Reduced second Zagreb index has been defined recently. In this paper we characterized extremal bicyclic graphs with pendent vertices with respect to this novel index

Ediz, Süleyman

Le Matematiche (Dipartimento di Matematica e Informatica, Università degli Studi di Catania)

LE MATEMATICHEVol. LXXI (2016) – Fasc. I, pp. 135–147doi: 10.4418/2016.71.1.11REDUCED SECOND ZAGREB INDEX OF BICYCLIC GRAPHSWITH PENDENT VERTICESSU¨LEYMAN EDIZReduced second Zagreb index has been defined recently. In this pa-per we characterized extremal bicyclic graphs with pendent vertices withrespect to this novel index.1. IntroductionLet G be a simple connected graph with n vertices and m edges. dv is the numberof edges incident to the vertex v. A vertex of degree one is said to be a pendentvertex. Unicyclic graphs are connected graphs with n vertices and n edges.Bicyclic graphs are connected graphs with n vertices and n+1 edges.We write ∆and δ for the largest and the smallest of all degrees of vertices of G, respectively.The first Zagreb and the second Zagreb index of the graph G are defined as:M1 = M1(G) = ∑v∈V (G)d2v (1)andM2 = M2(G) = ∑uv∈E(G)du dv (2)Entrato in redazione: 31 marzo 2015AMS 2010 Subject Classification: 05C07, 05C38.Keywords: Degree based topological index, Reduced second Zagreb index, Bicyclic graph.136 SU¨LEYMAN EDIZrespectively. In formula (2), uv denotes an edge connecting the vertices u and v.In 1972, the quantities M1 and M2 were found to occur within certain approx-imate expressions for the total pi-electron energy [19]. The first Zagreb indexsatisfies the identityM1(G) = ∑uv∈E(G)(du +dv) (3)where the notation is same as in equation(2) [20]. In view of the extensiveresearch on the two Zagreb indices, in particular, their difference M2−M1 , thedifference between the equations (2) and (3 ), has been never examined. In [17],Furtula et al. examined M2−M1 and proposed a new degree based topologicalindex, named it ‘Reduced second Zagreb index’ and characterized the maximumtrees with respect to reduced second Zagreb index. Reduced second Zagrebindex is defined [17] as follows;RM2 = RM2(G) = ∑uv∈E(G)(du−1)(dv−1) = M2(G)−M1(G)+m (4)where m denotes the number of edges. Zagreb indices of bicyclic graphs areinvestigated in [2–4, 9]. For other topological indices of bicyclic graphs see in[1, 5–8, 10–16]. RM2 index of unicyclic graphs were investigated in [18]. Inthis paper we investigate maximum and minimum bicyclic graphs with respectto RM2 index.2. Minimum and maximum RM2 index of bicyclic graphsLet DC denote all bicyclic graphs with n vertex , n + 1 edges and k pendentvertices (here, DC stands for double cycle). The arrangement of cycles of DChas at most three possible cases.Case 1: DCa,b (k1,k2, . . . ,ka,s2,s3, . . . ,sb) is the set of G ∈ DC in which thecycles Ca and Cb have only one common vertex. Here,k1,k2, . . . ,ka,s2,s3, . . . ,sbdenote the number of pendent vertices of correspondingv1,v2, . . . ,va,v2′,v3′, . . . ,vb′vertices. See Figure 1.Case 2: DCla,b (k1,k2, . . . ,ka,r1,r2, . . . ,rl,s1,s2, . . . ,sb) is the set of G ∈ DCin which the cycles Ca and Cb have no common vertex for l ≥ 0. Herek1,k2, . . . ,ka,r1,r2, . . . ,rl,s1,s2, . . . ,sbREDUCED SECOND ZAGREB INDEX OF BICYCLIC GRAPHS 137Figure 1: The first class of bicyclic graphs with k pendent vertices:DCa,b (k1,k2, . . . ,ka,s2,s3, . . . ,sb)denote the number of pendent vertices of correspondingv1,v2, . . . ,va,n1,n2, . . . ,nl,u1,u2, . . . ,ubvertices. See Figure 2.Figure 2: The second class of bicyclic graphs with k pendent vertices:DCla,b (k1,k2, . . . ,ka,r1,r2, . . . ,rl,s1,s2, . . . ,sb)Case 3: DCla+b (k1,k2, . . . ,ka−l,r1,r2, . . . ,rl,s2, . . . ,sb−l−1) is the set of G ∈DC in which the cycles Ca and Cb have a common path of length l +1 for l ≥ 0.Herek1,k2, . . . ,ka−l,r1,r2, . . . ,rl,s2, . . . ,sb−l−1denote the number of pendent vertices of correspondingv1,v2, . . . ,va−l,n1,n2, . . . ,nl,v2′,v3′, . . . ,v′b−l−1vertices. See Figure 3.With direct calculations, we get the following propositions.138 SU¨LEYMAN EDIZFigure 3: The third class of bicyclic graphs with k pendent vertices:DCla+b (k1,k2, . . . ,ka,r1,r2, . . . ,rl,s2, . . . ,sb−l−1)Proposition 2.1. Let DCa,b (k1,k2, . . . ,ka,s2,s3, . . . ,sb) be the set of G ∈ DC.ThenRM2(G) = (k1 +3)(k2 + ka + s2 + sb +4)+(k2 +1)(k3 +1)+ . . .+(ka−1 +1)(ka +1)+(s2 +1)(s3 +1)+ · · ·+(sb−1 +1)(sb +1) .Proposition 2.2. Let DCla,b (k1,k2, . . . ,ka,r1,r2, . . . ,rl,s1,s2, . . . ,sb) be the set ofG ∈ DC then(a) RM2(G) = (ka + k2 +2)+(k1 +2)(s1 +2)+(k2 +1)(k3 +1)+ . . .+(ka−1 +1)(ka +1)+(s2 +1)(s3 +1)+ · · ·+(sb−1 +1)(sb +1)for l = 0.(b) RM2 (G) = (k1 +2)(k2 + ka + r1 +3)+(k2 +1)(k3 +1)+ . . .+(ka−1 +1)(ka +1)+(r1 +1)(r2 +1)+ · · ·+(rl−1 +1)(rl +1)+(s1 +2)(s2 + sb + rl +3)+(s2 +1)(s3 +1)+ · · ·+(sb−1 +1)(sb +1)for l ≥ 1.Proposition 2.3. Let DCla+b (k1,k2, . . . ,ka−l,r1,r2, . . . ,rl,s2, . . . ,sb−l−1) be theREDUCED SECOND ZAGREB INDEX OF BICYCLIC GRAPHS 139set of G ∈ DC then(a) RM2 (G) = (k1 +2)(k2 + s2 + r1 +3)+(ka−l +2)(ka−l−1 + sb−l−1 + rl +3)+(k2 +1)(k3 +1)+ · · ·+(ka−l−2 +1)(ka−l−1 +1)+(s2 +1)(s3 +1)+ · · ·+(sb−l−2 +1)(sb−l−1 +1)+(r1 +1)(r2 +1)+ . . . .+(rl−1 +1)(rl +1)for 1≤ l ≤ n−4.(b) RM2(G) = (k1 +2)(k2 + s2 +2)+(k2 +1)(k3 +1)+ . . .+(ka−2 +1)(ka−1 +1)+(ka +2)(ka−1 + sb−1 +2)+(s2 +1)(s3 +1)+ · · ·+(sb−2 +1)(sb−1 +1)+(k1 +2)(ka +2)for l = 0.Lemma 2.4. Let G1 ∈ DCa,b with no pendent vertex and n vertices. Let G2 ∈DCa,b with k ≥ 1 pendent vertices and n vertices. Then RM2(G1)< RM2(G2).Proof. Let k = 1. Let uvl be a path of G1 where all degrees are 2. Then, weobtain G2 from G1 by taking u attached to v as a pendent vertex. In this caseRM2(G2)−RM2(G1) = 1 > 0. On the other hand, let uvl be a path of G1 andv = v1 so that dv = 4. Then, we obtain G2 from G1 by taking u attached to v as apendent vertex. In this case RM2(G2)−RM2(G1) = 6. The other cases for k≥ 2are similar.Now, we give the following lemmas whose proofs are similar to that ofLemma 2.4.Lemma 2.5. Let G1 ∈ DCla,b with no pendent vertex and n vertices. Let G2 ∈DCla,b with k ≥ 1 pendent vertices and n vertices. Then RM2(G1)< RM2(G2).Lemma 2.6. Let G1 ∈ DCla+b with no pendent vertex and n vertices. Let G2 ∈DCla+b with k ≥ 1 pendent vertices and n vertices. Then RM2(G1)< RM2(G2).Corollary 2.7. (a) Let G ∈ DCa,b (k1,k2, . . . ,ka,s2,s3, . . . ,sb). Then the mini-mum RM2 index of G is DCa,b (0,0, . . . ,0).(b) Let G ∈ DCla,b (k1,k2, . . . ,ka,r1,r2, . . . ,rl,s1,s2, . . . ,sb). Then the minimumRM2 index of G is DCla,b (0,0, . . . ,0).(c) Let G∈DCla+b (k1,k2, . . . ,ka−l,r1,r2, . . . ,rl,s1,s2, . . . ,sb−l−1). Then the min-imum RM2 index of G is DCla+b (0,0, . . . ,0).Notice that in all three cases G has no pendent vertices.140 SU¨LEYMAN EDIZProposition 2.8. Let G ∈ DCa,b (0,0, . . . ,0). Then the minimum RM2 index isRM2(G) = n+9.Proof. From Proposition 2.1 and Corollary 2.7, RM2(G) = a + b + 8. Sincea+b = n+1, the desired result is acquired.Proposition 2.9. Let G ∈ DCla,b (0,0, . . . ,0). Then the minimum RM2 index isRM2(DCla,b) = n+7 for l ≥ 1.Proof. From Proposition 2.2b and Corollary 2.7, RM2(G) = a+b+ l+7. Sincen = a+b+ l, the desired result is acquired.Proposition 2.10. Let G ∈DCla+b (0,0, . . . ,0). Then the minimum RM2 index isRM2(DCla+b) = n+7 for l ≥ 1.Proof. From Proposition 2.3a and Corollary 2.7, RM2(G) = a+b− l+5. Sincea+b− l = n+2, the desired result is acquired.Definition 2.11. Let Ξ be a family of the set DC3,3 (k1,k2,k3,s2,s3) such thats2 = s3 = 0 and ki− k j = 0 or ki− k j = 1 for 1 ≤ i ≤ j ≤ 3. Or by symmetry,k2 = k3 = 0 and si− s j = 0 or si− s j = 1 for 1≤ i≤ j ≤ 3. See Figure 4 .Figure 4: G = DC3,3 (r−1,r−1,r−2,0,0) ∈ Ξ for n = 3r+1, r ≥ 2.Proposition 2.12. Let G ∈ Ξ with n vertices and n−5 pendent vertices. Then(a) RM2 (G) = 3r2 +2r+2 f or n = 3r,r ≥ 2.(b) RM2 (G) = 3r2 +4r+3 f or n = 3r+1,r ≥ 2.(c) RM2 (G) = 3r2 +6r+1 f or n = 3r+2,r ≥ 2.Proof. We only prove the case (b), the other cases are similar. From Proposition2.1 and Figure 4, we can directly writeRM2 (G) = (r+2)(r+ r−1+2)+ r (r−1)+1= (r+2)(2r+1)+ r2− r+1 = 3r2 +4r+3REDUCED SECOND ZAGREB INDEX OF BICYCLIC GRAPHS 141Lemma 2.13. Let a,b,c,d,e be non-negative integers and a+ b+ c+ d = e.Then ab+ cd takes its maximum value when a =⌈ e2⌉, b =⌊ e2⌋and c = d = 0.Or by symmetry c =⌈ e2⌉, d =⌊ e2⌋and a = b = 0.Lemma 2.14. Let a,b,c,d,e, f be non-negative integers and a+b+c+d+e =f . Then ab+ ac+ de takes its maximum value when e =⌈f2⌉, d =⌊f2⌋anda = b = c = 0. Or by symmetry a =⌈f2⌉, b+ c =⌊f2⌋and d = e = 0.Proof. Let b + c = x. Then ab + ac + ed = a(b + c) + ed = ax + ed. FromLemma 2.13, the desired is result acquired.Proposition 2.15. Let G ∈ DCa,b (k1,k2, . . . ,ka,s2,s3, . . . ,sb) with a,b ≥ 4 andk2 + ka + s2 + sb =Ω. Then, the maximum RM2 index of G isDCa,b (k1,k2, . . . ,ka,s2,s3, . . . ,sb) such that k3 = k4 = · · ·= ka−1 = 0 , s3 = s4 =· · ·= sb−1 = 0 and k1 =Ω or |k1−Ω|= 1.Proof. By Proposition 2.1,RM2(G) = (k1 +3)(k2 + ka + s2 + sb +4)+(k2 +1)(k3 +1)+ . . .+(ka−1 +1)(ka +1)+(s2 +1)(s3 +1)+ · · ·+(sb−1 +1)(sb +1)= k1k2 + k1ka + k1s2 + k1sb +4k1 +3k2 +3ka +3s2 +3sb +12+ k2k3 + k2 + k3 +1+ · · ·+ ka−1ka + ka−1 + ka +1+ s2s3 + s2 + s3 +1+ . . .+ sb−1sb + sb−1 + sb +1= k1k2 + k1ka + k1s2 + k1sb + k2k3 + k3k4 + · · ·+ ka−2ka−1 + ka−1ka+ s2s3 + s3s4 + · · ·+ sb−2sb−1 + sb−1sb +4k1 +4k2 +2k3 + · · ·+2ka−1+4ka +4s2 +2s3 + · · ·+2sa−1 +4sb +a+b+8 == k1k2 + k1ka + k1s2 + k1sb + k2k3 + k3k4 + · · ·+ ka−2ka−1 + ka−1ka+ s2s3 + s3s4 + · · ·+ sb−2sb−1 + sb−1sb+2(k1 + k2 + · · ·+ ka + s2 + s3 + · · ·+ sb)+2(k1 + k2 + ka + s2 + sb)+a+b+8Since k1 + k2 + · · ·+ ka + s2 + s3 + · · ·+ sb = n−a−b+1, then142 SU¨LEYMAN EDIZRM2 (G) = k1k2 + k1ka + k1s2 + k1sb + k2k3 + k3k4 + · · ·+ ka−2ka−1 + ka−1ka+ s2s3 + s3s4 + · · ·+ sb−2sb−1 + sb−1sb +2n−2a−2b+2+2(k1 + k2 + ka + s2 + sb)+a+b+8= k1 (k2 + ka + s2 + sb)+ k2k3 + k3k4 + · · ·+ ka−2ka−1 + ka−1ka+ s2s3 + s3s4 + · · ·+ sb−2sb−1 + sb−1sb+2(k1 + k2 + ka + s2 + sb)+2n−a−b+10= k1Ω+ k2k3 + k3k4 + · · ·+ ka−2ka−1 + ka−1ka+ s2s3 + s3s4 + · · ·+ sb−2sb−1 + sb−1sb +2(k1 +Ω)+2n−a−b+10.Clearly from the last equality by using Lemma 2.13, RM2 takes its maximumvalue when k3 = k4 = · · · = ka−1 = 0 , s3 = s4 = · · · = sb−1 = 0 and k1 = Ω or|k1−Ω|= 1.Theorem 2.16. Let G ∈ Ξ with n vertices and n− 5 pendent vertices. Then Ghas maximum RM2 value among the all graphs belong to DCa,b with n vertices.Proof. We only consider n = 3r + 1 for r ≥ 2. The other cases are similar.Firstly, we show that G has maximum RM2 value among all the graphs belong-ing to DC3,3 (k1,k2,k3,s2,s3). From the definition of RM2 index,RM2 (G) = (k1 +3)(k2 + k3 + s2 + s3 +4)+(k2 +1)(k3 +1)+(s2 +1)(s3 +1)= (k1 +3)(k2 + k3 + s2 + s3 +4)+(k2 + k3 + s2 + s3)+ k2k3 + s2s3 +2Since k2 + k3 + s2 + s3 = n− k1−5 = 3r− k1−4 thenRM2 (G) = (k1 +3)(3r− k1)+3r− k1−4+ k2k3 + s2s3 +2.By Lemma 2.13, k2k3 + s2s3 takes its maximum value when k2 = k3 or k2 =k3 +1 and s2 = s3 = 0. Or by symmetry s2 = s3 or s2 = s3 +1 and k2 = k3 = 0.We only consider the first part of the Lemma 2.13. The second part can behandled similarly. ThenRM2 (G) = f (k1,k2,k3) = (k1 +3)(3r− k1)+3r− k1−4+ k2k3 +2.Since k1 + k2 + k3 = 3r−4, theng(k1,k2,k3) = 3r−4− k1− k2− k3 = 0 (5)can be written. By using the Lagrange multipliers method, we obtain; 3r−2k1−4 = k2 and 3r− 2k1− 4 = k3. Thus, k2 + k3 = 6r− 4k1− 8. From Equation 5,REDUCED SECOND ZAGREB INDEX OF BICYCLIC GRAPHS 143k2 + k3 = 3r− k1−4. And from these last two equalities k1 = r−1. By Defini-tion 2.11, k2 = r−1 and k3 = r−2.Secondly, we show that G has maximum RM2 value among all the graphs be-longing to DCa,b (k1, . . . ,ka,s2, . . . ,sb) for a+ b ≥ 7 with n vertices. There aretwo cases in this situation.Case 1: Let a = 4 and b = 3. See Figure 5. From the definition of RM2 index,RM2 (G) = (k1 +3)(k2 + k4 + s2 + s3 +4)+(k2 +1)(k3 +1)+(k3 +1)(k4 +1)+(s2 +1)(s3 +1) .Since k2 + k4 + s2 + s3 = n−6− k1 = 3r−5− k1, thenRM2 (G) = (k1 +3)(3r− k1−1)+ k2 + k4 + s2 + s3+ k3 + k2k3 + k3k4 + s2s3 +3= (k1 +3)(3r− k1−1)+3r−5− k1 + k3 + k2k3 + k3k4 + s2s3 +3.By Lemma 2.14, k2k3 + k3k4 + s2s3 takes its maximum value when k2 = k3 =k4 = 0 and s2 = s3 or s2 = s3 +1. ThereforeRM2 (G) = f (k1,s2,s3) = (k1 +3)(3r− k1−1)+3r−5− k1 + s2s3 +3.Since s2 + s3 = 3r−5− k1, theng(k1,s2,s3) = 3r−5− k1− s2− s3 = 0 (6)can be written. By using the Lagrange multipliers method, we get s2 = 3r−k1− 5 and s3 = 3r− k1− 5. Thus, s2 + s3 = 6r− 4k1− 10. From Equation 6,s2 + s3 = 3r− k1− 5. And from these last two equalities k1 = r− 1, s2 = s3 =r−2 can be found. Thus,RM2 (G) = (r+2)(r−2+ r−2+4)+(r−1)2 +2= r2 +2r+ r2−2r+1+2 = 2r2 +3.This last value is smaller than that of Proposition 2.12 (b). For a≥ 5 and b = 3the proof is similar.Case 2: Let a≥ 4 and b≥ 4. By Proposition 2.15 the proof is clear.Now, we begin to investigate the maximum RM2-index of the second classof bicyclic graphs with k pendent vertices.Proposition 2.17. Let G ∈ DC03,3(k1,k2,k3,s1,s2,s3) with n vertices and n− 6pendent vertices. Then G =Ψ= DC03,3(k1,0,0,s1,0,0), with k1 = s1 or |k1− s1|= 1, has maximum RM2 index among all the graphs belonging toDC03,3(k1,k2,k3,s1,s2,s3).144 SU¨LEYMAN EDIZFigure 5: G = DC4,3 (k1,k2,k3,s2,s3) for the Case 1 of Theorem 2.16Proof. From Proposition 2.2 (a),RM2 (G) = (k1 +2)(s1 +2)+(k1 +2)(k2 + k3 +2)+(k2 +1)(k3 +1)+(s1 +2)(s2 + s3 +2)+(s2 +1)(s3 +1)= k1s1 + k1 (k2 + k3)+ s1 (s2 + s3)+ k2k3 + s2s3 +14.f (k1,k2,k3,s1,s2,s3) = k1s1 + k1 (k2 + k3)+ s1 (s2 + s3)+ k2k3 + s2s3 +14.Since k1 + k2 + k3 + s1 + s2 + s3 = n−6 then,g(k1,k2,k3,s1,s2,s3) = n− k1− k2− k3− s1− s2− s3−6 = 0.And by using the Lagrange multipliers method k2 = k3 = 0 , s2 = s3 = 0, k1 = s1or |k1− s1|= 1.Corollary 2.18. Let Ψ ∈ DC03,3(k1,k2,k3,s1,s2,s3). ThenRM2 (Ψ) =⌈n−22⌉⌊n−22⌋.Proof. Without loss of generality, from Proposition 2.17, let k1 =⌈n−62⌉ands1 =⌊n−62⌋. Then with direct calculations the desired result is acquired.Theorem 2.19. LetΨ∈DC03,3(k1,k2,k3,s1,s2,s3) with n vertices and n−6 pen-dent vertices. Then Ψ has maximum RM2 value among all the graphs belongingto DCla,b with n vertices and k pendent vertices.Proof. From Proposition 2.2, Lemma 2.14, Proposition 2.17 and Corollary 2.18the desired result is acquired.Now, we begin to investigate the maximum RM2-index of the third class ofbicyclic graphs with k pendent vertices.REDUCED SECOND ZAGREB INDEX OF BICYCLIC GRAPHS 145Proposition 2.20. Let G ∈ DC03+3(k1,k2,k3,s2) with n vertices and n− 4 pen-dent vertices. Then G = Z = DC03+3(k1,k2,k3,s2), with |k1− k3| ≤ 1 and∣∣k j− (k2 + s2)∣∣ ≤ 1 ( j = 1 or j = 3), has maximum RM2 index among all thegraphs belonging to DC03+3(k1,k2,k3,s2).Proof. From Proposition 2.3 (b),RM2 (G) = (k1 + k3 +4)(k2 + s2 +2)+(k1 +2)(k3 +2) .If we put k2 + s2 = x thenRM2 (G) = f (k1,k3,x) = k1x+ k3x+4x+ k1k3 +2k1 +2k3 +12.Since k1+k3+x= n−4 then g(k1,k3,x)= n−k1−k3−x−4= 0 can be written.And by using the Lagrange multipliers method we get k1 = k3 = x = n−43 . Thus,|k1− k3| ≤ 1 and |k1,3− (k2 + s2)| ≤ 1.Proposition 2.21. Let G ∈ Z with n vertices and n−4 pendent vertices. Then(a) RM2 (G) = 3r2 +4r+1 f or n = 3r,r ≥ 2.(b) RM2 (G) = 3r2 +6r+3 f or n = 3r+1,r ≥ 2.(c) RM2 (G) = 3r2 +8r+5 f or n = 3r+2,r ≥ 2.Proof. By Proposition 2.3 and Proposition 2.20 we get the desired result.Theorem 2.22. Let Z ∈ DC03+3(k1,k2,k3,s2) with n vertices and n− 4 pendentvertices. Then Z has maximum RM2 value among all the graphs belonging toDCla+b with n vertices and k pendent vertices.Proof. From Proposition 2.3, Lemma 2.14, Proposition 2.20 and Proposition2.21 the desired result is obtained.And now, from Theorem 2.16 , Theorem 2.19 and Theorem 2.22 we canstate the following corollary.Corollary 2.23. Among all the bicyclic graphs with n vertices and k pendentvertices Z = DC03+3(k1,k2,k3,s2), with |k1− k3| ≤ 1 and∣∣k j− (k2 + s2)∣∣ ≤ 1( j = 1 or j = 3), has maximum RM2 index.146 SU¨LEYMAN EDIZAcknowledgementsThe author would like to thank the referee for his/her careful reading and makingsome valuable comments which have essentially improved the presentation ofthis paper.REFERENCES[1] A. Chang - F. Tian - A. Yu, On the index of bicyclic graphs with perfect matchings,Discrete Math. 283 (1-3) (2004), 51–59.[2] H. Deng, A unified approach to the extremal Zagreb indices for trees, unicyclicgraphs and bicyclic graphs, MATCH Commun. Math. Comput. Chem. 57 (3)(2007), 597–616.[3] S. Li, Comparing the Zagreb indices for connected bicyclic graphs, MATCHCommun. Math. Comput. Chem. 62 (2009), 699–714.[4] Q. Zhao - S. Li, Sharp bounds for the Zagreb indices of bicyclic graphs withk-pendant vertices, Discrete Appl. Math. 158 (17) (2010), 1953–1962.[5] S. Chen - W. Liu, Extremal modified Schultz index of bicyclic graphs, MATCHCommun. Math. Comput. Chem. 64 (3) (2010), 767–782.[6] L. Feng - W. Liu - K. Xu, The hyper-Wiener index of bicyclic graphs, Util. Math.84 (2011), 97–104.[7] X. F. Pan - N. N. Lv, Sharp bounds of the zeroth-order general Randic´ index ofbicyclic graphs with given pendent vertices, Discrete Appl. Math. 159 (4) (2011),240–245.[8] L. Feng - W. Liu, The maximal Gutman index of bicyclic graphs, MATCH Com-mun. Math. Comput. Chem. 66 (2) (2011), 699–708.[9] S. Li - O. Zhao, Sharp upper bounds on Zagreb indices of bicyclic graphs witha given matching number, Math. Comput. Modelling 54 (11-12) (2011), 2869–2879.[10] K. Xu - I. Gutman, The greatest Hosoya index of bicyclic graphs with given max-imum degree, MATCH Commun. Math. Comput. Chem. 66 (3) (2011), 795–824.[11] K. Xu - K. C. Das, Trees, unicyclic, and bicyclic graphs extremal with respect tomultiplicative sum zagreb index, MATCH Commun. Math. Comput. Chem. 68 (1)(2012), 257–272.[12] Z. Du - B. Zhou, On sum-connectivity index of bicyclic graphs, Bull. Malays.Math. Sci. Soc. 35 (2012), 101–117.[13] R.Sun - Z. Zhu - L.Tan, On the Merrifield-Simmons index and Hosoya index ofbicyclic graphs with a given girth, Ars Combin. 103 (2012), 465–478.[14] C. Du, Minimum detour index of bicyclic graphs, MATCH Commun. Math. Com-put. Chem. 68 (1) (2012), 357–370.REDUCED SECOND ZAGREB INDEX OF BICYCLIC GRAPHS 147[15] X. Qi, Detour index of bicyclic graphs, Util. Math. 90 (2013), 101–113.[16] L.Feng - G. Yu - K. Xu - Z. Jiang, A note on the Kirchhoff index of bicyclic graphs,Ars Combin. 114 (2014), 33–40.[17] B. Furtula - I. Gutman - S. Ediz, On difference of Zagreb indices, Discrete Appl.Math. 178 (2014), 83–88.[18] S. Ediz, Reduced second Zagreb index of unicyclic graphs, (submitted).[19] I. Gutman - N. Trinajstic´, Graph theory and molecular orbitals. Total pi-electronenergy of alternant hydrocarbons, Chem. Phys. Lett. 17 (1971), 535–538.[20] T. Dosˇlic´ - B. Furtula - A. Graovac - I. Gutman - S. Moradi - Z. Yarahmadi, Onvertex degree based molecular structure descriptors, MATCH Commun. Math.Comput. Chem. 66 (2011), 613–626.SU¨LEYMAN EDIZFaculty of EducationYu¨zu¨ncu¨ Yıl UniversityVan, Turkeye-mail: suleymanediz@yyu.edu.tr

Reduced second Zagreb index of bicyclic graphs with pendent vertices

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Reduced second Zagreb index of bicyclic graphs with pendent vertices

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