Abstract For a (molecular) graph, the first Zagreb index M1 is equal to the sum of squaresof the vertex degrees, and the second Zagreb index M2 is equal to the sum of products ofdegrees of pairs of adjacent vertices. In this paper, we study the Zagreb indices of n-vertexconnected graphs with k cut vertices, the upper bound for M1- and M2-values of n-vertexconnected graphs with k cut vertices are determined, respectively. The corresponding ex-tremal graphs are characterized.
Keywords Randic connectivity index · Zagreb index · Cut vertex · Pendent path · Clique
The structure of a chemical compound is usually modeled as a polygonal shape, called themolecular graph of this compound. The biochemical community has been using topologicalindices to correlate a chemical compound’s molecular graph with experimentally gathereddata regarding the compound’s characteristics. Among various indices, the Randic index hasbeen one of the most widely used descriptors in quantitative structure activity relationships.
In 1975, Randic proposed a structural descriptor called branching index [48] that laterbecame well-known Randic connectivity index, which is the most used molecular descriptorin QSPR and QSAR; see [20, 32, 33, 46, 52]. The name connectivity index that replaced theoriginal Randic term branching index has been suggested by Kier as stated by Randic [49].The first paper in which the Randic connectivity index was used in QSAR appeared soon af-ter the original publication, also in [34]. Mathematicians too exhibited considerable interestin the properties of the Randic connectivity index; see [8, 9, 21, 22, 36, 37, 44, 45, 47]. TheRandic connectivity index has also evolved into several variants [20, 46, 49, 50]
The research is supported in part by National Science Foundation of China (Grant No. 10671081).
Q. Zhao · S. Li (�)Faculty of Mathematics and Statistics, Hubei Key Laboratory of Mathematical Sciences, Central ChinaNormal University, Wuhan 430079, Chinae-mail: [email protected]
The Randic connectivity index has been extended as the general Randic connectivity in-dex and general zeroth-order Randic connectivity index, and then the Zagreb indices appearto be the special cases of them [11, 13, 14, 19, 25, 29, 37]. The Zagreb indices have beenintroduced in 1972 in the report of Gutman and Trinajstic on the topological basis of the π -electron energy [23]—two terms appeared in the topological formula for the total π -energyof alternant hydrocarbons, which were in 1975 used by Gutman et al. [24] as branchingindices, denoted by M1 and M2, and later employed as molecular descriptors in QSPR andQSAR; see [4, 5]. The name Zagreb indices instead of the term branching indices was firstused by Balaban et al. [2].
There are three groups of closed related problems which have attracted the attention ofresearchers for a long time:
• How M1(G) (respectively, M2(G)) depends on the structure of G.• Given a set of molecular graph G, find upper and lower bounds for M1(G) and M2(G)
of graphs in G and characterize the graphs in which the maximal (respectively, minimal)M1-, M2-values are attained, respectively.
• A natural issue is to compare the values of Zagreb indices on the same graph;see [1, 26, 42].
In view of these problems, it is not surprising that in the chemical literature there are nu-merous studies of properties of the Zagreb indices of molecular graphs. In fact, investigationof the above problems mainly deal with graphs whose cyclomatic number is at most 2 as thesole objects [15, 18, 30, 35, 40, 47, 51, 55]; Mathematical and computational properties ofZagreb indices have also been considered [17, 27, 43, 56, 57]. Other direction of investiga-tion include studies of relation between M1(G) (respectively, M2(G)) and the correspondinginvariant of elements of the graph G (order, cut edges, pendent vertices, diameter, maximumdegree, girth, perfect matching); see [12, 15, 16, 30, 35, 38, 39, 51, 56, 57].
In addition to the myriad applications of the Zagreb indices in chemistry there are manysituations in communication, facility location, cryptology, etc., that are effectively modeledby a connected graph G satisfying certain restriction. In light of the information available forM1 and M2 of trees, unicyclic graphs, bicyclic graphs, et al., it is natural to consider otherclasses of graphs, and the n-vertex connected graphs with k cut vertices are a reasonablestarting point for such an investigation. The n-vertex connected graphs with k cut verticeshave been considered in mathematical literature [6, 28, 31, 41, 54], whereas to our bestknowledge, the Zagreb indices of n-vertex connected graphs with k cut vertices were, sofar, not considered in the chemical literature. On the other hand, connected graphs with n-vertices and k cut vertices represent important class of molecules; see [3]. Here we considern-vertex connected graphs with k cut vertices.
In this paper, we give some basic properties, especially upper bounds in terms of othergraph invariants, of the first and the second Zagreb indices. We determine the extremalconnected graph with fixed numbers of vertices and cut vertices having the maximumvalues of M1,M2, respectively. From our results, the maximum M1- (respectively, M2-)value is correlated with the order and the number cut vertices of the corresponding graph.In our exposition we will use the terminology and apparatus of (chemical) graph theory(see [7, 10, 53]).
2 Definitions and Notations
Let G = (V (G),E(G)) be a simple connected graph with the vertex set V (G) and the edgeset E(G). For a vertex x of the graph G, we denote the neighborhood and the degree of x by
On the Maximum Zagreb Indices of Graphs with k Cut Vertices 95
NG(x) and dG(x), respectively. The Randic connectivity index [48] R = R(G) (resp. zerothorder Randic connectivity index [37] R′ = R′(G)) of G is defined as
R = R(G) =∑
uv∈E(G)
(dG(u)dG(v))−1/2, R′ = R′(G) =∑
u∈V (G)
(dG(u))−1/2.
The first Zagreb index M1 = M1(G) and the second Zagreb index M2 = M2(G) [17, 23,24, 27, 43] of the graph G are given by
M1 = M1(G) =∑
u∈V (G)
(dG(u))2, M2 = M2(G) =∑
uv∈E(G)
dG(u)dG(v).
Given a graph G, if W ⊆ V (G), we denote by G − W the subgraph of G obtained bydeleting the vertices of W and the edges incident with them; if E′ ⊆ E(G), we denote byG − E′ the subgraph of G obtained by deleting the edges of E′. Specially, when W = {v}and E′ = {xy}, we write G − v and G − xy instead of G − {v} and G − {xy}, respectively.Similarly, G + xy is a graph that arises from G by adding an edge xy �∈ E(G), wherex, y ∈ V (G).
A cut vertex in a connected graph G is a vertex whose deletion breaks the graph intotwo (or more) connected components. A connected graph that has no cut vertices is called ablock. We also call a block an endblock of G if it has at most one cutvertex in the graph as awhole. A clique of a simple graph G is a subset S of V such that G[S] is complete.
Let P = v0v1 · · ·vs (s ≥ 1) be a path of G with d(v1) = · · · = d(vs−1) = 2 (unless s = 1).If d(v0), d(vs) ≥ 3, then we call P an internal path of G; if d(v0) ≥ 3 and d(vs) = 1, thenwe call P a pendent path of G. Let PG(u, v) be the shortest path in G starting from u tov. The distance between u and v in G, denoted by dG(u, v), is the length of PG(u, v), i.e.,dG(u, v) = |E(PG(u, v))|. A graph T is called a tree, if it is connected and contains no cycle.A connected graph G is called a unicyclic graph if it contains a unique cycle.
Let Gn,k := {G : G is a connected graphs on n vertices and k cut vertices, 0 < k ≤ n−3},Hn,k := {G1
n,k,G2n,k}
⋃H ′
n,k , where
• G1n,k is obtained from Kn−k by attaching at most one pendent edge to each vertex of Kn−k,
0 < k ≤ n2 .
• G2n,k is obtained from Kn−k by attaching exactly one pendent path, say Pi , to each vertex
of Kn−k , where |E(Pi)| = 1 or 2 and n2 < k ≤ 2n
3 .• H ′
n,k = {G3n,k : G3
n,k is obtained from Kn−k by attaching exactly one pendent path, say Pi ,to each vertex of Kn−k , where |E(Pi)| ≥ 2 and 3n
2 < k ≤ n − 3}.
The graphs G1n,k,G
2n,k and G3
n,k ∈ H ′n,k are depicted in Fig. 1.
Fig. 1 Graphs G1n,k
,G2n,k
and G3n,k
96 Q. Zhao, S. Li
3 The Graphs in Gn,k with the Maximum Zagreb Indices
In this section, we are to determine the sharp upper bounds, respectively, for M1,M2 ofconnected n-vertex graphs, each of which contains exactly k cut vertices.
By the definition of the first and the second Zagreb indices, we have the following lemma.
Lemma 3.1 Let G be a simple connected graph with x, y ∈ V (G). If xy �∈ E(G), then
Mi(G + xy) > Mi(G), i = 1,2.
In order to determine the upper bounds for Zagreb indices, we choose connected graphG ∈ Gn,k such that its M1- (resp. M2-) value is as large as possible.
Lemma 3.2 Choose G ∈ Gn,k such that its M1-, M2-values are as large as possible. ThenEach cut vertex of G connects exact two blocks and each of the blocks contained in G is aclique.
Proof By contradiction. Let v be a cut vertex of G, if v connects at least three con-nected components, say G1,G2,G3, . . . ,Gs , of G, s ≥ 3, then let G′ = G + uw, whereu ∈ V (G2) \ {v} and w ∈ V (G3) \ {v}. It is straightforward to check that G′ ∈ Gn,k and inview of Lemma 3.1, we have Mi(G
′) > Mi(G), i = 1,2, which contradict the assumptionof G. Hence, we obtain that each cut vertex connects exact two components of G. Further-more, if one of the components, say G1, is not a clique, then we may add an edge in G1
to connect two non-adjacent vertices and denote the resultant graph by G′′, then in view ofLemma 3.1, we get Mi(G
′′) > Mi(G), i = 1,2, a contradiction. This completes the proofof Lemma 3.2. �
By Lemma 3.2, in what follows, we might assume that Ka1 , . . . ,Kas are all of the cliquescontained in G. We call two cliques Ka1 ,Ka2 with a1, a2 ≥ 3 in G are adjacent, if Ka1
connects Ka2 by a path P such that P does not intersect some other clique of order atleast 3. By Lemma 3.2, we have the following claim.
Claim 1 Let G ∈ Gn,k such that its M1- (resp. M2-) value is as large as possible. If twocliques Ka1 ,Ka2 in G are adjacent, then the path connecting Ka1 and Ka2 is either of length0 or an internal path.
Lemma 3.3 Choose G ∈ Gn,k such that M1(G) and M2(G) are as large as possible. If Ka2
is an endblock of G, then a2 = 2.
Proof By contradiction. Assume that a2 ≥ 3. Let Ka1 ,Ka2 be two cliques such that V (Ka1),V (Ka2) have one cut vertex, say v, in common, and by Lemma 3.2, v is not a cutvertex of some other clique, where a1 ≥ 2. Let V (Ka1) = {v1, . . . , va1−1, v}, V (Ka2) ={u1, . . . , ua2−1, v}. Set
a contradiction to the choice of G.Let G0 = G − {v1, . . . , va1−1, u1, . . . , ua2−1} and denote Ai = ∑
x∈NG(vi )\V (Ka1 ) dG(x),where i = 1, . . . , a1 − 1. Then
M2(G∗) − M2(G)
= 1
2
[(a1 − 2)(a2 − 2)
a1−1∑
i=1
dG(vi) + (a1 − 1)(a1 − 2)(a2 − 2)2
]
+ 1
2
[(a1 − 2)(a2 − 3)
a2−1∑
j=2
dG(uj ) + (a1 − 2)2(a2 − 2)(a2 − 3)
]
+ (a2 − 2)
a1−1∑
i=1
Ai + 2(a1 + a2 − 2)(a1 − 2)(a2 − 2) − (a2 − 1)
a2−1∑
j=2
dG(uj )
+[
a1−1∑
i=1
dG(vi) + (a1 − 1)(a2 − 2)
][a2−1∑
j=2
dG(uj ) + (a1 − 2)(a2 − 2)
]
>
a1−1∑
i=1
a2−1∑
j=2
dG(vi)dG(uj ) + (a1 − 1)(a2 − 2)
a2−1∑
j=2
dG(uj ) − (a2 − 1)
a2−1∑
j=2
dG(uj )
=(
a1−1∑
i=1
dG(vi) − 1
)a2−1∑
j=2
dG(uj ) + (a1 − 2)(a2 − 2)
a2−1∑
j=2
dG(uj ) > 0,
a contradiction to the choice of G.Hence, the proof of Lemma 3.3 is completed. �
98 Q. Zhao, S. Li
Denote G∗n,k = {G : G(∈ Gn,k) is obtained by attaching at most one pendent path to each
vertex of Kn−k}. Then it is easy to check that Hn,k ⊆ G∗n,k .
Lemma 3.4 Of all the connected graphs in G∗n,k , the maximum Mi -value, i = 1,2, is ob-
tained at the graph(s) in Hn,k .
Proof Choose G ∈ G∗n,k such that Mi(G) is as large as possible, i = 1,2. If G ∈ Hn,k , the
theorem holds immediately. Otherwise, G ∈ G∗n,k \ Hn,k . Without loss of generality, we as-
sume that P1, which is attached to u1, is the shortest path of all the pendent paths in G andP2, which is attached to u2, is the longest one. Since G �∈ Hn,k , we consider the followingthree possible cases.
Case 1. |E(P1)| = 0 (i.e., G has no pendent path attached to u1) and |E(P2)| ≥ 3.Let P2 = u2 · · ·w3w2w1. Set G∗ = G − w1w2 + w1u1. Then G∗ ∈ G∗
On the Maximum Zagreb Indices of Graphs with k Cut Vertices 99
Case 3. |E(P1)| = 1 and |E(P2)| ≥ 3.Let P1 = u1w0, P2 = u2 · · ·w3w2w1. Set G∗ = G − w1w2 + w1w0. Then G∗ ∈ G∗
n,k , andby direct computing we have
M1(G∗) − M1(G) = 0, M2(G
∗) − M2(G) > 0,
a contradiction to the choice of G in this case.By Cases 1, 2 and 3, the proof of Lemma 3.4 is completed. �
Theorem 3.5 Of all the n-vertex connected graphs with k cut vertices, the maximum Mi -value, i = 1,2, is obtained at the graph(s) in Hn,k .
Proof Choose G ∈ Gn,k such that M1(G) (resp. M2(G)) is as large as possible. ByLemma 3.2, we assume that Ka1 , . . . ,Kas are the cliques of G. In the following, we willshow two facts.
Fact 1 There is only one clique Kaisuch that ai ≥ 3.
Proof of Fact 1 By contradiction. Assume that there exists two cliques Ka1 ,Ka2 such thatKa1 is adjacent to Ka2 , a1, a2 ≥ 3. By Lemma 3.3, we have that Ka1 ,Ka2 are not endblocks.Then, in view of Lemma 3.2, we choose two such blocks such that at least one of themhave a pendent path attached to one of its vertices. Without loss of generality, we assumethat Ka2 is one of such cliques. Without loss of generality, we assume that u1 is attached byone pendent path, say P1 = u1 · · ·v. By Claim 1, we might assume that Ka1 and Ka2 haveexactly one vertex in common or, Ka1 connects Ka2 by an internal path P of length l − 1 .We proceed by distinguishing the following possible cases to prove Fact 1.
Case 1. Ka1 and Ka2 have exactly one vertex in common (see Fig. 3).Then we assume that V (Ka1) = {v1, . . . , va1−1, t1}, V (Ka2) = {u1, . . . , ua2−1, t1}. Denote
Ai = ∑x∈NG(vi )\V (Ka1 ) dG(x), where i = 1, . . . , a1 − 1, and Bj = ∑
Since dG(vi) ≥ a1 − 1, dG(u1) = a2 and dG(uj ) ≥ a2 − 1, where i = 1, . . . , a1 − 1, andj = 2, . . . , a2 − 1, then
∑a1−1i=1 dG(vi) ≥ (a1 − 1)2 ≥ 4,
∑a2−1j=1 dG(uj ) > (a2 − 1)2 ≥ 4.
When |E(P1)| = 1, we have
M2(G∗) − M2(G)
= (a2 − 2)
a1−1∑
i=1
Ai + 1
2
[(a1 − 2)(a2 − 2)
a1−1∑
i=1
dG(vi) + (a1 − 1)(a1 − 2)(a2 − 2)2
]
+ (a1 − 2)
a2−1∑
j=2
Bj + 1
2
[(a1 − 2)(a2 − 2)
a2−1∑
j=1
dG(uj ) + (a1 − 2)2(a2 − 1)(a2 − 2)
]
+ 2(a1 + a2 − 2) + 2 +[
a1−1∑
i=1
dG(vi) + (a1 − 1)(a2 − 2)
]
×[
a2−1∑
j=1
dG(uj ) + (a1 − 2)(a2 − 1)
]
− a2 − (a1 + a2 − 2)
(a1−1∑
i=1
dG(vi) +a2−1∑
j=1
dG(uj )
)
>
(a1−1∑
i=1
dG(vi) − 2
)(a2−1∑
j=1
dG(uj ) − 2
)+ (a1 − 3)(a2 − 2)
a1−1∑
i=1
dG(vi)
On the Maximum Zagreb Indices of Graphs with k Cut Vertices 101
Fig. 4 G ⇒ G∗
+ (a1 − 2)(a2 − 3)
a2−1∑
j=1
dG(uj ) + (a1 − 1)(a1 − 2)(a2 − 1)(a2 − 2)
+ (2a1 + a2 − 6) > 0.
Similarly, when |E(P1)| ≥ 2, we also obtain M2(G∗) − M2(G) > 0, a contradiction to the
choice of G.
Case 2. The internal path P is of length 1 (see Fig. 4).Then we assume that V (Ka1) = {v1, . . . , va1−1, t1}, V (Ka2) = {u1, . . . , ua2−1, t2}. Simi-
Similarly, when |E(P1)| ≥ 2, we also have M2(G∗) − M2(G) > 0, a contradiction to the
choice of G.
Case 3. The internal path P is of length at least 2; see Fig. 5.In this case, let P = t1t2 · · · tl−1tl (l ≥ 3). Then we assume that V (Ka1) = {v1, . . . ,
va1−1, t1}, V (Ka2) = {u1, . . . , ua2−1, tl}. Set
Similarly, when |E(P1)| ≥ 2, we also have M2(G∗) − M2(G) > 0, a contradiction to the
choice of G. �
Fact 2 Let Ka be the only clique such that a ≥ 3, then a = n − k.
Proof By Lemma 3.2 and Fact 1, we have that there are k + 1 cliques and k of them areisomorphic to K2. Since there are k cut vertices, and each one belongs to two cliques, thenwe have 2k + a − k = n. Hence, a = n − k, as desired. �
By Facts 1 and 2, we have G ∈ G∗n,k . By Lemma 3.4, we have Mi(G) ≤ Mi(G
j
n,k), i =1,2, j ∈ {1,2,3}, with the equalities holding if and only if G ∼= G
j
n,k .This completes the proof of Theorem 3.5. �
4 Conclusion and Remarks
In view of Theorem 3.5, we can obtain the sharp up bounds on the first and the secondZagreb indices of n-vertex connected graph with k cut vertices, respectively. In fact, by anelementary calculation, we get
M1(G) ={
(n − 2k)(n − k − 1)2 + k(n − k)2 + k, if G ∼= G1n,k;
(n − k)3 + 7k − 3n, if G ∼= G2n,k or G ∈ H ′
n,k,
and
M2(G)
=⎧⎪⎨
⎪⎩
(k2)(n − k)2 + (n−2k
2)(n − k − 1)2 + k(n − k)[(n − 2k)(n − k − 1) + 1], if G ∼= G1
n,k;
(n−k2
)(n − k)2 + k(n − k) + 2(2k − n), if G ∼= G2
n,k;
(n−k2
)(n − k)2 + 2(n − k)2 + 2(5k − 3n), if G ∈ H ′
n,k.
Therefore, by Theorem 3.5, we obtain the following results.
Theorem 4.1 Let G ∈ Gn,k .
(i) If 0 < k ≤ n2 , then M1(G) ≤ (n − 2k)(n − k − 1)2 + k(n − k)2 + k, the equality holds if
and only if G ∼= G1n,k .
(ii) If n2 < k ≤ n − 3, then M1(G) ≤ (n − k)3 + 7k − 3n, the equality holds if and only if
G ∼= G2n,k or G ∈ H ′
n,k .
Theorem 4.2 Let G ∈ Gn,k .
(i) If 0 < k ≤ n2 , then M2(G) ≤ (
k
2
)(n − k)2 + (
n−2k
2
)(n − k − 1)2 + k(n − k)[(n − 2k)(n −
k − 1) + 1], the equality holds if and only if G ∼= G1n,k .
On the Maximum Zagreb Indices of Graphs with k Cut Vertices 105
In this paper, we determine the sharp upper bounds, respectively, for the first and the sec-ond Zagreb index of n-vertex connected graphs with k cut-vertices. It is natural to considerthe following research problem.
Problem 4.3 How can we determine the lower bound for the first and the second Zagrebindices of n-vertex connected graphs with k cut-vertices? What is the characterization of thecorresponding extremal graphs?
Acknowledgements The authors would like to express their sincere gratitude to the referees for a verycareful reading of the paper and for all their insightful comments and valuable suggestions, which make anumber of improvements on this paper.
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