The M-Polynomial of Line graph of Subdivision graphs Sourav Mondala , Nilanjan Deb , Anita Palc a
Department of mathematics, NIT Durgapur, India. Department of Basic Sciences and Humanities (Mathematics), Calcutta Institute of Engineering and Management, Kolkata, India. c Department of mathematics, NIT Durgapur, India. b
Abstract Three composite graphs Ladder graph (Ln ), Tadpole graph (T n,k )and Wheel graph (Wn ) are graceful graphs, which have different applications in electrical, electronics, wireless communication etc. In this report, we first determine M-polynomial of the Line graph of those three graphs using subdivision idea and then compute some degree based indices of the same. MSC (2010): Primary: 05C35; Secondary: 05C07, 05C40 Keywords: M-polynomial, Degree based topological index, Line graph, Subdivision graph. 1. Introduction Mathematical modellings play significant role to analyse important concepts in chemistry. Mathematical chemistry has excellent tools such as polynomials, functions which can predict properties of chemical compounds successfully. TopoP logical indices are functions T : → R+ with the property T (G) = T (H) for P every graph G isomorphic to H, where is the class of all graphs. These numerical quantity corresponding to a molecular graph are effective in correlating the structure with different physiochemical properties, chemical reactivity, and biological activities. These indices are evaluated by formal definitions. Instead of calculating different topological indices of a specific category [1, 2], we can use a compact general method related to polynomial for calculating the same. For instance, Wiener polynomial is a general polynomial in the field of distance-based Email addresses:
[email protected] (Sourav Mondal),
[email protected] (Nilanjan De),
[email protected] (Anita Pal)
topological indices whose derivatives at 1 yield Weiner and Hyper Weiner indices [2]. There are many such polynomials such as Pi polynomial [3], Theta polynomial [4] etc. In the area of degree-based topological indices, M-polynomial [1] perform similar role to compute closed expressions of many degree based topological indices [1, 5]. Thus computation of degree based topological indices reduce to evaluation of a single polynomial. Moreover, detailed analysis of this polynomial can yield new insights in the knowledge of degree based topological indices. The subdivision graph [6, 7, 8] S (G) is obtained by inserting an additional vertex to each edge of G [9]. The Line graph [10] L(G) is the simple graph whose vertices are the edges of G and e1 e2 ∈ E(L(G)) if e1 , e2 ∈ E(G) share a common end point in G. The Ladder graph Ln is obtained by the Cartesian product of the Path graph Pn and the complete graph K2 . The Tadpole graph T n,k [11] is obtained by joining a cycle Cn to a path Pk . Wheel graph is obtained by joining K1 to Cn . Ranjini et al.[12] investigate the Zagreb indices of the Line graphs of those three graphs using the subdivision concepts. G Su et al.[13] derived topological indices of the same graphs. Here we calculate the degree based topological indices of L(S (Ln )), L(S (T n,k )), L(S (Wn )) using M-polynomial. 2. Basic definitions and literature review Definition 1. Let G be a molecular graph. The M-polynomial of G is defined as, X M(G; x, y) = mi j (G)xi y j δ6i6 j6∆
Where δ =min{dv : v ∈ V(G)} and ∆ =max{dv : v ∈ V(G)}, and mi j (G) is the number of edges uv ∈ E(G) such that {du , dv } = {i, j}. Gutman and Trinajsti´c introduced Zagreb indices [14]. The first Zagreb index is defined as, X M1 (G) = (dv )2 v∈V(G)
The second Zagreb index is defined as, M2 (G) =
X uv∈E(G)
2
du dv .
For more details about this indices see. The second modified Zagreb index is defined as, X 1 m M2 (G) = dd uv∈E(G) u v Bollobas and Erdos [15] and Amic et al. [16] presented the idea of the generalized Randi´c index and discussed widely in both chemistry and mathematics [17]. For more discussion, readers are referred [19, 18]. The general Randi´c index is defined as, X Rα (G) = (du dv )α uv∈E(G)
The inverse Randi´c index is defined as, X RRα (G) =
1 (du dv )α uv∈E(G)
symmetric division index of a connected graph G, is defined as: X min(du , dv ) max(du , dv ) + }. S DD(G) = { max(d , d ) min(d , d ) u v u v uv∈E(G) The Harmonic index [20] is defined as, X H(G) =
2 . d + d u v uv∈E(G)
The inverse sum index [21] is given by: X I(G) =
du dv . d + d u v uv∈E(G)
The augmented Zagreb index of G proposed by Furtula et al. [22] is defined as, X du dv A(G) = { }3 . d + dv − 2 uv∈E(G) u The relations of some degree-based topological indices with the M-polynomial are shown in the table below.
3
Table 1: Derivation of some degree based topological indices Topological Index
f(x,y)
Derivation from M(G; x, y)
First Zagreb Index Second Zagreb Index Modified Second Zagreb Index Randi´c Index(α ∈ N) Inverse Randi´c Index(α ∈ N)
x+y xy
(D x + Dy )(M(G; x, y))| x=y=1 (D x Dy )(M(G; x, y))| x=y=1 (S x S y )(M(G; x, y))| x=y=1 (Dαx Dαy )(M(G; x, y))| x=y=1 (S αx S yα )(M(G; x, y))| x=y=1
Symmetric Division Index Harmonic Index Inverse sum Index Augumented Zagreb Index
1 xy
(xy)α 1 (xy)α x2 +y2 xy 2 x+y xy x+y xy (x+y−2)3
(D x S y + S x Dy )(M(G; x, y))| x=y=1 2S x J(M(G; x, y))| x=1 2S x JD x Dy (M(G; x, y))| x=1 2S 3x Q−2 JD3x D3y (M(G; x, y))| x=1
Where, D x ( f (x, y)) = x
S x ( f (x, y)) =
x
Z 0
∂( f (x, y)) ∂( f (x, y)) , Dy ( f (x, y)) = y , ∂x ∂y
∂( f (t, y)) dt, S y ( f (x, y)) = t
Z 0
y
∂( f (x, t)) dt t
J( f (x, y)) = f (x, x), Qα ( f (x, y)) = xα f (x, y).
3. Main Results In this part, we give our main computational results and divide the section in three subsections. 3.1. Computational aspects of the Line graph of subdivision of Ladder graph. We compute the M-polynomial of the line graph of subdivision graph of the Ladder graph in the following theorem. Theorem 1. Let L(S (Ln )) be the line graph of subdivision graph of the Ladder 4
Figure 1: The Line graph of subdivision graph of the Ladder graph. .
graph. Then we have, M(L(S (Ln ))) = 6x2 y2 + 4x2 y3 + (9n − 20)x3 y3 . Proof. From the figure of L(S (Ln )), we can say that its vertices have two partitions, |V{2} | = |{du ∈ V(L(S (Ln ))) : du = 2}| = 8. |V{3} | = |{u ∈ V(L(S (Ln ))) : du = 2}| = 6n − 12. By handshaking lemma, we have |E(L(S (Ln )))| = 9n−10. The edge set of L(S (Ln )) can be partitioned as, |E{2,2} | = |{uv ∈ E(L(S (Ln ))) : du = 2, dv = 2}| = 6. |E{2,3} | = |{uv ∈ E(L(S (Ln ))) : du = 2, dv = 3}| = 4. |E{3,3} | = |{uv ∈ E(L(S (Ln ))) : du = 3, dv = 3}| = 9n − 20. From the definition, the M-polynomial of L(S (Ln )) is obtained bellow, X M(L(S (Ln ))) = mi j (L(S (Ln )))xi y j i6 j
5
=
X
mi j (L(S (Ln )))x2 y2 +
262
+
X
X
mi j (L(S (Ln )))x2 y3 +
263
mi j (L(S (Ln )))x3 y3
363
= |E{2,2} |x2 y2 + |E{2,3} |x2 y3 + |E{3,3} |x3 y3 = 6x2 y2 + 4x2 y3 + (9n − 20)x3 y3 . This completes the proof. Now using this M-polynomial, we calculate some degree based topological index of the line graph of subdivision of the Ladder graph in the following theorem. Theorem 2. Let L(S (Ln )) be the line graph of subdivision of the Ladder graph. Then we have, 1. 2. 3. 4. 5. 6. 7. 8. 9.
M1 (L(S (Ln ))) = 54n − 76. M2 (L(S (Ln ))) = 81n − 132. M2m (L(S (Ln ))) = 18n−1 . 18 α Rα (L(S (Ln ))) = 6(4) + 4(6)α + (9n − 20)(9)α . RRα (L(S (Ln ))) = 46α + 64α + 9n−20 . 9α 58 S DD(L(S (Ln ))) = 18n − 3 . H(L(S (Ln ))) = 3n − 31 . 15 27n+48 I(L(S (Ln ))) = 10 . A(L(S (Ln ))) = 6561n−9460 . 64
Proof. Let M(L(S (Ln )); x, y) = f (x, y) = 6x2 y2 + 4x2 y3 + (9n − 20)x3 y3 . Then we have, D x ( f (x, y)) = 12x2 y2 + 8x2 y3 + 3(9n − 20)x3 y3 Dy ( f (x, y)) = 12x2 y2 + 12x2 y3 + 3(9n − 20)x3 y3 . D x Dy ( f (x, y)) = 24x2 y2 + 24x2 y3 + 9(9n − 20)x3 y3 . 9n − 20 3 3 S x ( f (x, y)) = 3x2 y2 + 2x2 y3 + xy. 3 4 9n − 20 3 3 S y ( f (x, y)) = 3x2 y2 + x2 y3 + xy. 3 3 3 2 2 2 2 3 9n − 20 3 3 S x S y ( f (x, y)) = xy + xy + xy. 2 3 9 Dαx Dαy ( f (x, y)) = 6(4α )x2 y2 + 4(6α )x2 y3 + (9n − 20)(9α )x3 y3 . 6
S αx S yα ( f (x, y)) = D x S y ( f (x, y)) = S x Dy ( f (x, y)) = S x J( f (x, y)) = S x JD x Dy ( f (x, y)) = S 3x Q−2 JD3x D3y ( f (x, y)) =
6 2 2 4 2 3 9n − 20 3 3 x y + αx y + xy. 4α 6 9α 8 6x2 y2 + x2 y3 + (9n − 20)x3 y3 . 3 2 2 6x y + 6x2 y3 + (9n − 20)x3 y3 . 24x4 y2 + 20x5 + 6(9n − 20)x6 . 24 3(9n − 20) 6 6x4 + x5 + x. 5 2 729 (9n − 20)x4 . 48x2 + 32x3 + 64
Using table 1.we have, 1. 2. 3. 4. 5. 6. 7. 8. 9.
M1 (L(S (Ln ))) = (D x + Dy )( f (x, y))| x=y=1 = 54n − 76. M2 (L(S (Ln ))) = (D x Dy )( f (x, y))| x=y=1 = 81n − 132. M2m (L(S (Ln ))) = (S x S y )( f (x, y))| x=y=1 = 18n−1 . 18 Rα (L(S (Ln ))) = (Dαx Dαy )( f (x, y))| x=y=1 = 6(4)α + 4(6)α + (9n − 20)(9)α . RRα (L(S (Ln ))) = (S αx S yα )( f (x, y))| x=y=1 = 46α + 64α + 9n−20 . 9α . S DD(L(S (Ln ))) = (D x S y + S x Dy )( f (x, y))| x=y=1 = 18n − 58 3 31 H(L(S (Ln ))) = 2(S x )( f (x, y))| x=1 = 3n − 15 . I(L(S (Ln ))) = S x JD x Dy ( f (x, y))| x=1 = 27n+48 . 10 6561n−9460 3 3 3 . A(L(S (Ln ))) = S x Q−2 JD x Dy ( f (x, y))| x=1 = 64
This completes the proof.
3.2. Computational aspects of the Line graph of subdivision of Tadpole graph. We compute the M-polynomial of the line graph of subdivision graph of the Tadpole graph in the following theorem. Theorem 3. Let L(S (T n,k )) be the line graph of subdivision graph of the Tadpole graph. Then we have, M(L(S (T n,k ))) = xy2 + 2(n + k − 3)x2 y2 + 3x2 y3 + 3x3 y3 . Proof. The graph of L(S (T n,k ) has 2(n + k) vertices, having three partitions of vertices, |V{1} | = |{u ∈ V(L(S (Ln ))) : du = 1}| = 1. |V{2} | = |{u ∈ V(L(S (Ln ))) : du = 2}| = 2n + 2k − 4. 7
Figure 2: The Line graph of subdivision graph of the Tadpole graph. . |V{3} | = |{u ∈ V(L(S (Ln ))) : du = 3}| = 3. This shows that L(S (T n,k ) has 9n-10 edges. The edge set of L(S (T n,k )) has partitions as follows, |E{1,2} | = |{uv ∈ E(L(S (T n,k ))) : du = 1, dv = 2}| = 1. |E{2,2} | = |{uv ∈ E(L(S (T n,k ))) : du = 2, dv = 2}| = 2(n + k − 3). |E{2,3} | = |{uv ∈ E(L(S (T n,k ))) : du = 2, dv = 3}| = 3. |E{3,3} | = |{uv ∈ E(L(S (T n,k ))) : du = 3, dv = 3}| = 3. From the definition, the M-polynomial of L(S (T n,k )) is obtained bellow, X M(L(S (T n,k ))) = mi j (L(S (T n,k )))xi y j i6 j
=
X
mi j (L(S (T n,k )))x1 y2 +
162
+
X
X
mi j (L(S (T n,k )))x2 y2 +
262 2 3
mi j (L(S (T n,k )))x y
263
+
X
mi j (L(S (T n,k )))x3 y3
363
= |E{1,2} |xy2 + |E{2,2} |x2 y2 + |E{2,3} |x2 y3 + |E{3,3} |x3 y3 = xy2 + 2(n + k − 3)x2 y2 + 3x2 y3 + 3x3 y3 . 8
This completes the proof. Now using this M-polynomial, we calculate some degree based topological index of the Line graph of subdivision of the Tadpole graph in the following theorem. Theorem 4. Let L(S (T n,k )) be the line graph of subdivision of the Ladder graph. Then we have, 1. 2. 3. 4. 5. 6. 7. 8. 9.
M1 (L(S (T n,k ))) = 8n + 8k + 12. M2 (L(S (T n,k ))) = 8n + 8k + 23. . M2m (L(S (T n,k ))) = 3n+3k−1 6 α Rα (L(S (T n,k ))) = 2 + (n + k − 3)22α+1 + 3(6α ) + 32α+1 . 1 RRα (L(S (T n,k ))) = 21α + n+k−3 + 63α + 32α−1 . 22α−1 S DD(L(S (T n,k ))) = 4n + 4k − 1. 2 . H(L(S (T n,k ))) = n + k − 15 17 I(L(S (T n,k ))) = 2n + 2k − 30 . . A(L(S (T n,k ))) = 16n + 16k + 1163 64
Proof. Let M(L(S (T n,k )); x, y) = f (x, y) = xy2 + 2(n + k − 3)x2 y2 + 3x2 y3 + 3x3 y3 . Then we have, D x ( f (x, y)) = xy2 + 4(n + k − 3)x2 y2 + 6x2 y3 + 9x3 y3 . Dy ( f (x, y)) = 2xy2 + 4(n + k − 3)x2 y2 + 9x2 y3 + 9x3 y3 . D x Dy ( f (x, y)) = 2xy2 + 8(n + k − 3)x2 y2 + 18x2 y3 + 27x3 y3 3 S x ( f (x, y)) = xy2 + (n + k − 3)x2 y2 + x2 y3 + x3 y3 . 2 xy2 S y ( f (x, y)) = + (n + k − 3)x2 y2 + x2 y3 + x3 y3 2 xy2 (n + k − 3) 2 2 x2 y3 x3 y3 S x S y ( f (x, y)) = + xy + + . 2 2 2 3 Dαx Dαy ( f (x, y)) = 2α xy2 + (n + k − 3)(22α+1 )x2 y2 + 3(6α )x2 y3 + 32α+1 x3 y3 . xy2 (n + k − 3) 2 2 3 2 3 1 + x y + α x y + 2α−1 x3 y3 . α 2α−1 2 2 6 3 xy2 D x S y ( f (x, y)) = + 2(n + k − 3)x2 y2 + 2x2 y3 + 3x3 y3 . 2 9 S x Dy ( f (x, y)) = 2xy2 + 2(n + k − 3)x2 y2 + x2 y3 + 3x3 y3 . 2
S αx S yα ( f (x, y)) =
9
Figure 3: The Line graph of subdivision graph of the wheel graph. . 1 3 n+k−3 4 3 5 1 6 x + x + x + x. 3 2 5 2 2 3 18 27 S x JD x Dy ( f (x, y)) = x + 2(n + k − 3)x4 + x5 + x6 . 3 5 6 2187 x4 . S 3x Q−2 JD3x D3y ( f (x, y)) = 8x + 16(n + k − 3)x2 + 24x3 + 64 Using table 1.we have, S x J( f (x, y)) =
1. 2. 3. 4. 5. 6. 7. 8. 9.
M1 (L(S (T n,k ))) = (D x + Dy )( f (x, y))| x=y=1 = 8n + 8k + 12. M2 (L(S (T n,k ))) = (D x Dy )( f (x, y))| x=y=1 = 8n + 8k + 23. . M2m (L(S (T n,k ))) = (S x S y )( f (x, y))| x=y=1 = 3n+3k−1 6 α α α Rα (L(S (T n,k ))) = (D x Dy )( f (x, y))| x=y=1 = 2 +(n+k −3)22α+1 +3(6α )+32α+1 . 1 RRα (L(S (T n,k ))) = (S αx S yα )( f (x, y))| x=y=1 = 21α + n+k−3 + 63α + 32α−1 . 22α−1 S DD(L(S (T n,k ))) = (D x S y + S x Dy )( f (x, y))| x=y=1 = 4n + 4k − 1. 2 H(L(S (T n,k ))) = 2(S x )( f (x, y))| x=1 = n + k − 15 . 17 I(L(S (T n,k ))) = S x JD x Dy ( f (x, y))| x=1 = 2n + 2k − 30 . 3 3 3 A(L(S (T n,k ))) = S x Q−2 JD x Dy ( f (x, y))| x=1 = 16n + 16k + 1163 . 64
This completes the proof.
3.3. Computational aspects of the Line graph of subdivision of Wheel graph. We compute the M-polynomial of the line graph of subdivision graph of the Wheel graph in the following theorem. 10
Theorem 5. Let L(S (Wn+1 )) be the line graph of subdivision graph of the Wheel graph. Then we have, M(L(S (Wn+1 ))) = 4nx3 y3 + nx3 yn +
n(n − 1) n n x y + 3x3 y3 . 2
Proof. It is easy to see that the order of the graph L(S (Wn+1 ) is 4n out of which 3n vertices are of degree 3 and n vertices are of degree n. By handshaking lemma, 2 the size of the graph L(S (Wn+1 )) is n +9n . The edge set of L(S (Wn+1 )) can be 2 partitioned as, |E{3,3} | = |{uv ∈ E(L(S (Wn+1 ))) : du = 3, dv = 3}| = 4n. |E{n,3} | = |{uv ∈ E(L(S (Wn+1 ))) : du = n, dv = 3}| = n. |E{n,n} | = |{uv ∈ E(L(S (Wn+1 ))) : du = n, dv = n}| =
n(n − 1) . 2
From the definition, the M-polynomial of L(S (Wn+1 )) is obtained bellow, X M(L(S (Wn+1 ))) = mi j (L(S (Wn+1 )))xi y j i6 j
=
X
mi j (L(S (Wn+1 )))x3 y3 +
363
+
X
X
mi j (L(S (Wn+1 )))x3 yn +
36n n n
mi j (L(S (Wn+1 )))x y
n6n
= |E{3,3} |x3 y3 + |E{3,n} |x3 yn + |E{n,n} |xn yn n(n − 1) n n = 4nx3 y3 + nx3 yn + x y + 3x3 y3 . 2 This completes the proof. Now using this M-polynomial, we calculate some degree based topological index of the Line graph of subdivision of the Wheel graph in the following theorem. Theorem 6. Let L(S (Wn+1 )) be the line graph of subdivision of the Wheel graph. Then we have, 1. M1 (L(S (Wn+1 ))) = n3 + 27n. 11
n(n3 −n2 +6n+72) . 2 8n2 +15n−9 = 18n . 2α
2. M2 (L(S (Wn+1 ))) = 3. M2m (L(S (Wn+1 )))
4. Rα (L(S (Wn+1 ))) = 4n(3 ) + n(3n)α + . 5. RRα (L(S (Wn+1 ))) = 34n2α + 3nnα + n(n−1) 2(n2α ) 4n2 +21n+9 . 3 11n2 +42n−9 H(L(S (Wn+1 ))) = 6(n+3) . 4 3 +33n2 +72n I(L(S (Wn+1 ))) = n +2n4(n+3) . 7 n 27n4 A(L(S (Wn+1 ))) = 16(n−1) 2 + (n+1)3
n2α+1 (n−1) . 2
6. S DD(L(S (Wn+1 ))) = 7. 8. 9.
+
729n . 16
Proof. Let M(L(S (Wn+1 )); x, y) = f (x, y) = 4nx3 y3 + nx3 yn + Then we have, D x ( f (x, y)) = Dy ( f (x, y)) = D x Dy ( f (x, y)) = S x ( f (x, y)) = S y ( f (x, y)) = S x S y ( f (x, y)) = Dαx Dαy ( f (x, y)) = S αx S yα ( f (x, y)) = D x S y ( f (x, y)) = S x Dy ( f (x, y)) = S x J( f (x, y)) = S x JD x Dy ( f (x, y)) =
n(n−1) n n xy 2
+ 3x3 y3 .
n2 (n − 1) n n xy. 12nx y + 3nx y + 2 n2 (n − 1) n n xy. 12nx3 y3 + n2 x3 yn + 2 n3 (n − 1) n n 3 3 2 3 n 36nx y + 3n x y + xy 2 4n 3 3 n 3 n n − 1 n n xy + xy + xy. 3 3 2 4n 3 3 n−1 n n x y + x3 yn + xy 3 2 4n 3 3 x3 yn n − 1 n n xy + + xy. 3 3 2n n2α+1 (n − 1) n n 4n(32α )x3 y3 + n(3n)α x3 yn + xy. 2 4n 3 3 n 3 n n(n − 1) n n xy + xy + xy. 2α 3 (3n)α 2n2α n(n − 1) n n 4nx3 y3 + 3x3 yn + xy. 2 n2 n(n − 1) n n 4nx3 y3 + x3 yn + xy. 3 2 2n 6 n n+3 n − 1 2n x + x + x . 3 n+3 4 3n2 n+3 n2 (n − 1) 2n 6 6nx + x + x . n+3 4 3 3
12
3 n
S 3x Q−2 JD3x D3y ( f (x, y)) =
27n4 n+1 n7 729n 4 x + x + x2n−2 . 16 (n + 1)3 16(n − 1)2
Using table 1.we have, 1. M1 (L(S (Wn+1 ))) = (D x + Dy )( f (x, y))| x=y=1 = n3 + 27n. 2. M2 (L(S (Wn+1 ))) = (D x Dy )( f (x, y))| x=y=1 = 3. M2m (L(S (Wn+1 ))) = (S x S y )( f (x, y))| x=y=1 =
n(n3 −n2 +6n+72) . 2 2 8n +15n−9 . 18n 2α
4. Rα (L(S (Wn+1 ))) = (Dαx Dαy )( f (x, y))| x=y=1 = 4n(3 ) + n(3n)α + 5. RRα (L(S (Wn+1 ))) = (S αx S yα )( f (x, y))| x=y=1 = 34n2α + 3nnα + n(n−1) . 2(n2α ) 6. S DD(L(S (Wn+1 ))) = (D x S y + S x Dy )( f (x, y))| x=y=1 =
n2α+1 (n−1) . 2
4n2 +21n+9 . 3
11n2 +42n−9 . 6(n+3) 4 3 +33n2 +72n . I(L(S (Wn+1 ))) = S x JD x Dy ( f (x, y))| x=1 = n +2n4(n+3) 7 27n4 n A(L(S (Wn+1 ))) = S 3x Q−2 JD3x D3y ( f (x, y))| x=1 = 16(n−1) 2 + (n+1)3
7. H(L(S (Wn+1 ))) = 2(S x )( f (x, y))| x=1 = 8. 9.
+
729n . 16
This completes the proof. 4. Conclusion
we obtained many topological indices for line graph of Ladder graph, Tadpole graph, Wheel graph using subdivision concept.Firstly, we comuted M-polynomial of these graphs and later recovered many degree-based topological indices applying it. These results can play an important role in Electrical, Electronics, Wireless communication, Cryptography etc.
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