π with degree sequence π as follows Select a vertex v 01 as a root and begin with v 01 of the zeroth layer Put s 1 = d 1 and select s 1 vertices {v 11,, v 1,s1 } of the first layer such that they are adjacent to v 01, and v 11 is adjacent to v 12 and v 13 Thus d(v 01 ) = s 1 = d 1 Next we construct the second layer Put d(v 1i ) = d i+1 (i = 1,, s 1 ) and select s 2 = s 1 d(v i=1 1i) s 1 4 vertices {v 21,, v 2,s2 } of the second layer such that d(v 11 ) 3 vertices are adjacent to v 11, d(v 12 ) 2 (resp d(v 13 ) 2) vertices are adjacent to v 12 (resp d(v 13 )), and d(v 1i ) 1 vertices are adjacent to v 1i for i = 4,, s 1 In general, assume that all vertices of the tth (t 2) layer have been constructed and are denoted by {v t1,, v t,st } Now using the induction
4 Y Huang et al / Discrete Mathematics 311 (2011) Fig 1 B π hypothesis, we construct all the vertices of the (t + 1)st layer Put d(v ti ) = d i+1+ t 1 j=1 s (i = 1,, s t ) and select j s t+1 = s t d(v i=1 ti) s t vertices {v t+1,1,, v t+1,st+1 } of the (t + 1)st layer such that d(v ti ) 1 vertices are adjacent to v ti for i = 1,, s t In this way, we obtain only one connected bicyclic graph with degree sequence π For example, for a given bicyclic degree sequence π = (6, 5, 4, 3, 2, 1 (8) ), B π is the bicyclic graph of order 13 shown in Fig 1 It can be seen that for a prescribed bicyclic degree sequence π, the graph B π constructed above has a BFSordering Now we show that B π is the only bicyclic graph that has the largest signless Laplacian spectral radius in B π Lemma 33 Let π = (d 1, d 2,, d n ) be a nonincreasing bicyclic degree sequence with d n = 2 Then B π is the only connected bicyclic graph that has the largest signless Laplacian spectral radius in B π Proof Let G = (V, E) be a connected graph that has the largest signless Laplacian spectral radius in B π, and f be the Perron vector of Q (G) By Lemma 24, there exists a BFSordering of G with root v 1 such that v 1 v 2 v n, f (v 1 ) f (v 2 ) f (v n ), h(v 1 ) h(v 2 ) h(v n ), and d(v 1 ) d(v 2 ) d(v n ) Let V i = {v V h(v) = i} for i = 0, 1,, h(v n ) (=p) Hence we can relabel the vertices of G in such a way that V i = {v i1,, v i,si } with f (v il ) f (v ij ) f (v rk ) and d(v il ) d(v ij ) d(v rk ) for 0 i < r p, 1 l < j s i, and 1 k s r Clearly, s 1 = d(v 1 ) = d(v 01 ) = d 1 On the other hand, note that d n = n 2 and i=1 d i = 2n + 2 by Proposition 32 Then we need to consider the following two bicyclic degree sequences Case 1 π = (4, 2,, 2), where the frequency of 2 in π is n 1 Hence d(v 01 ) = 4 and d(v iji ) = 2 for 1 i p and 1 j i s i If v 1i v 1j E for all 1 i < j 4, it follows from π = (4, 2,, 2) that we can suppose there exist two cycles (v 01, v 11, w 1,, w n1, v 12, v 01 ) and (v 01, v 13, u 1,, u n2, v 14, v 01 ) without loss of generality, where n 1, n 2 1 and n 1 + n = n Since d(v 01 ) > d(v 11 ), by Corollary 23, we have f (v 01 ) > f (v 11 ) It follows from Corollary 28 that f (v 13 ) > f (w 1 ) Let G 1 = G v 11 w 1 v 13 u 1 + v 11 v 13 + w 1 u 1 Note that G 1 B π, f (v 11 ) f (u 1 ) (u 1 V 2 ) and f (v 13 ) > f (w 1 ), then by Lemma 27, we have µ(g 1 ) > µ(g), a contradiction If there exists an edge v 1i v 1j E (1 i < j 4), according to the degree sequence π = (4, 2,, 2), then G is isomorphic to C(3, n 2), that is, B π Case 2 π = (3, 3, 2,, 2), where the frequency of 2 in π is n 2 Thus d(v 01 ) = d(v 11 ) = 3 and d(v 12 ) = d(v 13 ) = d(v iji ) = 2 for 2 i p and 1 j i s i If v 11 v 12 E (or v 11 v 13 E), then it follows from π = (3, 3, 2,, 2) that G is isomorphic to θ(1, 2, n 2), namely, G = B π If v 11 v 12, v 11 v 13 E, according to the degree sequence π = (3, 3, 2,, 2), we consider the following two subcases Subcase 21 There exist two cycles (v 01, v 12, w 1,, w n1, v 13, v 01 ) and (v 11, v 21, u 1,, u n2, v 22, v 11 ), where n 1, n 2 0 and n 1 + n = n Since d(v 11 ) > d(w 1 ), by Corollary 23, we have f (v 11 ) > f (w 1 ) Note that w 1 v 21, v 11 v 12 E Let G 1 = G v 11 v 21 v 12 w 1 + v 11 v 12 + v 21 w 1 Then G 1 B π Since f (v 11 ) > f (w 1 ) and f (v 12 ) f (v 21 ), by Lemma 27, we get µ(g 1 ) > µ(g) This contradicts G having the largest signless Laplacian spectral radius in B π Subcase 22 There exist two cycles (v 01, v 11, w 1,, w n1, v 12, v 01 ) and (v 01, v 11, u 1,, u n2, v 13, v 01 ), where n 1, n 2 1 and n 1 + n = n Since d(v 11 ) > d(u n2 ), it follows from Corollary 23 that f (v 11 ) > f (u n2 ) Note that w 1 u n2, v 11 v 13 E Let G 1 = G v 11 w 1 v 13 u n2 + v 11 v 13 + u n2 w 1 Then G 1 B π Since f (v 11 ) > f (u n2 ) and f (v 13 ) f (w 1 ) (w 1 V 2 ), then by Lemma 27, µ(g 1 ) > µ(g), which is a contradiction All in all, we conclude that G is isomorphic to B π Lemma 34 Let π = (d 1, d 2,, d n ) be a nonincreasing bicyclic degree sequence with d n = 1 and d 2 = 2 Then B π is the only connected bicyclic graph that has the largest signless Laplacian spectral radius in B π Proof Let H be a connected bicyclic graph that has the largest signless Laplacian spectral radius in B π By Proposition 32, we have d 1 > 4 and d i 2 (2 i n) Then H must be the graph of order n obtained from C(n 1, n 2 ) and d 1 4 paths P i of order p i for i = 1, 2,, d 1 4 by identifying the maximum degree vertex of C(n 1, n 2 ) with one end vertex of each P i
5 508 Y Huang et al / Discrete Mathematics 311 (2011) Claim 1 n 1 = n 2 = 3 In fact, if n 1 > 3 (or n 2 > 3), let H 1 be a graph of order n 1 obtained from H by the contraction of an edge of C n1 (or C n2 ) in H By Lemma 29, we have µ(h 1 ) > µ(h) Let H 2 be a graph of order n obtained from H 1 by adding a new vertex v and adding a new edge incident to v and a pendant vertex of H 1 It follows from Lemma 25 that µ(h 2 ) > µ(h 1 ), which implies µ(h 2 ) > µ(h) Note that H 2 B π, a contradiction Hence n 1 = n 2 = 3 Claim 2 p i p j 1 for all 1 i, j d 1 4 If there exist s, t such that p s p t 2, then H can be expressed as G(p s, p t ) Hence by Lemma 210, µ(h) = µ(g(p s, p t )) < µ(g(p s 1, p t + 1)) and G(p s 1, p t + 1) B π, a contradiction Hence the d 1 4 paths have almost equal lengths Consequently, H is isomorphic to B π Let uv be a cut edge of G = (V, E) If one component of G uv is a tree T (suppose u V(T)), then the induced subgraph G[V(T) {v}] is called a hanging tree on vertex v Lemma 35 Let G = (V, E) be a connected graph with pendant vertices that has the largest signless Laplacian spectral radius in B π Let P = w 0 w 1 w k w k+1 (k 0) be a path with d(w 0 ) > d(w k+1 ) = 1 and u 1 u 2 E be an edge of a cycle If u 1 w j (0 j k) and u 1 w t, u 2 w t E for j < t k + 1, then f (u 2 ) > f (w j ) > f (w k+1 ) Moreover, let T be a hanging tree on a vertex v and u 1 u 2 E (u 1, u 2 v) be an edge of a cycle If u 1 v E, then f (u 2 ) > f (v) Proof Note that d(w k+1 ) = 1 < d(w j ), where 0 j k By Corollary 23, we have f (w k+1 ) < f (w j ) Now we show that f (u 2 ) > f (w j ) By contradiction, suppose f (u 2 ) f (w j ) Let G = G w j w j+1 u 1 u 2 + u 1 w j + u 2 w j+1 Then G B π If f (w j+1 ) < f (u 1 ), then by Lemma 27, we have µ(g ) > µ(g), a contradiction Hence f (w j+1 ) f (u 1 ) Let G = G w j+1 w j+2 u 1 u 2 + u 1 w j+2 + u 2 w j+1, then we get f (w j+2 ) f (u 2 ) by Lemma 27 and the fact that G B π By repeating similar discussion as above, we arrive at f (w k+1 ) min{f (u 1 ), f (u 2 )} It follows from Corollary 23 that d(w k+1 ) min{d(u 1 ), d(u 2 )} 2, a contradiction Moreover, since T is a hanging tree on vertex v, there exists a path P = vw 1 w k w k+1 (k 0) such that d(v) > d(w k+1 ) = 1 Note that u 1 v E Hence by the previous discussion, we immediately get f (u 2 ) > f (v) Corollary 36 Let G = (V, E) be a connected graph with pendant vertices that has the largest signless Laplacian spectral radius in B π Then the vertex of G having the largest µweight lies on a cycle Proof Let v 1 be the vertex of G having the largest µweight By contradiction, suppose v 1 does not lie on any cycles By Lemma 22, we have d(v 1 ) 3, and then there exists a hanging tree on v 1 Note that there is an edge u 1 u 2 E of a cycle such that u 1 v 1 E Hence by Lemma 35, f (u 2 ) > f (v 1 ), a contradiction Lemma 37 Let π = (d 1, d 2,, d n ) be a nonincreasing bicyclic degree sequence with d 1 d 2 3 and d n = 1 Then B π is the only connected bicyclic graph that has the largest signless Laplacian spectral radius in B π Proof Let G = (V, E) be a connected graph that has the largest signless Laplacian spectral radius in B π, and f be the Perron vector of Q (G) By Lemma 24, there exists a BFSordering of G with root v 1 such that v 1 v 2 v n, f (v 1 ) f (v 2 ) f (v n ), h(v 1 ) h(v 2 ) h(v n ), and d(v 1 ) d(v 2 ) d(v n ) Let V i = {v V h(v) = i} for i = 0, 1,, h(v n ) (=p) Hence we can relabel the vertices of G in such a way that V i = {v i1,, v i,si } with f (v il ) f (v ij ) f (v rk ) and d(v il ) d(v ij ) d(v rk ) for 0 i < r p, 1 l < j s i, and 1 k s r Clearly, s 1 = d(v 1 ) = d(v 01 ) = d 1 d(v 11 ) 3, and d(v 12 ) d(v 13 ) 2 Since G is a connected bicyclic graph, let C 1, C 2 be two cycles of G (C 1 and C 2 may have some common vertices) and P (if exists) be the unique path joining C 1 and C 2 By Corollary 36, assume that v 01 V(C 1 ) without loss of generality Claim 1 V(C 1 ) V(C 2 ) 2 By contradiction, we have the following two cases Case 1 V(C 1 ) V(C 2 ) = 0 If there exists a hanging tree on v 01, we can take u 1 u 2 E(C 2 ) such that u 1 v 01 E By Lemma 35, we get that f (u 2 ) > f (v 01 ), a contradiction If there exists a hanging tree on v 11, observe that there is an edge u 1 u 2 of a cycle such that u 1, u 2 v 01 and u 1 v 11 E, then by Lemma 35, we have f (u 2 ) > f (v 11 ), which is a contradiction Otherwise, there exist no hanging trees on v 01 and v 11 Hence v 01 V(C 1 ) V(P), v 11 V(C 2 ) V(P), and d(v 01 ) = d(v 11 ) = d(v 12 ) = 3 (since d n = 1) Thus there is a hanging tree on v 12 Without loss of generality, suppose v 12 V(C 1 ) Then there is an edge u 1 u 2 E(C 1 ) such that u 1, u 2 v 01 and u 1 v 12 E By Lemma 35, we have f (u 2 ) > f (v 12 ), also a contradiction Case 2 V(C 1 ) V(C 2 ) = 1 If V(C 1 ) V(C 2 ) = {v 01 } (or V(C 1 ) V(C 2 ) {v 01 }), then there is a hanging tree on v 11 (or v 01 ), and an edge u 1 u 2 E(C i ) such that u 1, u 2 v 01 and u 1 v 11 E (or u 1 v 01 E) By Lemma 35, f (u 2 ) > f (v 11 ) (or f (u 2 ) > f (v 01 )), a contradiction
6 Y Huang et al / Discrete Mathematics 311 (2011) From Claim 1, we conclude that G contains a θgraph θ(p, q, r) as its induced subgraph Let x, y be the vertices such that d θ(p,q,r) (x) = d θ(p,q,r) (y) = 3 If v 01 {x, y} (or v 11 {x, y}), then there is a hanging tree on v 01 (or v 11 ), and an edge u 1 u 2 E(C i ) such that u 1, u 2 v 01 and u 1 v 01 E (or u 1 v 11 E) By Lemma 35, f (u 2 ) > f (v 01 ) (or f (u 2 ) > f (v 11 )), which is a contradiction Hence {v 01, v 11 } = {x, y} Claim 2 V(C 1 ) = V(C 2 ) = 3 By contradiction, suppose V(C i ) > 3 (i = 1 or 2) Note that there exists a hanging tree on v 01 (or on v 11, or a hanging tree on v 12 since d n = 1) Take an edge u 1 u 2 E(C i ) such that u 1, u 2 v 01, u 2 v 11, and u 1 v 01 E (or u 1 v 11 E, or u 1 v 12 E) By Lemma 35, f (u 2 ) > f (v 01 ) (or f (u 2 ) > f (v 11 ), or f (u 2 ) > f (v 12 )), a contradiction It follows from Claim 2 that G contains θ(1, 2, 2) as its induced subgraph Finally, we need to show that v 12, v 13 θ(1, 2, 2) By contradiction, suppose v 1h θ(1, 2, 2) (h = 2 or 3) Then there is a hanging tree on v 1h Take an edge u 1 u 2 θ(1, 2, 2) such that u 2 v 01, v 11 and u 1 v 1h E Then by Lemma 35, f (u 2 ) > f (v 1h ), which is a contradiction Therefore, v 11 v 12, v 11 v 13 E Combining this with the BFSordering of G given above, we obtain that G = B π The main result of this section follows from Lemmas 33, 34 and 37 Theorem 38 Let π be a bicyclic degree sequence Then B π Laplacian spectral radius in B π is the only connected bicyclic graph that has the largest signless Now the signless Laplacian majorization theorem is proved to be true for bicyclic graphs in the following theorem Theorem 39 Let π and π be two different nonincreasing bicyclic degree sequences with the same order If π π, then µ(b π ) < µ(b π ) Proof By Lemma 211, without loss of generality, assume that π = (d 1,, d n ) and π = (d,, 1 d ) n with d i = d i (i p, q), and d p = d 1, p d q = d + q 1 (1 p < q n) Let f be the Perron vector of Q (B π ) By Lemma 24, there exists a BFSordering of B π with root v 1 such that v 1 v 2 v n, f (v 1 ) f (v 2 ) f (v n ), h(v 1 ) h(v 2 ) h(v n ), and d(v i ) = d i for i = 1, 2,, n If π = (4, 2,, 2), then π = (5, 2,, 2, 1) or (4, 3, 2,, 2, 1); if π = (3, 3, 2,, 2), then π = (3, 3, 3, 2,, 2, 1) or (4, 3, 2,, 2, 1) or (4, 2,, 2) By Theorem 38, it can be directly checked that there exists a vertex v k (k p, q) such that v k v q E(B π ) and v kv p E(B π ) Let B 1 = B π + v kv p v k v q Then by Lemma 26, µ(b π ) < µ(b 1) Since B 1 B π, it follows from Theorem 38 that µ(b π ) < µ(b 1) µ(b π ) In the following, we may assume that π (4, 2,, 2) and (3, 3, 2,, 2) Note that d q 1, if 5 q n, 2, if 2 q 4 Then d q = d q + 1 2, if 5 q n, Case 1 d q 3, if 2 q 4 2, if 6 q n, 3, if 3 q 5, 4, if q = 2 By Theorem 38, there exists a vertex v k with k > q such that v k v q E(B π ) and v kv p E(B π ) Let B 1 = B π + v kv p v k v q Then by Lemma 26, µ(b π ) < µ(b 1) Since B 1 B π, by Theorem 38, we have µ(b π ) < µ(b 1) µ(b π ) Case 2 q = 2 and d 2 = 3 Then d = 2 2 Subcase 21 d = 1 4 Then d 1 = 3 and n 5 Hence π = (4, 2,, 2), and then π = (3, 3, 2,, 2) Such a subcase has been proved Subcase 22 d > 1 4 Suppose that B π has k pendant vertices It follows that π = (k + 4, 2,, 2, 1,, 1), and then π = (k + 3, 3, 2,, 2, 1,, 1) By Theorem 38 and Lemmas 212 and 213, we obtain that d(u)(d(u) + m(u)) + d(v)(d(v) + m(v)) µ(b π ) max uv E(B π ) d(u) + d(v) k + 5 = (B π ) + 1 < µ(b π ) Case 3 q = 5 and d 5 = 2 Then d = 5 1 Subcase 31 d n = 1 and d 2 = 2 If p = 1, then d > 1 d 1 > 4 and d i = d i = 2 (2 i 4), but there does not exist such graph with degree sequence π by Proposition 32 Hence p = 2, and then d = 2 d = 3 By Theorem 38, v 4 v 5 E(B π ) but v 2 v 4 E(B π ) Let B 1 = B π + v 2v 4 v 4 v 5 Then by Lemma 26, µ(b π ) < µ(b 1) Since B 1 B π, by Theorem 38, µ(b π ) < µ(b 1) µ(b π ) Subcase 32 d n = 1 and d 2 3 By Theorem 38, there exists a vertex u such that uv 5 E(B π ) but uv p E(B π ) Let B 1 = B π + uv p uv 5 Then by Lemma 26, µ(b π ) < µ(b 1) Since B 1 B π, by Theorem 38, µ(b π ) < µ(b 1) µ(b π ) Let B n denote the set of all connected bicyclic graphs of order n Denote by B n (s) the set of all connected bicyclic graphs of order n with s pendant vertices, and B n, the set of all connected bicyclic graphs of order n with maximum degree, respectively
7 510 Y Huang et al / Discrete Mathematics 311 (2011) Corollary 310 Let G B n, where n 4 Then µ(g) µ(b π 1 ) with equality if and only if G is isomorphic to B π 1 with π 1 = (n 1, 3, 2, 2, 1,, 1), where the frequency of 1 in π 1 is n 4 (In other words, B π 1 is obtained from θ(1, 2, 2) and (n 4) K 2 by identifying one of the maximum degree vertices of θ(1, 2, 2) with one vertex of each K 2 ) Proof Let G B n with the nonincreasing degree sequence π If π = π 1, then by Theorem 38, µ(g) µ(b π 1 ) with equality if and only if G is isomorphic to B π 1 If π π 1, it is easy to see that π, π 1 are two different bicyclic degree sequences, and π π 1 By Theorems 38 and 39, we have µ(g) µ(b π ) µ(b π 1 ) with equality if and only if G is isomorphic to B π 1 Corollary 311 Let G B n (s), where n 4 (1) If n = 4, then G = θ(1, 2, 2) (2) If s = 0 and n 5, then µ(g) µ(c(3, n 3)) with equality if and only if G = C(3, n 2) (3) If n = 5 and s 1, then µ(g) µ(g ) with equality if and only if G is isomorphic to G, where G is obtained from θ(1, 2, 2) and K 2 by identifying one of the maximum degree vertices of θ(1, 2, 2) with one vertex of K 2 (4) If 1 s n 5, then µ(g) µ(b π 2 ) with equality if and only if G is isomorphic to B π 2 with π 2 = (s+4, 2,, 2, 1,, 1), where the frequency of 2 in π 2 is n s 1, and the frequency of 1 in π 2 is s (In other words, suppose n 5 = sq+t, 0 t < s, and B π 2 is obtained from C(3, 3), t paths of order q + 2, and s t paths of order q + 1 by identifying the maximum degree vertex of C(3, 3) with one end of each path of the s paths) Proof The result of (1) is obvious Now let G B n (s) with the nonincreasing degree sequence π If s = 0 and n 5, then π = (4, 2,, 2) or (3, 3, 2,, 2) If n = 5 and s 1, then π = (4, 3, 2, 2, 1) or (3, 3, 3, 2, 1) Note that (3, 3, 2,, 2) (4, 2,, 2), and (3, 3, 3, 2, 1) (4, 3, 2, 2, 1) By Theorems 38 and 39, we obtain the results of (2) and (3) as desired (4) If 1 s n 5, let π = (d 1,, d n ) be the nonincreasing degree sequence of G B n (s) Thus d n s > 1 and d n s+1 = = d n = 1 If π = π 2, then by Theorem 38, µ(g) µ(b π 2 ) with equality if and only if G is isomorphic to B π 2 If π π 2, it is easy to see that π, π 2 are two different bicyclic degree sequences, and π π 2 By Theorems 38 and 39, we have µ(g) µ(b π ) µ(b π 2 ) with equality if and only if G = B π 2 Hence the assertion holds Corollary 312 Let G B n,, where n 1 3 Then µ(g) µ(b π 3 ) with equality if and only if G = B π 3, where π 3 is defined as follows: n+2 (1) If 2 n 1, then π3 = (, n + 2, 2, 2, 1,, 1), where the frequency of 1 in π 3 is n 4 n+4 (2) If 3 < n+2 2, then π3 = (,, n 2 + 4, 2, 1,, 1), where the frequency of 1 in π 3 is n 4 (3) If < n+4 3, then let p = log 1 n( 2) 2 ( 1) 4 n ( 1)p 1 [ ( 1) 4] + 2 = Denote by ( 1)r 4 + q, if p = 1, ( 1)r + q, if p 2, where 0 q < 1 Hence π 3 = (,,, q + 1, 1,, 1), where the frequency of in π 3 is m = 1 + r, if p = 1, ( 1) p 2 [ ( 1) 4] r, if p 2 Proof Let π be the nonincreasing degree sequence of G B n, If π = π 3, then by Theorem 38, µ(g) µ(b π 3 ) with equality if and only if G = B π 3 If π π 3, it is easy to see that π, π 3 are two different bicyclic degree sequences, and n+2 (1) if 2 n 1, then π π3 = (, n + 2, 2, 2, 1,, 1); n+4 (2) if 3 < n+2 2, then π π3 = (,, n 2 + 4, 2, 1,, 1) Then for (1) and (2), by Theorems 38 and 39, we have µ(g) µ(b π ) µ(b π 3 ) with equality if and only if G = B π 3 (3) If < n+4 3, by Theorem 38, B π3 can be constructed as follows Assume that B π 3 has (p + 2) layers Then there is one vertex in the layer 0 (ie, the root vertex); there are vertices in layer 1; there are ( 1) 4 vertices in layer 2;
8 Y Huang et al / Discrete Mathematics 311 (2011) ; there are [ ( 1) 4]( 1) p 2 vertices in layer p (p 2); there are at most [ ( 1) 4]( 1) p 1 vertices in layer p + 1 Hence for p 1, ( 1) p 1 [ ( 1) 4] + 2 < n ( 1)p [ ( 1) 4] + 2 Let p = Since < n+4 3, then p 1, and there exist an integer r and 0 q < 1 such that log 1 n( 2) 2 ( 1) 4 n ( 1)p 1 [ ( 1) 4] + 2 = ( 1)r 4 + q, if p = 1, ( 1)r + q, if p 2 Therefore, the degrees of all vertices from layer 0 to layer p 1 are, and there are r vertices in layer p with degree Let 1 + r, if p = 1, m = ( 1) p 2 [ ( 1) 4] r, if p 2 Then there are m vertices with degree in B π 3 Therefore, by Equality ( ), π 3 = (,,, q + 1, 1,, 1), where the frequency of in π 3 is m Moreover, it can be seen that π, π 3 are two different bicyclic degree sequences and π π 3 It follows from Theorems 38 and 39 that µ(g) µ(b π ) µ(b π 3 ) ( ) with equality if and only if G = B π 3 Acknowledgements The authors would like to thank the referees for the valuable comments, and for the suggestions to improve the readability of this paper References [1] JA Bondy, USR Murty, Graph Theory with Applications, Macmillan Press, New York, 1976 [2] RA Brualdi, ES Solheid, On the spectral radius of connected graphs, Publ Inst Math (Beograd) 39 (53) (1986) [3] D Cvetković, Signless Laplacian and line graphs, Bull Acad Serbe Sci Arts Cl Sci Math Natur Sci Math 131 (30) (2005) [4] D Cvetković, M Doob, H Sachs, Spectra of Graphs Theory and Applications, third ed, Academic Press, New York, 1980, 1995 [5] D Cvetković, P Rowlinson, S Simić, Signless Laplacian of finite graphs, Linear Algebra Appl 423 (2007) [6] D Cvetković, P Rowlinson, S Simić, An Introduction to the Theory of Graph Spectra, Cambridge University Press, Cambridge, 2010 [7] D Cvetković, SK Simić, Towards a spectral theory of graphs based on the signless Laplacian I, Publ Inst Math (Beograd) 85 (99) (2009) [8] D Cvetković, SK Simić, Towards a spectral theory of graphs based on the signless Laplacian II, Linear Algebra Appl 432 (2010) [9] D Cvetković, SK Simić, Towards a spectral theory of graphs based on the signless Laplacian III, Appl Anal Discrete Math 4 (2010) [10] M Desai, V Rao, A characterization of the smallest eigenvalue of a graph, J Graph Theory 18 (1994) [11] P Erdös, T Gallai, Graphs with prescribed degrees of vertices, Mat Lapok 11 (1960) (in Hungarian) [12] Y Fan, BS Tam, J Zhou, Maximizing spectral radius of unoriented Laplacian matrix over bicyclic graphs of a given order, Linear Multilinear Algebra 56 (2008) [13] Y Fan, D Yang, The signless Laplacian spectral radius of graphs with given number of pendant vertices, Graphs Combin 25 (2009) [14] L Feng, Q Li, XD Zhang, Minimizing the Laplacian spectral radius of trees with given matching number, Linear Multilinear Algebra 55 (2007) [15] RA Horn, CR Johnson, Matrix Analysis, Cambridge University Press, 1990, Reprinted with corrections [16] J Li, XD Zhang, On the Laplacian eigenvalues of a graph, Linear Algebra Appl 285 (1998) [17] R Merris, Laplacian matrices of graphs: a survey, Linear Algebra Appl (1994) [18] BS Tam, Y Fan, J Zhou, Unoriented Laplacian maximizing graphs are degree maximal, Linear Algebra Appl 429 (4) (2008) [19] WD Wei, The class R(R, S) of (0, 1)matrices, Discrete Math 39 (1982) [20] XD Zhang, The Laplacian spectral radii of trees with degree sequences, Discrete Math 308 (2008) [21] XD Zhang, The signless Laplacian spectral radius of graphs with given degree sequences, Discrete Appl Math 157 (2009)
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