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The second Zagreb indices of unicyclic graphs with given degree sequences
Discrete Applied Mathematics 167 (2014) 217–221
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Discrete Applied Mathematics journal homepage: www.elsevier.com/locate/dam
The second Zagreb indices of unicyclic graphs with given degree sequences Muhuo Liu a,b,c , Bolian Liu b,∗ a
Institute of Mathematics, School of Mathematical Science, Nanjing Normal University, Nanjing, 210046, China
b
School of Mathematical Science, South China Normal University, Guangzhou, 510631, China
c
Department of Mathematics, South China Agricultural University, Guangzhou, 510642, China
article
abstract
info
Let π = (d1 , d2 , . . . , dn ) and π ′ = (d′1 , d′2 , . . . , d′n ) be two different non-increasing degree
Article history: Received 16 October 2012 Received in revised form 21 October 2013 Accepted 25 October 2013 Available online 22 November 2013
j
j
′ ′ sequences. We write π ▹ π ′ , if and only if i=1 di = i=1 di ≤ i=1 di for i=1 di , and all j = 1, 2, . . . , n. Let Γ (π ) be the class of connected graphs with degree sequence π . The second Zagreb index of a graph G is denoted by M2 (G) = uv∈E (G) d(u)d(v). In this paper, we characterize an extremal unicyclic graph that achieves the maximum second Zagreb index in the class of unicyclic graphs with given degree sequence, and we also prove that if π ▹ π ′ , π and π ′ are unicyclic degree sequences and U ∗ and U ∗∗ have the maximum second Zagreb indices in Γ (π ) and Γ (π ′ ), respectively, then M2 (U ∗ ) < M2 (U ∗∗ ). Furthermore, we determine the first to ninth largest second Zagreb indices together with the corresponding extremal unicyclic graphs in the class of unicyclic graphs on n ≥ 17 vertices. © 2013 Elsevier B.V. All rights reserved.
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Keywords: Second Zagreb index Degree sequence Majorization theorem
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1. Introduction Throughout this paper, we are concerned with connected undirected simple graph only. Let Sn , Cn and Pn be the star, cycle and path of order n, respectively. The second Zagreb index of G is denoted by [2] M2 (G) =
(d(u)d(v)) .
(1.1)
uv∈E (G)
For the chemical applications and mathematical properties of the second Zagreb index, we refer the readers to [1,2,5,6] and the references therein. The sequence π = (d1 , d2 , . . . , dn ) is called the degree sequence of G if di = d(vi ) holds for 1 ≤ i ≤ n. Throughout this paper, we use di to denote the ith largest degree of G and we suppose that d(vi ) = di , where 1 ≤ i ≤ n. Let Γ (π ) be the class of connected graphs with degree sequence π . Suppose π = (d1 , d2 , . . . , dn ) and π ′ = (d′1 , d′2 , . . . , d′n ) are two different non-increasing degree sequences, we write
j
j
′ i=1 di for all j = 1, 2, . . . , n. Such an ordering is sometimes called majorization (see [4]). A unicyclic graph is a connected graph with n vertices and n edges. In [3], an extremal tree with the maximum second Zagreb index in the class of trees with given degree sequence was characterized. In this note, we identify an extremal unicyclic graph with the maximum second Zagreb index in the class of unicyclic graphs with given degree sequence, and we also prove
π ▹ π ′ if and only if
∗
n
i=1
di =
n
i=1
d′i , and
i=1
di ≤
Corresponding author. Tel.: +86 20 85212613. E-mail addresses: [email protected] (M. Liu), [email protected], [email protected] (B. Liu).
0166-218X/$ – see front matter © 2013 Elsevier B.V. All rights reserved. http://dx.doi.org/10.1016/j.dam.2013.10.033
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M. Liu, B. Liu / Discrete Applied Mathematics 167 (2014) 217–221
that if π ▹ π ′ , π and π ′ are two different unicyclic degree sequences, U ∗ and U ∗∗ have the maximum second Zagreb indices in Γ (π ) and Γ (π ′ ), respectively, then M2 (U ∗ ) < M2 (U ∗∗ ). Furthermore, we determine the first to ninth largest second Zagreb indices together with the corresponding extremal unicyclic graphs in the class of unicyclic graphs on n ≥ 17 vertices. 2. The main results Let G − uv and G + uv , respectively, denote the new graph obtained from G by deleting and adding the edge uv . Lemma 2.1 ([3]). Let G = (V , E ) be a connected graph with v1 u1 ∈ E , v2 u2 ∈ E , v1 v2 ̸∈ E and u1 u2 ̸∈ E. Let G′ = G − u1 v1 − u2 v2 + v1 v2 + u1 u2 . If d(v1 ) ≥ d(u2 ) and d(v2 ) ≥ d(u1 ), then M2 (G′ ) ≥ M2 (G), where M2 (G′ ) > M2 (G) if and only if both two inequalities are strict. Lemma 2.2 ([3]). Suppose G ∈ Γ (π ), and there exist three vertices u, v, w of a connected graph G such that uv ∈ E (G), uw ̸∈ E (G), d(v) < d(w) ≤ d(u), and d(u) > d(x) for all x ∈ N (w). Then, there exists another connected graph G′ ∈ Γ (π ) such that M2 (G) < M2 (G′ ). Lemma 2.3. For any degree sequence π with n ≥ 3, there exists an extremal graph G ∈ Γ (π ) with the maximum second Zagreb index in Γ (π ) such that {v2 , v3 } ⊆ N (v1 ). Proof. If v1 v2 ̸∈ E (G), then there is some vertex v such that v1 v ∈ E (G) and d(v1 ) ≥ d(v2 ) > d(v) and d(v1 ) > d(x) holds for all x ∈ N (v2 ), which contradicts Lemma 2.2. Thus, v1 v2 ∈ E (G). Now, we assume that v1 v3 ̸∈ E (G). Then, d(v3 ) > d(v) holds for every v ∈ N (v1 )\{v2 }. By Lemma 2.2, we may assume that there exists some vertex u ∈ N (v3 ) such that d(u) = d1 . If u = v2 , the result already holds. If u ̸= v2 , then d(u) = d1 = d2 = d3 . Let P be a shortest path from v1 to v3 . If v2 ̸∈ V (P ), choose x ∈ N (v1 ) ∩ V (P ), since v1 ∈ N (x) \ N (v3 ), there must exist some vertex y ∈ N (v3 ) \ V (P ) such that y ̸∈ N (x). Let G1 = G + v1 v3 + xy − v1 x − v3 y. By Lemma 2.1, G1 ∈ Γ (π ) and M2 (G1 ) ≥ M2 (G), the result holds. If v2 ∈ V (P ), we may suppose that v2 v3 ̸∈ E (G) (otherwise, the result already holds). Choose x ∈ N (v2 ) ∩ V (P ) such that x ̸= v1 . It can be proved similarly with the case v2 ̸∈ V (P ). Lemma 2.4. If π is a unicyclic degree sequence with dn = 1, then there exists an extremal graph G ∈ Γ (π ) with the maximum second Zagreb index in Γ (π ) such that the unique cycle of G is a triangle with V (C3 ) = {v1 , v2 , v3 }. Proof. Suppose Ct is the unique cycle of G. Now, it suffices to prove the following claims. Claim 1. There is an extremal graph G ∈ Γ (π ) such that v1 ∈ V (Ct ) and {v2 , v3 } ⊆ N (v1 ). Assume that Claim 1 does not hold for any extremal graph G ∈ Γ (π ). By Lemma 2.3, we may suppose that G is an extremal graph of Γ (π ) such that {v2 , v3 } ⊆ N (v1 ). Then, v1 ̸∈ Ct . Suppose that P = u · · · v1 · · · xy is the unique path connecting u and y such that v1 is on the path P, where u ∈ Ct and d(y) = 1. Let w ∈ N (u) ∩ V (Ct ). If d(w) ≤ d(x), let G1 = G + ux +w y −w u − xy. By Lemma 2.1, M2 (G1 ) ≥ M2 (G). Note that G1 ∈ Γ (π ), v1 is in the unique cycle of G1 and {v2 , v3 } ⊆ N (v1 ), a contradiction. If d(u) ≤ d(x), let G2 = G + uy + w x − w u − xy. By Lemma 2.1, M2 (G2 ) ≥ M2 (G). Note that G2 ∈ Γ (π ), v1 is in the unique cycle of G2 and {v2 , v3 } ⊆ N (v1 ), a contradiction. Thus, min{d(u), d(w)} > d(x). Now, choose z ∈ (N (x) ∩ V (P )) \ {y}. Similarly, min{d(u), d(w)} > d(z ). Repeating the above process, we will yield that min{d(u), d(w)} > d(v1 ), a contradiction. Thus, Claim 1 holds. Claim 2. There is an extremal graph G ∈ Γ (π ) such that v1 v2 ∈ E (Ct ) and v3 ∈ N (v1 ). Assume that Claim 2 does not hold for any extremal graph G ∈ Γ (π ). By Claim 1, there exists an extremal graph G ∈ Γ (π ) such that v1 ∈ V (Ct ) and {v2 , v3 } ⊆ N (v1 ). Then, v2 ̸∈ V (Ct ). Choose {u, v} ⊆ V (Ct ) \ {v1 } such that uv ∈ E (Ct ). Suppose that P = v1 v2 xy · · · z is the unique path connecting v1 and z such that v2 is on the path P, where d(z ) = 1. Suppose that max{d(u), d(v)} = d(u). If d(u) ≥ d(x), let G1 = G + uv2 + v x − uv − v2 x. By Lemma 2.1, M2 (G1 ) ≥ M2 (G). But Claim 2 holds for G1 , a contradiction. Thus, max{d(u), d(v)} = d(u) < d(x). Similarly, max{d(u), d(v)} < d(y). Repeating the above process, we will yield that max{d(u), d(v)} < d(z ) = 1, a contradiction. Thus, Claim 2 holds. Claim 3. There is an extremal graph G ∈ Γ (π ) such that {v1 v2 , v1 v3 } ⊆ E (Ct ). By Claim 2, there is an extremal graph G ∈ Γ (π ) such that v1 v2 ∈ E (Ct ) and v3 ∈ N (v1 ). If v3 ̸∈ V (Ct ), then v2 v3 ̸∈ E (G). Choose u ∈ (V (Ct ) ∩ N (v2 )) \ {v1 } and v ∈ N (v3 ) \ {v1 }. Let G1 = G + v2 v3 + uv − vv3 − uv2 . By Lemma 2.1, M2 (G1 ) ≥ M2 (G) and G1 ∈ Γ (π ). It is easily checked that Claim 3 holds for G1 . Claim 4. There is an extremal graph G ∈ Γ (π ) such that {v1 , v2 , v3 } = V (Ct ). If d2 = 2, it is easily checked that the result holds. So, we may suppose that d2 ≥ 3. By Claim 3, there exists an extremal graph G ∈ Γ (π ) such that {v1 v2 , v1 v3 } ⊆ E (Ct ). If {v1 , v2 , v3 } ̸= V (Ct ), then v2 v3 ̸∈ E (G). Choose u ∈ N (v2 ) \ V (Ct ), and choose v ∈ (N (v3 ) ∩ V (Ct )) \ {v1 }. Let G1 = G + v2 v3 + uv − vv3 − uv2 . By Lemma 2.1, M2 (G1 ) ≥ M2 (G). Since G1 ∈ Γ (π ) and {v1 , v2 , v3 } = V (Ct ), Claim 4 holds. To give our main result, we have to introduce the following notation. Suppose π = (d1 , d2 , . . . , dn ), where dn = 1. Let UM (π ) be the unique unicyclic graph such that the unique cycle of UM (π) is a triangle with V (C3 ) = {v1 , v2 , v3 }, and the remaining vertices appear in BFS-ordering with respect to C3 starting from v4 that is adjacent to v1 . It means that, UM (π ) can be constructed by the breadth-first-search method as follows: select the vertices {v1 , v2 , v3 } as the root vertices and begin with {v1 , v2 , v3 } of the zeroth layer. Select the vertices v4 , v5 , . . . , vd1 +d2 +d3 −3 as the first layer such that N (v1 ) = {v2 , v3 , v4 , v5 , . . . , vd1 +1 }, N (v2 ) = {v1 , v3 , vd1 +2 , vd1 +3 , . . . , vd1 +d2 −1 },
M. Liu, B. Liu / Discrete Applied Mathematics 167 (2014) 217–221
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Fig. 1. The unicyclic graphs UM (π1 ) and F1 .
N (v3 ) = {v1 , v2 , vd1 +d2 , . . . , vd1 +d2 +d3 −3 }. Then, append d4 − 1 vertices to v4 such that N (v4 ) = {v1 , vd1 +d2 +d3 −2 , . . . , vd1 +d2 +d3 +d4 −4 } · · ·. Informally, for a given unicyclic degree sequence π1 = (5, 4, 3(3) , 2(10) , 1(8) ), UM (π1 ) is the unicyclic graph as shown in Fig. 1. Theorem 2.1. If dn = 1, then UM (π ) achieves the maximum second Zagreb index in the class of unicyclic graphs with given degree sequence π . Proof. By Lemma 2.4, we may suppose that G is an extremal graph in Γ (π ) such that the unique cycle of G is a triangle with V (C3 ) = {v1 , v2 , v3 }. Now, we create an ordering ≺ of V (G) by the breadth-first search as follows: firstly, let v1 ≺ v2 ≺ v3 ; secondly, append all neighbors u4 , . . . , ud1 +1 of N (v1 ) \ {v2 , v3 } to the ordered list, these neighbors are ordered such that u ≺ v whenever d(u) > d(v) (in the remaining case the ordering can be arbitrary); thirdly, append all neighbors ud1 +2 , . . . , ud1 +d2 −1 of N (v2 )\{v1 , v3 } to the ordered list, these neighbors are ordered such that u ≺ v whenever d(u) > d(v) (in the remaining case the ordering can be arbitrary). Then, with the same method we can append the vertices N (v3 )\{v1 , v2 } in the ordered list, and then to the vertices N (x) \ {v1 }, where d(x) = max{d(y): where y ∈ N (v1 ) \ {v2 , v3 }}. Continue recursively with all vertices v1 , v2 , . . ., until all vertices of G are processed. Let us use the notation Hi (G) to denote the set of vertices of G at distance i ≥ 0 from C3 , where V (C3 ) = {v1 , v2 , v3 }. Then, H0 = {v1 , v2 , v3 }. Suppose the last layer of G is p. It suffices to show the following claims. Claim 1. d(u) ≥ d(v) holds for every u ∈ Hi (G) and every y ∈ Hj (G), where j > i ≥ 0. Suppose that there exist vertices u and v with u ∈ Hi (G) and v ∈ Hj (G) such that d(u) < d(v), where i < j. Furthermore, for convenience, we may choose u as the first vertex in the ordering ≺ with such property and let d(v) = max{d(x) : x ∈ Ht , where t ≥ i + 1}. Choose u′ ∈ Hi−1 (G) such that u′ u ∈ E (G). Then, 1 ≤ i < j, since u ̸∈ H0 . If u lies on the unique shortest path connecting v to C3 , suppose that P = v · · · y is a path of G, where d(y) = 1. Choose z ∈ V (P ) such that d(z ) > d(u) and the distance between z and y is as small as possible. Without loss of generality, suppose that z = v . Choose v ′ ∈ Hj+1 ∩ V (P ) such that vv ′ ∈ E (G). By the choice of u and z , d(u′ ) ≥ d(v) > d(u) ≥ d(v ′ ). Let G1 = G + uv ′ + u′ v − u′ u − vv ′ . Then, G1 ∈ Γ (π ). By Lemma 2.1, M2 (G1 ) > M2 (G), which contradicts the choice of G. Thus, u does not lie on the unique shortest path connecting v to C3 . Choose w ∈ N (v) ∩ Hj−1 (G). Then, uw ̸∈ E (G) and u ̸= w . By the choice of u and j − 1 > i − 1, d(u′ ) ≥ d(w). Recall that d(v) > d(u). Let G2 = G + uw + u′ v − uu′ − vw . By Lemma 2.1, M2 (G2 ) ≥ M2 (G) and G2 ∈ Γ (π ). Now, we construct a new ordering ≺′ of V (G2 ) with the similar method as ≺. We suppose that v1 ≺ v2 ≺ v3 ≺ u1 ≺ · · · ≺ ut ≺ u is the first t + 4 elements in the ordering ≺ of V (G). By the choice of u and v , we have v1 ≺ v2 ≺ v3 ≺ u1 ≺ · · · ≺ ut ≺ v is the first t + 4 elements in the ordering ≺′ of V (G2 ). By the choice of u and v , for every x ∈ {v1 , v2 , v3 , u1 , . . . , ut , v}, if x ∈ Ht (G2 ), then d(x) ≥ d(x′ ) holds for every x′ ∈ Hs (G2 ), where t < s. Repeating the above process finitely many times, we can achieve a graph G∗ and an ordering ≺∗ such that Claim 1 holds for G∗ . Thus, we may suppose that Claim 1 holds for G. Claim 2. For 0 ≤ i ≤ p − 1, if {u, v} ⊆ Hi (G) and u ≺ v , then d(x) ≥ d(y) holds for every x ∈ N (u) ∩ Hi+1 (G) and every y ∈ N (v) ∩ Hi+1 (G). If Claim 2 does not hold, we may suppose that u is the first vertex in the ordering ≺ with the property that there exist vertices v ∈ Hi (G) (suppose that u ∈ Hi (G)), x ∈ N (u)∩ Hi+1 (G) and y ∈ N (v)∩ Hi+1 (G) such that u ≺ v , but d(x) < d(y). Furthermore, according to this, we may suppose that x is the first vertex in the ordering ≺ and suppose that d(y) = max{d(z ) : z ∈ Hi+1 (G), where x ≺ z }. Let G1 = G + uy + v x − ux − v y. Then, G1 ∈ Γ (π ). By the choice of u, we have d(u) ≥ d(v). Thus, M2 (G1 ) ≥ M2 (G) by Lemma 2.1. Now, we construct a new ordering ≺′ of V (G1 ) with the similar method as ≺. We suppose that v1 ≺ v2 ≺ v3 ≺ u1 ≺ · · · ≺ ut ≺ x is the first t + 4 elements in the ordering ≺ of V (G). By the choice of x and y, v1 ≺ v2 ≺ v3 ≺ u1 ≺ · · · ≺ ut ≺ y is the first t + 4 elements in the ordering ≺′ of V (G1 ). By the choice of x and y, for every z ∈ {v1 , v2 , v3 , u1 , . . . , ut , y} ∩ Hi+1 (G1 ), if w ∈ Hi+1 (G1 ) and z ≺′ w , then d(z ) ≥ d(w). Repeating the above process finitely many times, we can obtain a graph G∗ and an ordering ≺∗ of V (G∗ ) such that M2 (G∗ ) = M2 (G) and Claim 2 holds for G∗ . Furthermore, we can easily see that Claim 1 also holds for G∗ .
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M. Liu, B. Liu / Discrete Applied Mathematics 167 (2014) 217–221
Fig. 2. The unicyclic graphs U1 , U2 , . . . , U10 .
Thus, we may suppose that Claims 1 and 2 hold for G. By Claims 1 and 2, G = UM (π ). This completes the proof of this result. Remark 2.1. For a given unicyclic graph degree sequence π , it is easy to see that UM (π ) is unique, but it cannot deduce that the UM (π ) is the unique unicyclic graph with the maximum second Zagreb index in Γ (π ). For instance, when π1 = (5, 4, 3(3) , 2(10) , 1(8) ), then M2 (F1 ) = M2 (UM (π1 )), but F1 ̸= UM (π1 ), where F1 and UM (π1 ) are shown in Fig. 1. Lemma 2.5 ([4]). Let π and π ′ be two different non-increasing degree sequences. If π ▹ π ′ , then there exists a series of nonincreasing degree sequences π1 , . . . , πk such that (π =) π0 ▹ π1 ▹ · · · ▹ πk ▹ πk+1 (=π ′ ), and πi and πi+1 differ only in two positions, where the differences are 1 for 0 ≤ i ≤ k. Lemma 2.6 ([3]). Let u, v be two vertices of a connected graph G, and w1 , w2 , . . . , wk (1 ≤ k ≤ d(v)) be some vertices of N (v) \ (N (u) ∪ {u}). Let G′ = G + w1 u + w2 u + · · · + wk u − w1 v − w2 v − · · · − wk v . If d(u) ≥ d(v) and y∈N (u) d(y) ≥ ′ x∈N (v) d(x), then M2 (G ) > M2 (G). Theorem 2.2. Let π and π ′ be two different non-increasing unicyclic degree sequences with π ▹ π ′ . Let U ∗ and U ∗∗ be the unicyclic graphs with maximum second Zagreb indices in Γ (π ) and Γ (π ′ ), respectively. Then, M2 (U ∗ ) < M2 (U ∗∗ ). Proof. Set π = (d1 , d2 , . . . , dn ) and π ′ = (d′1 , d′2 , . . . , d′n ). Since π ▹ π ′ , by Lemma 2.5 we may suppose that π and π ′ differ only in two positions, where the differences are 1. Thus, we may assume that di = d′i for i ̸= p, q, and dp + 1 = d′p , dq − 1 = d′q . If dn = 2, then π = (2, 2, . . . , 2) and π ′ = (3, 2, . . . , 2, 1). It is easily checked that M2 (Cn ) < M2 (UM (π ′ )), and hence the result follows. If dn = 1, by Theorem 2.1 it suffices to show that M2 (UM (π )) < M2 (UM (π ′ )). Since π ▹ π ′ , we have 1 ≤ p < q ≤ n, and hence of V (G)) and d(vp ) ≥ d(vq ). Now, by vp ≺ vq (in the BFS-ordering Claims 1 and 2 in the proof of Theorem 2.1, we have y∈N (vp ) d(y) ≥ x∈N (vq ) d(x). Let P be a shortest path from UM (π)
UM (π)
vp to vq in UM (π ). If 2 ≤ q ≤ 3, then dq ≥ 3 since UM (π ′ ) ∈ Γ (π ′ ). If q ≥ 4, then dq ≥ 2. In both cases, there exists a vertex vk (k > q) such that vk ∈ NUM (π) (vq ) \ NUM (π) (vp ) and vk ̸∈ V (P ). Let G = UM (π ) − vq vk + vp vk . Note that G ∈ Γ (π ′ ) and d(vp ) ≥ d(vq ). By Lemma 2.6, M2 (UM (π )) < M2 (G) ≤ M2 (UM (π ′ )).
Paths Pl1 , . . . , Plk are said to have almost equal lengths if l1 , . . . , lk satisfy |li − lj | ≤ 1 for 1 ≤ i ≤ j ≤ k. Suppose G is a unicyclic graph with π = (d1 , d2 , . . . , dn ) as its degree sequence, where dn−k+1 = · · · = dn = 1 and dn−k ≥ 2. Then, π ▹ π ′ = (k + 2, 2, . . . , 2, 1, . . . , 1), where π ′ contains exactly k elements of 1. By Theorem 2.1, it immediately follows that Corollary 2.1 ([7]). If G is a unicyclic graph with n vertices and k pendant vertices, then M2 (G) ≤ M2 (Fn (k)), where Fn (k) is the unicyclic graph on n vertices obtained by attaching k paths of almost equal lengths to one vertex of C3 . Let U1 , U2 , . . . , U10 be the unicyclic graphs on n vertices as shown in Fig. 2, and let H1 , H2 , . . . , H7 be the unicyclic graphs on n vertices as shown in Fig. 3. Theorem 2.3. Let U be a unicyclic graph on n ≥ 17. If U ̸∈ {U1 , U2 , . . . , U10 }, then M2 (U ) < M2 (U10 ) < M2 (U9 ) < M2 (U8 ) < M2 (U7 ) = M2 (U6 ) < M2 (U5 ) < M2 (U4 ) < M2 (U3 ) < M2 (U2 ) < M2 (U1 ). Proof. It is easily checked that U1 is the unique unicyclic graph with d1 = n − 1, U2 , U3 , U6 are all the unicyclic graphs with d1 = n − 2, and U4 , U5 , U7 , U8 , U9 and H1 , H2 , . . . , H7 are all the unicyclic graphs with d1 = n − 3. Suppose the degree sequence of U is (a) = (d1 , d2 , d3 , . . . , dn ) and (b) = (n − 4, 5, 2, 1, . . . , 1). It is easily checked that U10 is the unique unicyclic graph with (b) as its degree sequence. If d1 ≤ n − 4, since U ̸= U10 , we have (a) ▹ (b). By Theorem 2.2, M2 (U ) < M2 (U10 ).
M. Liu, B. Liu / Discrete Applied Mathematics 167 (2014) 217–221
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Fig. 3. The unicyclic graphs H1 , H2 , . . . , H7 .
By an elementary computation, we have M2 (H7 ) = n2 − 4n + 15,
M2 (H6 ) = n2 − 4n + 18,
M2 (H5 ) = n2 − 4n + 19,
M2 (H4 ) = M2 (H3 ) = n2 − 3n + 10,
M2 (H2 ) = n − 3n + 13,
M2 (H1 ) = n2 − 3n + 14,
M2 (U10 ) = n − 3n + 21,
M2 (U9 ) = n2 − 2n + 5,
2
2
M2 (U8 ) = n2 − 2n + 7, M2 (U5 ) = n − 2n + 12, 2
M2 (U3 ) = n − n + 4, 2
M2 (U7 ) = M2 (U6 ) = n2 − 2n + 8, M2 (U4 ) = n2 − 2n + 13, M2 (U2 ) = n2 − n + 7,
M2 (U1 ) = n2 + 3. Since n ≥ 17, we have M2 (H7 ) < M2 (H6 ) < M2 (H5 ) < M2 (H4 ) = M2 (H3 ) < M2 (H2 ) < M2 (H1 ) < M2 (U10 ) < M2 (U9 ) < M2 (U8 ) < M2 (U7 ) = M2 (U6 ) < M2 (U5 ) < M2 (U4 ) < M2 (U3 ) < M2 (U2 ) < M2 (U1 ). This completes the proof of this result. Acknowledgments The authors are very grateful to the anonymous referees for their valuable comments and corrections, which led to an improvement of the original manuscript. Partially supported by National Natural Science Foundation of China (Nos. 11071088, 11201156), Natural Science Foundation of Jiangsu Province (No. BK20131357), and Project of Graduate Education Innovation of Jiangsu Province (No. CXZZ12–0378), and Program on International Cooperation and Innovation, Department of Education, Guangdong Province (No. 2012gjhz0007). References [1] I. Gutman, B. Ruščić, N. Trinajstić, C.F. Wilcox, Graph theory and molecular orbitals, Part XII. Acyclic polyenes, J. Chem. Phys. 62 (1975) 3399–3405. [2] I. Gutman, N. Trinajstić, Graph theory and molecular orbitals. Total π -electron energy of alternant hydrocarbons, Chem. Phys. Lett. 17 (1972) 535–538. [3] M. Liu, B. Liu, The second Zagreb indices and Wiener polarity indices of trees with given degree sequences, MATCH Commun. Math. Comput. Chem. 67 (2012) 439–450. [4] A.W. Marshall, I. Olkin, Inequalities: Theory of Majorization and its Applications, Academic Press, New York, 1979. [5] S. Nikolić, G. Kovačević, A. Miličević, N. Trinajstić, The Zagreb indices 30 years after, Croat. Chem. Acta 76 (2003) 113–124. [6] S. Nikolić, N. Trinajstić, I.M. Tolić, G. Rücker, C. Rücker, On molecular complexity indices, in: D. Bonchev, D.H. Rouvray (Eds.), Complexity in Chemistry, Francis & Taylor, London, 2003, pp. 29–89. [7] Z. Yan, H. Liu, H. Liu, Sharp bounds for the second Zagreb index of unicyclic graphs, J. Math. Chem. 42 (3) (2006) 565–574.
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Discrete Applied Mathematics journal homepage: www.elsevier.com/locate/dam
The second Zagreb indices of unicyclic graphs with given degree sequences Muhuo Liu a,b,c , Bolian Liu b,∗ a
Institute of Mathematics, School of Mathematical Science, Nanjing Normal University, Nanjing, 210046, China
b
School of Mathematical Science, South China Normal University, Guangzhou, 510631, China
c
Department of Mathematics, South China Agricultural University, Guangzhou, 510642, China
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abstract
info
Let π = (d1 , d2 , . . . , dn ) and π ′ = (d′1 , d′2 , . . . , d′n ) be two different non-increasing degree
Article history: Received 16 October 2012 Received in revised form 21 October 2013 Accepted 25 October 2013 Available online 22 November 2013
j
j
′ ′ sequences. We write π ▹ π ′ , if and only if i=1 di = i=1 di ≤ i=1 di for i=1 di , and all j = 1, 2, . . . , n. Let Γ (π ) be the class of connected graphs with degree sequence π . The second Zagreb index of a graph G is denoted by M2 (G) = uv∈E (G) d(u)d(v). In this paper, we characterize an extremal unicyclic graph that achieves the maximum second Zagreb index in the class of unicyclic graphs with given degree sequence, and we also prove that if π ▹ π ′ , π and π ′ are unicyclic degree sequences and U ∗ and U ∗∗ have the maximum second Zagreb indices in Γ (π ) and Γ (π ′ ), respectively, then M2 (U ∗ ) < M2 (U ∗∗ ). Furthermore, we determine the first to ninth largest second Zagreb indices together with the corresponding extremal unicyclic graphs in the class of unicyclic graphs on n ≥ 17 vertices. © 2013 Elsevier B.V. All rights reserved.
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Keywords: Second Zagreb index Degree sequence Majorization theorem
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1. Introduction Throughout this paper, we are concerned with connected undirected simple graph only. Let Sn , Cn and Pn be the star, cycle and path of order n, respectively. The second Zagreb index of G is denoted by [2] M2 (G) =
(d(u)d(v)) .
(1.1)
uv∈E (G)
For the chemical applications and mathematical properties of the second Zagreb index, we refer the readers to [1,2,5,6] and the references therein. The sequence π = (d1 , d2 , . . . , dn ) is called the degree sequence of G if di = d(vi ) holds for 1 ≤ i ≤ n. Throughout this paper, we use di to denote the ith largest degree of G and we suppose that d(vi ) = di , where 1 ≤ i ≤ n. Let Γ (π ) be the class of connected graphs with degree sequence π . Suppose π = (d1 , d2 , . . . , dn ) and π ′ = (d′1 , d′2 , . . . , d′n ) are two different non-increasing degree sequences, we write
j
j
′ i=1 di for all j = 1, 2, . . . , n. Such an ordering is sometimes called majorization (see [4]). A unicyclic graph is a connected graph with n vertices and n edges. In [3], an extremal tree with the maximum second Zagreb index in the class of trees with given degree sequence was characterized. In this note, we identify an extremal unicyclic graph with the maximum second Zagreb index in the class of unicyclic graphs with given degree sequence, and we also prove
π ▹ π ′ if and only if
∗
n
i=1
di =
n
i=1
d′i , and
i=1
di ≤
Corresponding author. Tel.: +86 20 85212613. E-mail addresses: [email protected] (M. Liu), [email protected], [email protected] (B. Liu).
0166-218X/$ – see front matter © 2013 Elsevier B.V. All rights reserved. http://dx.doi.org/10.1016/j.dam.2013.10.033
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that if π ▹ π ′ , π and π ′ are two different unicyclic degree sequences, U ∗ and U ∗∗ have the maximum second Zagreb indices in Γ (π ) and Γ (π ′ ), respectively, then M2 (U ∗ ) < M2 (U ∗∗ ). Furthermore, we determine the first to ninth largest second Zagreb indices together with the corresponding extremal unicyclic graphs in the class of unicyclic graphs on n ≥ 17 vertices. 2. The main results Let G − uv and G + uv , respectively, denote the new graph obtained from G by deleting and adding the edge uv . Lemma 2.1 ([3]). Let G = (V , E ) be a connected graph with v1 u1 ∈ E , v2 u2 ∈ E , v1 v2 ̸∈ E and u1 u2 ̸∈ E. Let G′ = G − u1 v1 − u2 v2 + v1 v2 + u1 u2 . If d(v1 ) ≥ d(u2 ) and d(v2 ) ≥ d(u1 ), then M2 (G′ ) ≥ M2 (G), where M2 (G′ ) > M2 (G) if and only if both two inequalities are strict. Lemma 2.2 ([3]). Suppose G ∈ Γ (π ), and there exist three vertices u, v, w of a connected graph G such that uv ∈ E (G), uw ̸∈ E (G), d(v) < d(w) ≤ d(u), and d(u) > d(x) for all x ∈ N (w). Then, there exists another connected graph G′ ∈ Γ (π ) such that M2 (G) < M2 (G′ ). Lemma 2.3. For any degree sequence π with n ≥ 3, there exists an extremal graph G ∈ Γ (π ) with the maximum second Zagreb index in Γ (π ) such that {v2 , v3 } ⊆ N (v1 ). Proof. If v1 v2 ̸∈ E (G), then there is some vertex v such that v1 v ∈ E (G) and d(v1 ) ≥ d(v2 ) > d(v) and d(v1 ) > d(x) holds for all x ∈ N (v2 ), which contradicts Lemma 2.2. Thus, v1 v2 ∈ E (G). Now, we assume that v1 v3 ̸∈ E (G). Then, d(v3 ) > d(v) holds for every v ∈ N (v1 )\{v2 }. By Lemma 2.2, we may assume that there exists some vertex u ∈ N (v3 ) such that d(u) = d1 . If u = v2 , the result already holds. If u ̸= v2 , then d(u) = d1 = d2 = d3 . Let P be a shortest path from v1 to v3 . If v2 ̸∈ V (P ), choose x ∈ N (v1 ) ∩ V (P ), since v1 ∈ N (x) \ N (v3 ), there must exist some vertex y ∈ N (v3 ) \ V (P ) such that y ̸∈ N (x). Let G1 = G + v1 v3 + xy − v1 x − v3 y. By Lemma 2.1, G1 ∈ Γ (π ) and M2 (G1 ) ≥ M2 (G), the result holds. If v2 ∈ V (P ), we may suppose that v2 v3 ̸∈ E (G) (otherwise, the result already holds). Choose x ∈ N (v2 ) ∩ V (P ) such that x ̸= v1 . It can be proved similarly with the case v2 ̸∈ V (P ). Lemma 2.4. If π is a unicyclic degree sequence with dn = 1, then there exists an extremal graph G ∈ Γ (π ) with the maximum second Zagreb index in Γ (π ) such that the unique cycle of G is a triangle with V (C3 ) = {v1 , v2 , v3 }. Proof. Suppose Ct is the unique cycle of G. Now, it suffices to prove the following claims. Claim 1. There is an extremal graph G ∈ Γ (π ) such that v1 ∈ V (Ct ) and {v2 , v3 } ⊆ N (v1 ). Assume that Claim 1 does not hold for any extremal graph G ∈ Γ (π ). By Lemma 2.3, we may suppose that G is an extremal graph of Γ (π ) such that {v2 , v3 } ⊆ N (v1 ). Then, v1 ̸∈ Ct . Suppose that P = u · · · v1 · · · xy is the unique path connecting u and y such that v1 is on the path P, where u ∈ Ct and d(y) = 1. Let w ∈ N (u) ∩ V (Ct ). If d(w) ≤ d(x), let G1 = G + ux +w y −w u − xy. By Lemma 2.1, M2 (G1 ) ≥ M2 (G). Note that G1 ∈ Γ (π ), v1 is in the unique cycle of G1 and {v2 , v3 } ⊆ N (v1 ), a contradiction. If d(u) ≤ d(x), let G2 = G + uy + w x − w u − xy. By Lemma 2.1, M2 (G2 ) ≥ M2 (G). Note that G2 ∈ Γ (π ), v1 is in the unique cycle of G2 and {v2 , v3 } ⊆ N (v1 ), a contradiction. Thus, min{d(u), d(w)} > d(x). Now, choose z ∈ (N (x) ∩ V (P )) \ {y}. Similarly, min{d(u), d(w)} > d(z ). Repeating the above process, we will yield that min{d(u), d(w)} > d(v1 ), a contradiction. Thus, Claim 1 holds. Claim 2. There is an extremal graph G ∈ Γ (π ) such that v1 v2 ∈ E (Ct ) and v3 ∈ N (v1 ). Assume that Claim 2 does not hold for any extremal graph G ∈ Γ (π ). By Claim 1, there exists an extremal graph G ∈ Γ (π ) such that v1 ∈ V (Ct ) and {v2 , v3 } ⊆ N (v1 ). Then, v2 ̸∈ V (Ct ). Choose {u, v} ⊆ V (Ct ) \ {v1 } such that uv ∈ E (Ct ). Suppose that P = v1 v2 xy · · · z is the unique path connecting v1 and z such that v2 is on the path P, where d(z ) = 1. Suppose that max{d(u), d(v)} = d(u). If d(u) ≥ d(x), let G1 = G + uv2 + v x − uv − v2 x. By Lemma 2.1, M2 (G1 ) ≥ M2 (G). But Claim 2 holds for G1 , a contradiction. Thus, max{d(u), d(v)} = d(u) < d(x). Similarly, max{d(u), d(v)} < d(y). Repeating the above process, we will yield that max{d(u), d(v)} < d(z ) = 1, a contradiction. Thus, Claim 2 holds. Claim 3. There is an extremal graph G ∈ Γ (π ) such that {v1 v2 , v1 v3 } ⊆ E (Ct ). By Claim 2, there is an extremal graph G ∈ Γ (π ) such that v1 v2 ∈ E (Ct ) and v3 ∈ N (v1 ). If v3 ̸∈ V (Ct ), then v2 v3 ̸∈ E (G). Choose u ∈ (V (Ct ) ∩ N (v2 )) \ {v1 } and v ∈ N (v3 ) \ {v1 }. Let G1 = G + v2 v3 + uv − vv3 − uv2 . By Lemma 2.1, M2 (G1 ) ≥ M2 (G) and G1 ∈ Γ (π ). It is easily checked that Claim 3 holds for G1 . Claim 4. There is an extremal graph G ∈ Γ (π ) such that {v1 , v2 , v3 } = V (Ct ). If d2 = 2, it is easily checked that the result holds. So, we may suppose that d2 ≥ 3. By Claim 3, there exists an extremal graph G ∈ Γ (π ) such that {v1 v2 , v1 v3 } ⊆ E (Ct ). If {v1 , v2 , v3 } ̸= V (Ct ), then v2 v3 ̸∈ E (G). Choose u ∈ N (v2 ) \ V (Ct ), and choose v ∈ (N (v3 ) ∩ V (Ct )) \ {v1 }. Let G1 = G + v2 v3 + uv − vv3 − uv2 . By Lemma 2.1, M2 (G1 ) ≥ M2 (G). Since G1 ∈ Γ (π ) and {v1 , v2 , v3 } = V (Ct ), Claim 4 holds. To give our main result, we have to introduce the following notation. Suppose π = (d1 , d2 , . . . , dn ), where dn = 1. Let UM (π ) be the unique unicyclic graph such that the unique cycle of UM (π) is a triangle with V (C3 ) = {v1 , v2 , v3 }, and the remaining vertices appear in BFS-ordering with respect to C3 starting from v4 that is adjacent to v1 . It means that, UM (π ) can be constructed by the breadth-first-search method as follows: select the vertices {v1 , v2 , v3 } as the root vertices and begin with {v1 , v2 , v3 } of the zeroth layer. Select the vertices v4 , v5 , . . . , vd1 +d2 +d3 −3 as the first layer such that N (v1 ) = {v2 , v3 , v4 , v5 , . . . , vd1 +1 }, N (v2 ) = {v1 , v3 , vd1 +2 , vd1 +3 , . . . , vd1 +d2 −1 },
M. Liu, B. Liu / Discrete Applied Mathematics 167 (2014) 217–221
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Fig. 1. The unicyclic graphs UM (π1 ) and F1 .
N (v3 ) = {v1 , v2 , vd1 +d2 , . . . , vd1 +d2 +d3 −3 }. Then, append d4 − 1 vertices to v4 such that N (v4 ) = {v1 , vd1 +d2 +d3 −2 , . . . , vd1 +d2 +d3 +d4 −4 } · · ·. Informally, for a given unicyclic degree sequence π1 = (5, 4, 3(3) , 2(10) , 1(8) ), UM (π1 ) is the unicyclic graph as shown in Fig. 1. Theorem 2.1. If dn = 1, then UM (π ) achieves the maximum second Zagreb index in the class of unicyclic graphs with given degree sequence π . Proof. By Lemma 2.4, we may suppose that G is an extremal graph in Γ (π ) such that the unique cycle of G is a triangle with V (C3 ) = {v1 , v2 , v3 }. Now, we create an ordering ≺ of V (G) by the breadth-first search as follows: firstly, let v1 ≺ v2 ≺ v3 ; secondly, append all neighbors u4 , . . . , ud1 +1 of N (v1 ) \ {v2 , v3 } to the ordered list, these neighbors are ordered such that u ≺ v whenever d(u) > d(v) (in the remaining case the ordering can be arbitrary); thirdly, append all neighbors ud1 +2 , . . . , ud1 +d2 −1 of N (v2 )\{v1 , v3 } to the ordered list, these neighbors are ordered such that u ≺ v whenever d(u) > d(v) (in the remaining case the ordering can be arbitrary). Then, with the same method we can append the vertices N (v3 )\{v1 , v2 } in the ordered list, and then to the vertices N (x) \ {v1 }, where d(x) = max{d(y): where y ∈ N (v1 ) \ {v2 , v3 }}. Continue recursively with all vertices v1 , v2 , . . ., until all vertices of G are processed. Let us use the notation Hi (G) to denote the set of vertices of G at distance i ≥ 0 from C3 , where V (C3 ) = {v1 , v2 , v3 }. Then, H0 = {v1 , v2 , v3 }. Suppose the last layer of G is p. It suffices to show the following claims. Claim 1. d(u) ≥ d(v) holds for every u ∈ Hi (G) and every y ∈ Hj (G), where j > i ≥ 0. Suppose that there exist vertices u and v with u ∈ Hi (G) and v ∈ Hj (G) such that d(u) < d(v), where i < j. Furthermore, for convenience, we may choose u as the first vertex in the ordering ≺ with such property and let d(v) = max{d(x) : x ∈ Ht , where t ≥ i + 1}. Choose u′ ∈ Hi−1 (G) such that u′ u ∈ E (G). Then, 1 ≤ i < j, since u ̸∈ H0 . If u lies on the unique shortest path connecting v to C3 , suppose that P = v · · · y is a path of G, where d(y) = 1. Choose z ∈ V (P ) such that d(z ) > d(u) and the distance between z and y is as small as possible. Without loss of generality, suppose that z = v . Choose v ′ ∈ Hj+1 ∩ V (P ) such that vv ′ ∈ E (G). By the choice of u and z , d(u′ ) ≥ d(v) > d(u) ≥ d(v ′ ). Let G1 = G + uv ′ + u′ v − u′ u − vv ′ . Then, G1 ∈ Γ (π ). By Lemma 2.1, M2 (G1 ) > M2 (G), which contradicts the choice of G. Thus, u does not lie on the unique shortest path connecting v to C3 . Choose w ∈ N (v) ∩ Hj−1 (G). Then, uw ̸∈ E (G) and u ̸= w . By the choice of u and j − 1 > i − 1, d(u′ ) ≥ d(w). Recall that d(v) > d(u). Let G2 = G + uw + u′ v − uu′ − vw . By Lemma 2.1, M2 (G2 ) ≥ M2 (G) and G2 ∈ Γ (π ). Now, we construct a new ordering ≺′ of V (G2 ) with the similar method as ≺. We suppose that v1 ≺ v2 ≺ v3 ≺ u1 ≺ · · · ≺ ut ≺ u is the first t + 4 elements in the ordering ≺ of V (G). By the choice of u and v , we have v1 ≺ v2 ≺ v3 ≺ u1 ≺ · · · ≺ ut ≺ v is the first t + 4 elements in the ordering ≺′ of V (G2 ). By the choice of u and v , for every x ∈ {v1 , v2 , v3 , u1 , . . . , ut , v}, if x ∈ Ht (G2 ), then d(x) ≥ d(x′ ) holds for every x′ ∈ Hs (G2 ), where t < s. Repeating the above process finitely many times, we can achieve a graph G∗ and an ordering ≺∗ such that Claim 1 holds for G∗ . Thus, we may suppose that Claim 1 holds for G. Claim 2. For 0 ≤ i ≤ p − 1, if {u, v} ⊆ Hi (G) and u ≺ v , then d(x) ≥ d(y) holds for every x ∈ N (u) ∩ Hi+1 (G) and every y ∈ N (v) ∩ Hi+1 (G). If Claim 2 does not hold, we may suppose that u is the first vertex in the ordering ≺ with the property that there exist vertices v ∈ Hi (G) (suppose that u ∈ Hi (G)), x ∈ N (u)∩ Hi+1 (G) and y ∈ N (v)∩ Hi+1 (G) such that u ≺ v , but d(x) < d(y). Furthermore, according to this, we may suppose that x is the first vertex in the ordering ≺ and suppose that d(y) = max{d(z ) : z ∈ Hi+1 (G), where x ≺ z }. Let G1 = G + uy + v x − ux − v y. Then, G1 ∈ Γ (π ). By the choice of u, we have d(u) ≥ d(v). Thus, M2 (G1 ) ≥ M2 (G) by Lemma 2.1. Now, we construct a new ordering ≺′ of V (G1 ) with the similar method as ≺. We suppose that v1 ≺ v2 ≺ v3 ≺ u1 ≺ · · · ≺ ut ≺ x is the first t + 4 elements in the ordering ≺ of V (G). By the choice of x and y, v1 ≺ v2 ≺ v3 ≺ u1 ≺ · · · ≺ ut ≺ y is the first t + 4 elements in the ordering ≺′ of V (G1 ). By the choice of x and y, for every z ∈ {v1 , v2 , v3 , u1 , . . . , ut , y} ∩ Hi+1 (G1 ), if w ∈ Hi+1 (G1 ) and z ≺′ w , then d(z ) ≥ d(w). Repeating the above process finitely many times, we can obtain a graph G∗ and an ordering ≺∗ of V (G∗ ) such that M2 (G∗ ) = M2 (G) and Claim 2 holds for G∗ . Furthermore, we can easily see that Claim 1 also holds for G∗ .
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Fig. 2. The unicyclic graphs U1 , U2 , . . . , U10 .
Thus, we may suppose that Claims 1 and 2 hold for G. By Claims 1 and 2, G = UM (π ). This completes the proof of this result. Remark 2.1. For a given unicyclic graph degree sequence π , it is easy to see that UM (π ) is unique, but it cannot deduce that the UM (π ) is the unique unicyclic graph with the maximum second Zagreb index in Γ (π ). For instance, when π1 = (5, 4, 3(3) , 2(10) , 1(8) ), then M2 (F1 ) = M2 (UM (π1 )), but F1 ̸= UM (π1 ), where F1 and UM (π1 ) are shown in Fig. 1. Lemma 2.5 ([4]). Let π and π ′ be two different non-increasing degree sequences. If π ▹ π ′ , then there exists a series of nonincreasing degree sequences π1 , . . . , πk such that (π =) π0 ▹ π1 ▹ · · · ▹ πk ▹ πk+1 (=π ′ ), and πi and πi+1 differ only in two positions, where the differences are 1 for 0 ≤ i ≤ k. Lemma 2.6 ([3]). Let u, v be two vertices of a connected graph G, and w1 , w2 , . . . , wk (1 ≤ k ≤ d(v)) be some vertices of N (v) \ (N (u) ∪ {u}). Let G′ = G + w1 u + w2 u + · · · + wk u − w1 v − w2 v − · · · − wk v . If d(u) ≥ d(v) and y∈N (u) d(y) ≥ ′ x∈N (v) d(x), then M2 (G ) > M2 (G). Theorem 2.2. Let π and π ′ be two different non-increasing unicyclic degree sequences with π ▹ π ′ . Let U ∗ and U ∗∗ be the unicyclic graphs with maximum second Zagreb indices in Γ (π ) and Γ (π ′ ), respectively. Then, M2 (U ∗ ) < M2 (U ∗∗ ). Proof. Set π = (d1 , d2 , . . . , dn ) and π ′ = (d′1 , d′2 , . . . , d′n ). Since π ▹ π ′ , by Lemma 2.5 we may suppose that π and π ′ differ only in two positions, where the differences are 1. Thus, we may assume that di = d′i for i ̸= p, q, and dp + 1 = d′p , dq − 1 = d′q . If dn = 2, then π = (2, 2, . . . , 2) and π ′ = (3, 2, . . . , 2, 1). It is easily checked that M2 (Cn ) < M2 (UM (π ′ )), and hence the result follows. If dn = 1, by Theorem 2.1 it suffices to show that M2 (UM (π )) < M2 (UM (π ′ )). Since π ▹ π ′ , we have 1 ≤ p < q ≤ n, and hence of V (G)) and d(vp ) ≥ d(vq ). Now, by vp ≺ vq (in the BFS-ordering Claims 1 and 2 in the proof of Theorem 2.1, we have y∈N (vp ) d(y) ≥ x∈N (vq ) d(x). Let P be a shortest path from UM (π)
UM (π)
vp to vq in UM (π ). If 2 ≤ q ≤ 3, then dq ≥ 3 since UM (π ′ ) ∈ Γ (π ′ ). If q ≥ 4, then dq ≥ 2. In both cases, there exists a vertex vk (k > q) such that vk ∈ NUM (π) (vq ) \ NUM (π) (vp ) and vk ̸∈ V (P ). Let G = UM (π ) − vq vk + vp vk . Note that G ∈ Γ (π ′ ) and d(vp ) ≥ d(vq ). By Lemma 2.6, M2 (UM (π )) < M2 (G) ≤ M2 (UM (π ′ )).
Paths Pl1 , . . . , Plk are said to have almost equal lengths if l1 , . . . , lk satisfy |li − lj | ≤ 1 for 1 ≤ i ≤ j ≤ k. Suppose G is a unicyclic graph with π = (d1 , d2 , . . . , dn ) as its degree sequence, where dn−k+1 = · · · = dn = 1 and dn−k ≥ 2. Then, π ▹ π ′ = (k + 2, 2, . . . , 2, 1, . . . , 1), where π ′ contains exactly k elements of 1. By Theorem 2.1, it immediately follows that Corollary 2.1 ([7]). If G is a unicyclic graph with n vertices and k pendant vertices, then M2 (G) ≤ M2 (Fn (k)), where Fn (k) is the unicyclic graph on n vertices obtained by attaching k paths of almost equal lengths to one vertex of C3 . Let U1 , U2 , . . . , U10 be the unicyclic graphs on n vertices as shown in Fig. 2, and let H1 , H2 , . . . , H7 be the unicyclic graphs on n vertices as shown in Fig. 3. Theorem 2.3. Let U be a unicyclic graph on n ≥ 17. If U ̸∈ {U1 , U2 , . . . , U10 }, then M2 (U ) < M2 (U10 ) < M2 (U9 ) < M2 (U8 ) < M2 (U7 ) = M2 (U6 ) < M2 (U5 ) < M2 (U4 ) < M2 (U3 ) < M2 (U2 ) < M2 (U1 ). Proof. It is easily checked that U1 is the unique unicyclic graph with d1 = n − 1, U2 , U3 , U6 are all the unicyclic graphs with d1 = n − 2, and U4 , U5 , U7 , U8 , U9 and H1 , H2 , . . . , H7 are all the unicyclic graphs with d1 = n − 3. Suppose the degree sequence of U is (a) = (d1 , d2 , d3 , . . . , dn ) and (b) = (n − 4, 5, 2, 1, . . . , 1). It is easily checked that U10 is the unique unicyclic graph with (b) as its degree sequence. If d1 ≤ n − 4, since U ̸= U10 , we have (a) ▹ (b). By Theorem 2.2, M2 (U ) < M2 (U10 ).
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Fig. 3. The unicyclic graphs H1 , H2 , . . . , H7 .
By an elementary computation, we have M2 (H7 ) = n2 − 4n + 15,
M2 (H6 ) = n2 − 4n + 18,
M2 (H5 ) = n2 − 4n + 19,
M2 (H4 ) = M2 (H3 ) = n2 − 3n + 10,
M2 (H2 ) = n − 3n + 13,
M2 (H1 ) = n2 − 3n + 14,
M2 (U10 ) = n − 3n + 21,
M2 (U9 ) = n2 − 2n + 5,
2
2
M2 (U8 ) = n2 − 2n + 7, M2 (U5 ) = n − 2n + 12, 2
M2 (U3 ) = n − n + 4, 2
M2 (U7 ) = M2 (U6 ) = n2 − 2n + 8, M2 (U4 ) = n2 − 2n + 13, M2 (U2 ) = n2 − n + 7,
M2 (U1 ) = n2 + 3. Since n ≥ 17, we have M2 (H7 ) < M2 (H6 ) < M2 (H5 ) < M2 (H4 ) = M2 (H3 ) < M2 (H2 ) < M2 (H1 ) < M2 (U10 ) < M2 (U9 ) < M2 (U8 ) < M2 (U7 ) = M2 (U6 ) < M2 (U5 ) < M2 (U4 ) < M2 (U3 ) < M2 (U2 ) < M2 (U1 ). This completes the proof of this result. Acknowledgments The authors are very grateful to the anonymous referees for their valuable comments and corrections, which led to an improvement of the original manuscript. Partially supported by National Natural Science Foundation of China (Nos. 11071088, 11201156), Natural Science Foundation of Jiangsu Province (No. BK20131357), and Project of Graduate Education Innovation of Jiangsu Province (No. CXZZ12–0378), and Program on International Cooperation and Innovation, Department of Education, Guangdong Province (No. 2012gjhz0007). References [1] I. Gutman, B. Ruščić, N. Trinajstić, C.F. Wilcox, Graph theory and molecular orbitals, Part XII. Acyclic polyenes, J. Chem. Phys. 62 (1975) 3399–3405. [2] I. Gutman, N. Trinajstić, Graph theory and molecular orbitals. Total π -electron energy of alternant hydrocarbons, Chem. Phys. Lett. 17 (1972) 535–538. [3] M. Liu, B. Liu, The second Zagreb indices and Wiener polarity indices of trees with given degree sequences, MATCH Commun. Math. Comput. Chem. 67 (2012) 439–450. [4] A.W. Marshall, I. Olkin, Inequalities: Theory of Majorization and its Applications, Academic Press, New York, 1979. [5] S. Nikolić, G. Kovačević, A. Miličević, N. Trinajstić, The Zagreb indices 30 years after, Croat. Chem. Acta 76 (2003) 113–124. [6] S. Nikolić, N. Trinajstić, I.M. Tolić, G. Rücker, C. Rücker, On molecular complexity indices, in: D. Bonchev, D.H. Rouvray (Eds.), Complexity in Chemistry, Francis & Taylor, London, 2003, pp. 29–89. [7] Z. Yan, H. Liu, H. Liu, Sharp bounds for the second Zagreb index of unicyclic graphs, J. Math. Chem. 42 (3) (2006) 565–574.