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General sum-connectivity index, general product-connectivity index, general Zagreb index and coindices of line graph of subdivision graphs
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General sum-connectivity index, general product-connectivity index, general Zagreb index and coindices of line graph of subdivision graphs Harishchandra S. Ramane a,∗ , Vinayak V. Manjalapur a , Ivan Gutman b a Department of Mathematics, Karnatak University, Dharwad - 580003, India b Faculty of Science, University of Kragujevac, P. O. Box 60, 34000, Kragujevac, Serbia
Received 10 May 2016; accepted 21 January 2017 Available online xxxx
Abstract The general sum-connectivity index, general product-connectivity index, general Zagreb index and coindices of line graphs of subdivision graphs of tadpole graphs, wheels and ladders have been reported in the literature. In this paper, we obtain general expressions for these topological indices for the line graph of the subdivision graphs, thus generalizing the existing results. c 2017 Kalasalingam University. Publishing Services by Elsevier B.V. This is an open access article under the CC BY-NC-ND ⃝ license (http://creativecommons.org/licenses/by-nc-nd/4.0/). Keywords: General sum-connectivity index; General product-connectivity index; Zagreb index; Randi´c index
1. Introduction Topological indices are numerical quantities of a graph that are invariant under graph isomorphism. The interest in topological indices is mainly related to their use in quantitative structure–property relationship (QSPR) and quantitative structure–activity relationship (QSAR) [1,2]. Let G be a simple graph without loops and multiple edges. Let V (G) be the vertex set and E(G) be the edge set of G, respectively. The degree of a vertex u in G is the number of edges incident to it and is denoted by dG (u). One of the first degree-based indices is Randi´c index [3], defined as R(G) = [dG (u) dG (v)]−1/2 . uv∈E(G)
For convenience, we may call R(G) the product-connectivity index.
Peer review under responsibility of Kalasalingam University. ∗ Corresponding author.
E-mail addresses: [email protected] (H.S. Ramane), [email protected] (V.V. Manjalapur), [email protected] (I. Gutman). http://dx.doi.org/10.1016/j.akcej.2017.01.002 c 2017 Kalasalingam University. Publishing Services by Elsevier B.V. This is an open access article under the CC BY-NC-ND license 0972-8600/⃝ (http://creativecommons.org/licenses/by-nc-nd/4.0/).
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The sum-connectivity index of a graph G is defined as [4] χ (G) = [dG (u) + dG (v)]−1/2 . uv∈E(G)
The product-connectivity and sum-connectivity indices are highly intercorrelated quantities [5]. Basic properties of the sum-connectivity index were established in [4]. The Randi´c index has been extended to the general product-connectivity index defined as [6] Rα (G) = [dG (u)dG (v)]α , uv∈E(G)
where α is any real number. The mathematical properties of the product-connectivity index and its general version may be found in [7,8]. Similarly the general sum-connectivity index is defined as [9] [dG (u) + dG (v)]α . χα (G) = uv∈E(G)
The first and second Zagreb indices are defined as [1,10] M1 (G) = [dG (u)]2 and M2 (G) = [dG (u)dG (v)]. u∈V (G)
uv∈E(G)
Details on their chemical applications can be found in [11–13] whereas mathematical properties are reported in [14–17]. It is easy to observe that M1 (G) satisfies the expression [18]. M1 (G) = [dG (u) + dG (v)]. uv∈E(G)
Li and Zhao [19] introduced the first general Zagreb index as follows M1α (G) = [dG (u)]α . u∈V (G)
It is easy to see that M1α (G) = [(dG (u))α−1 + (dG (v))α−1 ]. uv∈E(G)
The first and second Zagreb coindices are defined as [18,20,21] M1 (G) = [dG (u) + dG (v)] and M2 (G) = [dG (u)dG (v)]. uv̸∈ E(G)
uv̸∈ E(G)
Su and Xu [22] introduced the general sum-connectivity coindex as χα (G) = [dG (u) + dG (v)]α . uv̸∈ E(G)
The general product-connectivity coindex is defined as Rα (G) [dG (u)dG (v)]α . uv̸∈ E(G)
The sum of cubes of vertex degrees was encountered in [10]. In [23], Furtula and Gutman named it forgotten index and established its some basic properties. This index is defined as [23] F(G) = [dG (u)]3 . u∈V (G)
Note that,
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R1 (G) = M2 (G), R1 (G) = M2 (G), χ1 (G) = M1 (G), χ1 (G) = M1 (G), R−1/2 (G) = R(G), χ−1/2 (G) = χ (G), M12 (G) = M1 (G), M13 (G) = F(G). The subdivision graph S(G) is the graph obtained from G by replacing each of its edge by a path of length 2, or equivalently, by inserting an additional vertex into each edge of G. The line graph of a graph G, denoted by L(G) is the graph whose vertices are in a one-to-one correspondence with the edges of G and two vertices in L(G) are adjacent if and only if the corresponding edges are adjacent in G. The tadpole graph Tn,k is the graph obtained by joining a cycle Cn to a path of length k. By starting with a disjoint union of two graphs G 1 and G 2 and adding edges joining each vertex of G 1 to all vertices of G 2 , one obtains the sum G 1 + G 2 of G 1 and G 2 . The sum Cn + K 1 of a cycle Cn and a single vertex is referred to as a wheel Wn+1 of order n + 1. The Cartesian product G 1 G 2 of graphs G 1 and G 2 is a graph with vertex set V (G 1 ) × V (G 2 ), and two vertices (u 1 , v1 ) and (u 2 , v2 ) are adjacent in G 1 G 2 if and only if either u 1 = u 2 and v1 v2 ∈ E(G 2 ), or v1 = v2 and u 1 u 2 ∈ E(G 1 ). The ladder L n is given by L n = K 2 Pn , where Pn is the path of length n and K n is a complete graph on n vertices. For additional graph theoretic terminology we refer the book [24]. Ranjini et al. [25] calculated the Zagreb indices and coindices of the line graph of subdivision graph of tadpole, wheel, and ladder graphs. Su and Xu [26] generalized the results of Ranjini et al. [25] by calculating the general sum-connectivity index and general product-connectivity index of the line graph of subdivision graph of the tadpole, wheel, and ladder graphs. In this paper we obtain expressions for the general sum-connectivity index and general product-connectivity index, and coindices of the line graph of subdivision graph of any graph, which generalizes the results of Ranjini et al. [25] and Su and Xu [26]. 2. Topological indices of line graph of subdivision graphs If e = uv is an edge of G, then d L(G) (e) = dG (u) + dG (v) − 2. If G has n vertices and m edges, then L(G) has m vertices and −m + 12 u∈V (G) [dG (u)]2 edges. Observation 1. If u is the vertex of G, then d S(G) (u) = dG (u). Observation 2. If e = uv is a subdivision edge of S(G) where u ∈ V (G) and v is the subdivision vertex in S(G), then d L(S(G)) (e) = d S(G) (u) + d S(G) (v) − 2 = dG (u) + 2 − 2 = dG (u). Theorem 2.1. For any graph G and α ∈ R (i) χα (L(S(G))) = χα (G) + 2α−1
(dG (u))α+1 (dG (u) − 1) u∈V (G)
1 (dG (u))2α+1 (dG (u) − 1) . (ii) Rα (L(S(G))) = Rα (G) + 2 u∈V (G) Proof. The edges of S(G) will be the vertices of L(S(G)). Without loss of generality, let e and f be adjacent edges, adjacent at u in G. Let e′ and e′′ be the subdivision edges of e whereas f ′ and f ′′ be the subdivision edges of f in S(G). Let ve and v f be the subdivision vertices on e and f in S(G) (see Fig. 2). Partition the edge set E(L(S(G))) into sets E 1 and E 2 , so that E 1 = {e′ e′′ } where e′ and e′′ are subdivision edges with common end vertex ve in S(G), and E 2 = { f ′ e′′ } where f ′ and e′′ are subdivision edges with common end vertex u in S(G), where u ∈ V (G).
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Fig. 1. Graph, its subdivision graph and line graph of subdivision graph.
Fig. 2. Schematic representation of the graph G and S(G) used for the proof of Theorem 2.1.
It is easy to check that |E 1 | = m and 1 1 |E 2 | = dG (u) (dG (u) − 1) = −m + (dG (u))2 . 2 u∈V (G) 2 u∈V (G) (i) χα (L(S(G))) =
α d L(S(G)) (e) + d L(S(G)) ( f )
e f ∈E(L(S(G)))
=
α α d L(S(G)) (e) + d L(S(G)) ( f ) + d L(S(G)) (e) + d L(S(G)) ( f ) e f ∈E 1
=
e f ∈E 2 α
[dG (u) + dG (v)] +
=
[dG (u) + dG (u)]α
e f ∈E 2
uv∈E(G)
α
[dG (u) + dG (v)] +
[2dG (u)]α
e f ∈E 2
uv∈E(G)
= χα (G) +
α
(2dG (u))
u∈V (G)
dG (u) 2
(dG (u))α+1 (dG (u) − 1) .
= χα (G) + 2α−1
u∈V (G)
(ii) Rα [L(S(G))] =
(d L(S(G)) (e))(d L(S(G)) ( f ))
α
e f ∈E(L(S(G)))
=
α α (d L(S(G)) (e))(d L(S(G)) ( f )) + (d L(S(G)) (e))(d L(S(G)) ( f )) e f ∈E 1
=
e f ∈E 2 α
[dG (u)dG (v)] +
= Rα (G) +
[dG (u)dG (u)]α
e f ∈E 2
uv∈E(G)
(dG (u))2α
e f ∈E 2
= Rα (G) +
u∈V (G)
= Rα (G) +
(dG (u))
2α
dG (u) 2
1 (dG (u))2α+1 (dG (u) − 1) . 2 u∈V (G)
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The following is an immediate consequence of Theorem 2.1. Corollary 2.2 ([26]). Let Tn,k be the tadpole graph, Wn+1 be the wheel and L n be the ladder. Then (i) (ii) (iii) (iv) (v) (vi)
χα (L(S(Tn,k ))) = 22α+1 (n + k − 3) + 3 · 6α + 3 · 5α + 3α Rα (L(S(Tn,k ))) = 22α+1 (n + k − 3) + 3 · 6α + 32α+1 + 2α χα (L(S(Wn+1 ))) = 4n · 6α + n(n + 3)α + (n − 1) · 2α−1 n α+1 Rα (L(S(Wn+1 ))) = 4n · 32α + 3α n α+1 + 2−1 (n − 1)n 2α+1 χα (L(S(L n ))) = (9n − 20) · 6α + 4 · 5α + 4α+1 Rα (L(S(L n ))) = (9n − 20) · 9α + 4 · 6α + 6 · 4α .
If α = 1, then χ1 (G) = M1 (G) and R1 (G) = M2 (G). Therefore, by Theorem 2.1 we have following corollary. Corollary 2.3. For any graph G, (i) M1 (L(S(G))) = [dG (u)]3 = F(G) u∈V (G)
(ii) M2 (L(S(G))) = M2 (G) +
1 (dG (u))3 (dG (u) − 1) . 2 u∈V (G)
Corollary 2.4 follows from Corollary 2.3. Corollary 2.4 ([25]). Let Tn,k be the tadpole graph, Wn+1 be the wheel and L n be the ladder. Then (i) M1 (L(S(Tn,k ))) = 4(2n + 2k + 3) (ii) M2 (L(S(Tn,k ))) = 8n + 8k + 23 (iii) M1 (L(S(Wn+1 ))) = n(n 2 + 27) 3 2 (iv) M2 (L(S(Wn+1 ))) = n n −n +6n+72 2 (v) M1 (L(S(L n ))) = 54n − 66 (vi) M2 (L(S(L n ))) = 81n − 133. Theorem 2.5. For any graph G and α ∈ R M1α (L(S(G))) = [dG (u)]α+1 . u∈V (G)
Proof. For each vertex u ∈ V (G), there are dG (u) subdivided edges in S(G) and they contribute by (dG (u))α dG (u) to M1α (L(S(G))). Hence M1α (L(S(G))) = [dG (u)]α+1 . u∈V (G)
Corollary 2.6 ([26]). Let Tn,k be the tadpole graph, Wn+1 be the wheel and L n be the ladder. Then (i) M1α (L(S(Tn,k ))) = 2α+1 (n + k − 2) + 3α+1 + 1
(ii) M1α (L(S(Wn+1 ))) = n α+1 + n · 3α+1
(iii) M1α (L(S(L n ))) = 2α+3 + (6n − 12) · 3α .
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Fig. 3. Schematic representation of the subdivision graph S(G) used for the proof of Theorem 2.7.
Theorem 2.7. For any graph G with n vertices u 1 , u 2 , . . . , u n and α ∈ R, n 1 (dG (u) + dG (u i ))α + (dG (v) + dG (u i ))α dG (u i ) (i) χα (L(S(G))) = 2 uv∈E(G) i=1 [dG (u)]α+2 − χα (G) − 2α−1 u∈V (G)
1 (ii) Rα (L(S(G))) = 2
[dG (u)]
u∈V (G)
α+1
2 −
1 [dG (u)]α+3 − Rα (G). 2 u∈V (G)
Proof. Let u 1 , u 2 , . . . , u n be the vertices of G. Without loss of generality, let e = u 1 u 2 ∈ E(G). Let e′ = u 1 w and e′′ = wu 2 be the subdivided edges of e in S(G) (see Fig. 3). The vertex e′ is not adjacent to dG (u i ) vertices in L(S(G)) corresponding to the vertex u i for i = 3, 4, . . . , n and it is not adjacent to dG (u 2 ) − 1 vertices in L(S(G)) corresponding to the vertex u 2 . (i) Therefore e′ contributes the following quantity to χα (L(S(G))). α s(e′ ) = d L(S(G)) (e′ ) + d L(S(G)) ( f ) e′ f ̸∈ E(L(S(G)))
= [dG (u 1 ) + dG (u 2 )]α (dG (u 2 ) − 1) + [dG (u 1 ) + dG (u 3 )]α (dG (u 3 )) + · · · + [dG (u 1 ) + dG (u n )]α (dG (u n )) n = [dG (u 1 ) + dG (u i )]α dG (u i ) − [dG (u 1 ) + dG (u 2 )]α i=2 n = [dG (u 1 ) + dG (u i )]α dG (u i ) − [dG (u 1 ) + dG (u 1 )]α dG (u 1 ) − [dG (u 1 ) + dG (u 2 )]α i=1 n = [dG (u 1 ) + dG (u i )]α dG (u i ) − 2α (dG (u 1 ))α+1 − [dG (u 1 ) + dG (u 2 )]α . i=1
Similarly e′′ contributes the following quantity to χα (L(S(G))). s(e′′ ) =
n [dG (u 2 ) + dG (u i )]α dG (u i ) − 2α (dG (u 2 ))α+1 − [dG (u 2 ) + dG (u 1 )]α . i=1
Therefore, the total contribution of an edge e to χα (L(S(G))) is s(e) = s(e′ ) + s(e′′ ) n (dG (u 1 ) + dG (u i ))α + (dG (u 2 ) + dG (u i ))α dG (u i ) = i=1
− 2α (dG (u 1 ))α+1 + (dG (u 2 ))α+1 − 2[dG (u 1 ) + dG (u 2 )]α .
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Therefore, χα (L(S(G))) =
1 s(e) 2 e∈E(G)
n 1 α α = (dG (u 1 ) + dG (u i )) + (dG (u 2 ) + dG (u i )) dG (u i ) 2 e=u u ∈E(G) i=1 1 2 [dG (u 1 ) + dG (u 2 )]α − 2α−1 (dG (u 1 ))α+1 + (dG (u 2 ))α+1 − e=u 1 u 2 ∈E(G)
e=u 1 u 2 ∈E(G)
n α α (dG (u) + dG (u i )) + (dG (v) + dG (u i )) dG (u i )
1 2 uv∈E(G) i=1 − 2α−1 [dG (u)]α+2 − χα (G).
=
u∈V (G)
(ii) The edge e′ contributes the following quantity to Rα (L(S(G))). α s(e′ ) = d L(S(G)) (e′ ) d L(S(G)) ( f ) e′ f ̸∈ E(L(S(G)))
= (dG (u 1 ) dG (u 2 ))α (dG (u 2 ) − 1) + (dG (u 1 ) dG (u 3 ))α (dG (u 3 )) + · · · + (dG (u 1 ) dG (u n ))α (dG (u n )) α+1 α − (dG (u 1 ))α+2 − [dG (u 1 ) dG (u 2 )]α . (dG (u)) = (dG (u 1 )) u∈V (G)
Similarly e′′ contributes the following quantity to Rα (L(S(G))). ′′ α α+1 s(e ) = (dG (u 2 )) (dG (u)) − (dG (u 2 ))α+2 − [dG (u 2 )dG (u 1 )]α . u∈V (G)
Total contribution of an edge e to Rα (L(S(G))) is s(e) = s(e′ ) + s(e′′ ) = (dG (u 1 ))α + (dG (u 2 ))α − 2[dG (u 1 )dG (u 2 )]α .
α+1
(dG (u))
− (dG (u 1 ))α+2 + (dG (u 2 ))α+2
u∈V (G)
Therefore, Rα (L(S(G))) = =
1 s(e) 2 e∈E(G) 1 (dG (u))α+1 (dG (u 1 ))α + (dG (u 2 ))α 2 u∈V (G) e=u 1 u 2 ∈E(G) 1 − (dG (u 1 ))α+2 + (dG (u 2 ))α+2 − [dG (u 1 )dG (u 2 )]α 2 e=u u ∈E(G) e=u u ∈E(G) 1 2
1 2
1 1 = (dG (u))α+1 (dG (u))α+1 − (dG (u))α+3 − Rα (G) 2 u∈V (G) 2 u∈V (G) u∈V (G) 2 1 1 = [dG (u)]α+1 − [dG (u)]α+3 − Rα (G). 2 u∈V (G) 2 u∈V (G)
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Example 2.8. For a graph G given in Fig. 1, with α = 2, we get from Theorem 2.7(i) 4 1 2 2 χ2 (L(S(G))) = (dG (u) + dG (u i )) + (dG (v) + dG (u i )) dG (u i ) 2 uv∈E(G) i=1 −2 [dG (u)]4 − χ2 (G) u∈V (G)
1 = [(dG (u) + 1)2 + (dG (v) + 1)2 ] · 1 + [(dG (u) + 3)2 + (dG (v) + 3)2 ] · 3 2 uv∈E(G) + [(dG (u) + 2)2 + (dG (v) + 2)2 ] · 2 + [(dG (u) + 2)2 + (dG (v) + 2)2 ] · 2 − 2[1 + 81 + 16 + 1] − 82 1 2 = (2 + 42 ) + (42 + 62 ) · 3 + (32 + 52 ) · 2 + (32 + 52 ) · 2 2 + (42 + 32 ) + (62 + 52 ) · 3 + (52 + 42 ) · 2 + (52 + 42 ) · 2 + (42 + 32 ) + (62 + 52 ) · 3 + (52 + 42 ) · 2 + (52 + 42 ) · 2 + (32 + 32 ) + (52 + 52 ) · 3 + (42 + 42 ) · 2 + (42 + 42 ) · 2 − 2[1 + 81 + 16 + 1] − 82 = 366. The next Corollary follows from Theorem 2.7. Corollary 2.9 ([26]). Let Tn,k be the tadpole graph and Wn+1 be the wheel. Then (i) χα (L(S(Tn,k ))) = (2n 2 + 4nk + 2k 2 − 11n − 13k + 23) · 4α + (2n + 2k − 6) · 3α + (6n + 6k − 16) · 5α . (ii) χα (L(S(Wn+1 ))) = n(3n − 1)(n + 3)α + 2−1 n(9n − 11) · 6α . If α = 1, then χ1 (G) = M1 (G) and R1 (G) = M2 (G). Therefore, by Theorem 2.7 we have the following corollaries. Corollary 2.10. For any graph G,
(i) M1 (L(S(G))) = (2m − 1)M1 (G) −
[dG (u)]3
u∈V (G)
(ii) M2 (L(S(G))) =
1 1 [M1 (G)]2 − M2 (G) − [dG (u)]4 . 2 2 u∈V (G)
Corollary 2.11 ([25]). Let Tn,k be the tadpole graph, Wn+1 be the wheel, and L n be the ladder. Then 8(n + k)2 − 8(n + k + 2) + 2, when k > 1; (i) M1 (L(S(Tn,k ))) = 8n(n + 2k − 1) + 2(2k − 9), when k = 1 2 8(n + k) − 4(n + k + 6) − 1, when k > 1; (ii) M2 (L(S(Tn,k ))) = 8n(n 2 + 2nk + k) − 4n − 30, when k = 1 (iii) M1 (L(S(Wn+1 ))) = n(3n 2 + 35n − 36) (iv) M2 (L(S(Wn+1 ))) = 21 n(18n 2 + 75n − 99) (v) M1 (L(S(L n ))) = 108n 2 − 264n + 240 (vi) M2 (L(S(L n ))) = 162n 2 − 468n + 370.
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3. Conclusion Ranjini et al. [25] obtained the expression for the Zagreb indices and coindices of the line graph of the subdivision graph of tadpole graphs, wheel graphs, and ladder graphs. Su and Xu [26] investigated the general sum-connectivity index and general product-connectivity index of the line graph of subdivision graph of the tadpole graphs, wheel graphs, and ladder graphs. Here we obtained expressions for general sum-connectivity index, general product-connectivity index, general Zagreb index and coindices of the line graph of subdivision graph of any graph, which generalizes the results of Ranjini et al. [25] and Su and Xu [26]. Acknowledgment Authors H. S. Ramane and V. V. Manjalapur are grateful to the University Grants Commission (UGC), Govt. of India for support through research grant under UPE FAR-II Grant No. F 14-3/2012 (NS/PE). References [1] [2] [3] [4] [5] [6] [7] [8] [9] [10] [11] [12] [13] [14] [15] [16] [17] [18] [19] [20] [21] [22] [23] [24] [25] [26]
I. Gutman, Degree-based topological indices, Croat. Chem. Acta. 86 (2013) 351–361. J. Devillers, A.T. Balaban (Eds.), Topological Indices and Related Descriptors in QSAR and QSPR, Gordon & Breach, Amsterdam, 1999. M. Randi´c, On characterization of molecular branching, J. Am. Chem. Soc. 97 (1975) 6609–6615. B. Zhou, N. Trinajsti´c, On a novel connectivity index, J. Math. Chem. 46 (2009) 1252–1270. O. Ivanciuc, T. Ivanciuc, D. Cabrol-Bass, A.T. Balaban, Evaluation in quantitative structure–property relationship models of structural descriptors derived from information-theory operators, J. Chem. Inf. Comput. Sci. 40 (2000) 631–643. B. Bollab´as, P. Erd¨os, Graphs of extremal weights, Ars Combin. 50 (1998) 225–233. X. Li, I. Gutman, Mathematical Aspects of Randi´c Type Molecular Structure Descriptors, Univ. Kragujevac, Kragujevac, 2006. I. Gutman, B. Furtula (Eds.), Recent Results in the Theory of Randi´c index, Univ. Kragujevac, Kragujevac, 2008. B. Zhou, N. Trinajsti´c, On general sum-connectivity index, J. Math. Chem. 47 (2010) 210–218. I. Gutman, N. Trinajsti´c, Graph theory and molecular orbitals, total π-electron energy of alternant hydrocarbons, Chem. Phys. Lett. 17 (1972) 535–538. R. Todeschini, V. Consonni, Handbook of Molecular Descriptors, Wiley-VCH, Weinheim, 2000. R. Todeschini, V. Consonni, Handbook of Molecular Descriptors for Chemoinformatics, vols. I and II, Wiley-VCH, Weinheim, 2009. S. Nikoli´c, G. Kovaˇcevi´c, A. Miliˇcevi´c, N. Trinajsti´c, The Zagreb indices 30 years after, Croat. Chem. Acta. 76 (2003) 113–124. B. Furtula, I. Gutman, M. Dehmer, On structure-sensitivity of degree based topological indices, Appl. Math. Comput. 219 (2013) 8973–8978. I. Gutman, K.C. Das, The first Zagreb indices 30 years after, MATCH Commun. Math. Comput. Chem. 50 (2004) 83–92. I. Gutman, B. Furtula, Z.K. Vuki´cevi´c, G. Popivoda, On Zagreb indices and coindices, MATCH Commun. Math. Comput. Chem. 74 (2015) 5–16. S.M. Hosamani, I. Gutman, Zagreb indices of transformation graphs and total transformation graphs, Appl. Math. Comput. 247 (2014) 1156–1160. T. Doˇsli´c, Vertex-weighted Wiener polynomials for composite graphs, Ars Math. Contemp. 1 (2008) 66–80. X. Li, H. Zhao, Trees with the first three smallest and largest generalized topological indices, MATCH Commun. Math. Comput. Chem. 50 (2004) 57–62. A.R. Ashrafi, T. Doˇsli´c, A. Hamzeh, The Zagreb coindices of graph operations, Discrete Appl. Math. 158 (2010) 1571–1578. A.R. Ashrafi, T. Doˇsli´c, A. Hamzeh, Extremal graphs with respect to the Zagreb coindices, MATCH Commun. Math. Comput. Chem. 65 (2011) 85–92. G. Su, L. Xu, On the general sum-connectivity coindex of graphs, Iranian J. Math. Chem. 2 (2011) 89–98. B. Furtula, I. Gutman, A forgotten topological index, J. Math. Chem. 53 (2015) 1184–1190. F. Harary, Graph Theory, Addison Wesley, Reading, 1969. P.S. Ranjini, V. Lokesha, I.N. Cang¨ul, On the Zagreb indices of the line graphs of the subdivision graphs, Appl. Math. Comput. 218 (2011) 699–702. G. Su, L. Xu, Topological indices of the line graph of subdivision graphs and their Schur bounds, Appl. Math. Comput. 253 (2015) 395–401.
ScienceDirect AKCE International Journal of Graphs and Combinatorics (
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General sum-connectivity index, general product-connectivity index, general Zagreb index and coindices of line graph of subdivision graphs Harishchandra S. Ramane a,∗ , Vinayak V. Manjalapur a , Ivan Gutman b a Department of Mathematics, Karnatak University, Dharwad - 580003, India b Faculty of Science, University of Kragujevac, P. O. Box 60, 34000, Kragujevac, Serbia
Received 10 May 2016; accepted 21 January 2017 Available online xxxx
Abstract The general sum-connectivity index, general product-connectivity index, general Zagreb index and coindices of line graphs of subdivision graphs of tadpole graphs, wheels and ladders have been reported in the literature. In this paper, we obtain general expressions for these topological indices for the line graph of the subdivision graphs, thus generalizing the existing results. c 2017 Kalasalingam University. Publishing Services by Elsevier B.V. This is an open access article under the CC BY-NC-ND ⃝ license (http://creativecommons.org/licenses/by-nc-nd/4.0/). Keywords: General sum-connectivity index; General product-connectivity index; Zagreb index; Randi´c index
1. Introduction Topological indices are numerical quantities of a graph that are invariant under graph isomorphism. The interest in topological indices is mainly related to their use in quantitative structure–property relationship (QSPR) and quantitative structure–activity relationship (QSAR) [1,2]. Let G be a simple graph without loops and multiple edges. Let V (G) be the vertex set and E(G) be the edge set of G, respectively. The degree of a vertex u in G is the number of edges incident to it and is denoted by dG (u). One of the first degree-based indices is Randi´c index [3], defined as R(G) = [dG (u) dG (v)]−1/2 . uv∈E(G)
For convenience, we may call R(G) the product-connectivity index.
Peer review under responsibility of Kalasalingam University. ∗ Corresponding author.
E-mail addresses: [email protected] (H.S. Ramane), [email protected] (V.V. Manjalapur), [email protected] (I. Gutman). http://dx.doi.org/10.1016/j.akcej.2017.01.002 c 2017 Kalasalingam University. Publishing Services by Elsevier B.V. This is an open access article under the CC BY-NC-ND license 0972-8600/⃝ (http://creativecommons.org/licenses/by-nc-nd/4.0/).
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The sum-connectivity index of a graph G is defined as [4] χ (G) = [dG (u) + dG (v)]−1/2 . uv∈E(G)
The product-connectivity and sum-connectivity indices are highly intercorrelated quantities [5]. Basic properties of the sum-connectivity index were established in [4]. The Randi´c index has been extended to the general product-connectivity index defined as [6] Rα (G) = [dG (u)dG (v)]α , uv∈E(G)
where α is any real number. The mathematical properties of the product-connectivity index and its general version may be found in [7,8]. Similarly the general sum-connectivity index is defined as [9] [dG (u) + dG (v)]α . χα (G) = uv∈E(G)
The first and second Zagreb indices are defined as [1,10] M1 (G) = [dG (u)]2 and M2 (G) = [dG (u)dG (v)]. u∈V (G)
uv∈E(G)
Details on their chemical applications can be found in [11–13] whereas mathematical properties are reported in [14–17]. It is easy to observe that M1 (G) satisfies the expression [18]. M1 (G) = [dG (u) + dG (v)]. uv∈E(G)
Li and Zhao [19] introduced the first general Zagreb index as follows M1α (G) = [dG (u)]α . u∈V (G)
It is easy to see that M1α (G) = [(dG (u))α−1 + (dG (v))α−1 ]. uv∈E(G)
The first and second Zagreb coindices are defined as [18,20,21] M1 (G) = [dG (u) + dG (v)] and M2 (G) = [dG (u)dG (v)]. uv̸∈ E(G)
uv̸∈ E(G)
Su and Xu [22] introduced the general sum-connectivity coindex as χα (G) = [dG (u) + dG (v)]α . uv̸∈ E(G)
The general product-connectivity coindex is defined as Rα (G) [dG (u)dG (v)]α . uv̸∈ E(G)
The sum of cubes of vertex degrees was encountered in [10]. In [23], Furtula and Gutman named it forgotten index and established its some basic properties. This index is defined as [23] F(G) = [dG (u)]3 . u∈V (G)
Note that,
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R1 (G) = M2 (G), R1 (G) = M2 (G), χ1 (G) = M1 (G), χ1 (G) = M1 (G), R−1/2 (G) = R(G), χ−1/2 (G) = χ (G), M12 (G) = M1 (G), M13 (G) = F(G). The subdivision graph S(G) is the graph obtained from G by replacing each of its edge by a path of length 2, or equivalently, by inserting an additional vertex into each edge of G. The line graph of a graph G, denoted by L(G) is the graph whose vertices are in a one-to-one correspondence with the edges of G and two vertices in L(G) are adjacent if and only if the corresponding edges are adjacent in G. The tadpole graph Tn,k is the graph obtained by joining a cycle Cn to a path of length k. By starting with a disjoint union of two graphs G 1 and G 2 and adding edges joining each vertex of G 1 to all vertices of G 2 , one obtains the sum G 1 + G 2 of G 1 and G 2 . The sum Cn + K 1 of a cycle Cn and a single vertex is referred to as a wheel Wn+1 of order n + 1. The Cartesian product G 1 G 2 of graphs G 1 and G 2 is a graph with vertex set V (G 1 ) × V (G 2 ), and two vertices (u 1 , v1 ) and (u 2 , v2 ) are adjacent in G 1 G 2 if and only if either u 1 = u 2 and v1 v2 ∈ E(G 2 ), or v1 = v2 and u 1 u 2 ∈ E(G 1 ). The ladder L n is given by L n = K 2 Pn , where Pn is the path of length n and K n is a complete graph on n vertices. For additional graph theoretic terminology we refer the book [24]. Ranjini et al. [25] calculated the Zagreb indices and coindices of the line graph of subdivision graph of tadpole, wheel, and ladder graphs. Su and Xu [26] generalized the results of Ranjini et al. [25] by calculating the general sum-connectivity index and general product-connectivity index of the line graph of subdivision graph of the tadpole, wheel, and ladder graphs. In this paper we obtain expressions for the general sum-connectivity index and general product-connectivity index, and coindices of the line graph of subdivision graph of any graph, which generalizes the results of Ranjini et al. [25] and Su and Xu [26]. 2. Topological indices of line graph of subdivision graphs If e = uv is an edge of G, then d L(G) (e) = dG (u) + dG (v) − 2. If G has n vertices and m edges, then L(G) has m vertices and −m + 12 u∈V (G) [dG (u)]2 edges. Observation 1. If u is the vertex of G, then d S(G) (u) = dG (u). Observation 2. If e = uv is a subdivision edge of S(G) where u ∈ V (G) and v is the subdivision vertex in S(G), then d L(S(G)) (e) = d S(G) (u) + d S(G) (v) − 2 = dG (u) + 2 − 2 = dG (u). Theorem 2.1. For any graph G and α ∈ R (i) χα (L(S(G))) = χα (G) + 2α−1
(dG (u))α+1 (dG (u) − 1) u∈V (G)
1 (dG (u))2α+1 (dG (u) − 1) . (ii) Rα (L(S(G))) = Rα (G) + 2 u∈V (G) Proof. The edges of S(G) will be the vertices of L(S(G)). Without loss of generality, let e and f be adjacent edges, adjacent at u in G. Let e′ and e′′ be the subdivision edges of e whereas f ′ and f ′′ be the subdivision edges of f in S(G). Let ve and v f be the subdivision vertices on e and f in S(G) (see Fig. 2). Partition the edge set E(L(S(G))) into sets E 1 and E 2 , so that E 1 = {e′ e′′ } where e′ and e′′ are subdivision edges with common end vertex ve in S(G), and E 2 = { f ′ e′′ } where f ′ and e′′ are subdivision edges with common end vertex u in S(G), where u ∈ V (G).
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Fig. 1. Graph, its subdivision graph and line graph of subdivision graph.
Fig. 2. Schematic representation of the graph G and S(G) used for the proof of Theorem 2.1.
It is easy to check that |E 1 | = m and 1 1 |E 2 | = dG (u) (dG (u) − 1) = −m + (dG (u))2 . 2 u∈V (G) 2 u∈V (G) (i) χα (L(S(G))) =
α d L(S(G)) (e) + d L(S(G)) ( f )
e f ∈E(L(S(G)))
=
α α d L(S(G)) (e) + d L(S(G)) ( f ) + d L(S(G)) (e) + d L(S(G)) ( f ) e f ∈E 1
=
e f ∈E 2 α
[dG (u) + dG (v)] +
=
[dG (u) + dG (u)]α
e f ∈E 2
uv∈E(G)
α
[dG (u) + dG (v)] +
[2dG (u)]α
e f ∈E 2
uv∈E(G)
= χα (G) +
α
(2dG (u))
u∈V (G)
dG (u) 2
(dG (u))α+1 (dG (u) − 1) .
= χα (G) + 2α−1
u∈V (G)
(ii) Rα [L(S(G))] =
(d L(S(G)) (e))(d L(S(G)) ( f ))
α
e f ∈E(L(S(G)))
=
α α (d L(S(G)) (e))(d L(S(G)) ( f )) + (d L(S(G)) (e))(d L(S(G)) ( f )) e f ∈E 1
=
e f ∈E 2 α
[dG (u)dG (v)] +
= Rα (G) +
[dG (u)dG (u)]α
e f ∈E 2
uv∈E(G)
(dG (u))2α
e f ∈E 2
= Rα (G) +
u∈V (G)
= Rα (G) +
(dG (u))
2α
dG (u) 2
1 (dG (u))2α+1 (dG (u) − 1) . 2 u∈V (G)
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The following is an immediate consequence of Theorem 2.1. Corollary 2.2 ([26]). Let Tn,k be the tadpole graph, Wn+1 be the wheel and L n be the ladder. Then (i) (ii) (iii) (iv) (v) (vi)
χα (L(S(Tn,k ))) = 22α+1 (n + k − 3) + 3 · 6α + 3 · 5α + 3α Rα (L(S(Tn,k ))) = 22α+1 (n + k − 3) + 3 · 6α + 32α+1 + 2α χα (L(S(Wn+1 ))) = 4n · 6α + n(n + 3)α + (n − 1) · 2α−1 n α+1 Rα (L(S(Wn+1 ))) = 4n · 32α + 3α n α+1 + 2−1 (n − 1)n 2α+1 χα (L(S(L n ))) = (9n − 20) · 6α + 4 · 5α + 4α+1 Rα (L(S(L n ))) = (9n − 20) · 9α + 4 · 6α + 6 · 4α .
If α = 1, then χ1 (G) = M1 (G) and R1 (G) = M2 (G). Therefore, by Theorem 2.1 we have following corollary. Corollary 2.3. For any graph G, (i) M1 (L(S(G))) = [dG (u)]3 = F(G) u∈V (G)
(ii) M2 (L(S(G))) = M2 (G) +
1 (dG (u))3 (dG (u) − 1) . 2 u∈V (G)
Corollary 2.4 follows from Corollary 2.3. Corollary 2.4 ([25]). Let Tn,k be the tadpole graph, Wn+1 be the wheel and L n be the ladder. Then (i) M1 (L(S(Tn,k ))) = 4(2n + 2k + 3) (ii) M2 (L(S(Tn,k ))) = 8n + 8k + 23 (iii) M1 (L(S(Wn+1 ))) = n(n 2 + 27) 3 2 (iv) M2 (L(S(Wn+1 ))) = n n −n +6n+72 2 (v) M1 (L(S(L n ))) = 54n − 66 (vi) M2 (L(S(L n ))) = 81n − 133. Theorem 2.5. For any graph G and α ∈ R M1α (L(S(G))) = [dG (u)]α+1 . u∈V (G)
Proof. For each vertex u ∈ V (G), there are dG (u) subdivided edges in S(G) and they contribute by (dG (u))α dG (u) to M1α (L(S(G))). Hence M1α (L(S(G))) = [dG (u)]α+1 . u∈V (G)
Corollary 2.6 ([26]). Let Tn,k be the tadpole graph, Wn+1 be the wheel and L n be the ladder. Then (i) M1α (L(S(Tn,k ))) = 2α+1 (n + k − 2) + 3α+1 + 1
(ii) M1α (L(S(Wn+1 ))) = n α+1 + n · 3α+1
(iii) M1α (L(S(L n ))) = 2α+3 + (6n − 12) · 3α .
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Fig. 3. Schematic representation of the subdivision graph S(G) used for the proof of Theorem 2.7.
Theorem 2.7. For any graph G with n vertices u 1 , u 2 , . . . , u n and α ∈ R, n 1 (dG (u) + dG (u i ))α + (dG (v) + dG (u i ))α dG (u i ) (i) χα (L(S(G))) = 2 uv∈E(G) i=1 [dG (u)]α+2 − χα (G) − 2α−1 u∈V (G)
1 (ii) Rα (L(S(G))) = 2
[dG (u)]
u∈V (G)
α+1
2 −
1 [dG (u)]α+3 − Rα (G). 2 u∈V (G)
Proof. Let u 1 , u 2 , . . . , u n be the vertices of G. Without loss of generality, let e = u 1 u 2 ∈ E(G). Let e′ = u 1 w and e′′ = wu 2 be the subdivided edges of e in S(G) (see Fig. 3). The vertex e′ is not adjacent to dG (u i ) vertices in L(S(G)) corresponding to the vertex u i for i = 3, 4, . . . , n and it is not adjacent to dG (u 2 ) − 1 vertices in L(S(G)) corresponding to the vertex u 2 . (i) Therefore e′ contributes the following quantity to χα (L(S(G))). α s(e′ ) = d L(S(G)) (e′ ) + d L(S(G)) ( f ) e′ f ̸∈ E(L(S(G)))
= [dG (u 1 ) + dG (u 2 )]α (dG (u 2 ) − 1) + [dG (u 1 ) + dG (u 3 )]α (dG (u 3 )) + · · · + [dG (u 1 ) + dG (u n )]α (dG (u n )) n = [dG (u 1 ) + dG (u i )]α dG (u i ) − [dG (u 1 ) + dG (u 2 )]α i=2 n = [dG (u 1 ) + dG (u i )]α dG (u i ) − [dG (u 1 ) + dG (u 1 )]α dG (u 1 ) − [dG (u 1 ) + dG (u 2 )]α i=1 n = [dG (u 1 ) + dG (u i )]α dG (u i ) − 2α (dG (u 1 ))α+1 − [dG (u 1 ) + dG (u 2 )]α . i=1
Similarly e′′ contributes the following quantity to χα (L(S(G))). s(e′′ ) =
n [dG (u 2 ) + dG (u i )]α dG (u i ) − 2α (dG (u 2 ))α+1 − [dG (u 2 ) + dG (u 1 )]α . i=1
Therefore, the total contribution of an edge e to χα (L(S(G))) is s(e) = s(e′ ) + s(e′′ ) n (dG (u 1 ) + dG (u i ))α + (dG (u 2 ) + dG (u i ))α dG (u i ) = i=1
− 2α (dG (u 1 ))α+1 + (dG (u 2 ))α+1 − 2[dG (u 1 ) + dG (u 2 )]α .
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Therefore, χα (L(S(G))) =
1 s(e) 2 e∈E(G)
n 1 α α = (dG (u 1 ) + dG (u i )) + (dG (u 2 ) + dG (u i )) dG (u i ) 2 e=u u ∈E(G) i=1 1 2 [dG (u 1 ) + dG (u 2 )]α − 2α−1 (dG (u 1 ))α+1 + (dG (u 2 ))α+1 − e=u 1 u 2 ∈E(G)
e=u 1 u 2 ∈E(G)
n α α (dG (u) + dG (u i )) + (dG (v) + dG (u i )) dG (u i )
1 2 uv∈E(G) i=1 − 2α−1 [dG (u)]α+2 − χα (G).
=
u∈V (G)
(ii) The edge e′ contributes the following quantity to Rα (L(S(G))). α s(e′ ) = d L(S(G)) (e′ ) d L(S(G)) ( f ) e′ f ̸∈ E(L(S(G)))
= (dG (u 1 ) dG (u 2 ))α (dG (u 2 ) − 1) + (dG (u 1 ) dG (u 3 ))α (dG (u 3 )) + · · · + (dG (u 1 ) dG (u n ))α (dG (u n )) α+1 α − (dG (u 1 ))α+2 − [dG (u 1 ) dG (u 2 )]α . (dG (u)) = (dG (u 1 )) u∈V (G)
Similarly e′′ contributes the following quantity to Rα (L(S(G))). ′′ α α+1 s(e ) = (dG (u 2 )) (dG (u)) − (dG (u 2 ))α+2 − [dG (u 2 )dG (u 1 )]α . u∈V (G)
Total contribution of an edge e to Rα (L(S(G))) is s(e) = s(e′ ) + s(e′′ ) = (dG (u 1 ))α + (dG (u 2 ))α − 2[dG (u 1 )dG (u 2 )]α .
α+1
(dG (u))
− (dG (u 1 ))α+2 + (dG (u 2 ))α+2
u∈V (G)
Therefore, Rα (L(S(G))) = =
1 s(e) 2 e∈E(G) 1 (dG (u))α+1 (dG (u 1 ))α + (dG (u 2 ))α 2 u∈V (G) e=u 1 u 2 ∈E(G) 1 − (dG (u 1 ))α+2 + (dG (u 2 ))α+2 − [dG (u 1 )dG (u 2 )]α 2 e=u u ∈E(G) e=u u ∈E(G) 1 2
1 2
1 1 = (dG (u))α+1 (dG (u))α+1 − (dG (u))α+3 − Rα (G) 2 u∈V (G) 2 u∈V (G) u∈V (G) 2 1 1 = [dG (u)]α+1 − [dG (u)]α+3 − Rα (G). 2 u∈V (G) 2 u∈V (G)
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Example 2.8. For a graph G given in Fig. 1, with α = 2, we get from Theorem 2.7(i) 4 1 2 2 χ2 (L(S(G))) = (dG (u) + dG (u i )) + (dG (v) + dG (u i )) dG (u i ) 2 uv∈E(G) i=1 −2 [dG (u)]4 − χ2 (G) u∈V (G)
1 = [(dG (u) + 1)2 + (dG (v) + 1)2 ] · 1 + [(dG (u) + 3)2 + (dG (v) + 3)2 ] · 3 2 uv∈E(G) + [(dG (u) + 2)2 + (dG (v) + 2)2 ] · 2 + [(dG (u) + 2)2 + (dG (v) + 2)2 ] · 2 − 2[1 + 81 + 16 + 1] − 82 1 2 = (2 + 42 ) + (42 + 62 ) · 3 + (32 + 52 ) · 2 + (32 + 52 ) · 2 2 + (42 + 32 ) + (62 + 52 ) · 3 + (52 + 42 ) · 2 + (52 + 42 ) · 2 + (42 + 32 ) + (62 + 52 ) · 3 + (52 + 42 ) · 2 + (52 + 42 ) · 2 + (32 + 32 ) + (52 + 52 ) · 3 + (42 + 42 ) · 2 + (42 + 42 ) · 2 − 2[1 + 81 + 16 + 1] − 82 = 366. The next Corollary follows from Theorem 2.7. Corollary 2.9 ([26]). Let Tn,k be the tadpole graph and Wn+1 be the wheel. Then (i) χα (L(S(Tn,k ))) = (2n 2 + 4nk + 2k 2 − 11n − 13k + 23) · 4α + (2n + 2k − 6) · 3α + (6n + 6k − 16) · 5α . (ii) χα (L(S(Wn+1 ))) = n(3n − 1)(n + 3)α + 2−1 n(9n − 11) · 6α . If α = 1, then χ1 (G) = M1 (G) and R1 (G) = M2 (G). Therefore, by Theorem 2.7 we have the following corollaries. Corollary 2.10. For any graph G,
(i) M1 (L(S(G))) = (2m − 1)M1 (G) −
[dG (u)]3
u∈V (G)
(ii) M2 (L(S(G))) =
1 1 [M1 (G)]2 − M2 (G) − [dG (u)]4 . 2 2 u∈V (G)
Corollary 2.11 ([25]). Let Tn,k be the tadpole graph, Wn+1 be the wheel, and L n be the ladder. Then 8(n + k)2 − 8(n + k + 2) + 2, when k > 1; (i) M1 (L(S(Tn,k ))) = 8n(n + 2k − 1) + 2(2k − 9), when k = 1 2 8(n + k) − 4(n + k + 6) − 1, when k > 1; (ii) M2 (L(S(Tn,k ))) = 8n(n 2 + 2nk + k) − 4n − 30, when k = 1 (iii) M1 (L(S(Wn+1 ))) = n(3n 2 + 35n − 36) (iv) M2 (L(S(Wn+1 ))) = 21 n(18n 2 + 75n − 99) (v) M1 (L(S(L n ))) = 108n 2 − 264n + 240 (vi) M2 (L(S(L n ))) = 162n 2 − 468n + 370.
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3. Conclusion Ranjini et al. [25] obtained the expression for the Zagreb indices and coindices of the line graph of the subdivision graph of tadpole graphs, wheel graphs, and ladder graphs. Su and Xu [26] investigated the general sum-connectivity index and general product-connectivity index of the line graph of subdivision graph of the tadpole graphs, wheel graphs, and ladder graphs. Here we obtained expressions for general sum-connectivity index, general product-connectivity index, general Zagreb index and coindices of the line graph of subdivision graph of any graph, which generalizes the results of Ranjini et al. [25] and Su and Xu [26]. Acknowledgment Authors H. S. Ramane and V. V. Manjalapur are grateful to the University Grants Commission (UGC), Govt. of India for support through research grant under UPE FAR-II Grant No. F 14-3/2012 (NS/PE). References [1] [2] [3] [4] [5] [6] [7] [8] [9] [10] [11] [12] [13] [14] [15] [16] [17] [18] [19] [20] [21] [22] [23] [24] [25] [26]
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