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Scholarly Research Journal for Interdisciplinary Studies, Online ISSN 2278-8808, SJIF 2021 = 7.380, www.srjis.com PEER REVIEWED & REFEREED JOURNAL, JULY-AUGUST, 2021, VOL- 9/66

ON MULTIPLICATIVE K BANHATTI INDICES OF LINE GRAPHS B. Manjunath Department of Mathematics, Governement First Grade College, Mulbagal, Kolar-563 101

Paper Received On: 21 JULY 2021 Peer Reviewed On: 31 JULY 2021 Published On: 1 SEPT 2021

Let G = (V,E) be a connected graph. The multiplicative K Banhatti indices of G are defined as BΠ∗(G) = Que[dG(u) ∗ dG(e)], where ∗ is usual addition or multiplication and ue means that the vertex u and edge e are incident in G. In this paper, we compute the multiplicative K Banhatti indices of line graphs.. Mathematics Subject Classification: 05C05, 05C07, 05C35. Keywords: Multiplicative K Banhatti indices; Line graph.

Scholarly Research Journal's is licensed Based on a work at www.srjis.com 1 Introduction

By a graph, we mean a finite, undirected without loops and multiple edges. Let G be a connected graph with vertex set V (G) and edge set E(G). The degree dG(v) of a vertex v is the number of vertices adjacent to v. The edge connecting the vertices u and v will be denoted by uv. Let dG(e) denotes the degree of an edge e in G, which is defined by dG(e) = dG(u) + dG(v) − 2 with e = uv. For definitions and notions, the reader may refer to [7]. A molecular graph is a graph such that its vertices correspond to the atoms and the edges to the bonds. In Chemical Science, the physico- chemical properties of chemical compounds are often modeled by means of molecular graph based structure descriptors, which are also referred to as topological indices, see [3]. The K Banhatti indices of G are defined as B∗(G) = Pue[dG(u) ∗ dG(e)], where ∗ is usual addition or multiplication and ue means that the vertex u and edge e are incident in G. If ∗ is addition, then first K Banhatti index B+(G) = B1(G) = Pue[dG(u) + dG(e)], and if ∗ is multiplication, then the second K Banhatti index B×(G) = B2(G) =

P ue[dG(u)

× dG(e)].

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Todeschini et al. [12] proposed that multiplicative variants of molecular structure descriptors be considered. When this idea is applied to Zagreb indices, for example, in [2]. The multiplicative version of first and second K Banhatti indices were introduced by Kulli in [10] and [11] as follows. The multiplicative K Banhatti indices of G are defined as BΠ∗(G) = Que[dG(u) ∗ dG(e)], where ∗ is usual addition or multiplication and ue means that the vertex u and edge e are incident in G. If ∗ is addition, then first multiplicative K Banhatti index BΠ+(G) = BΠ1(G) = Que[dG(u) + dG(e)], and if ∗ is multiplication, then the second multiplicative K Banhatti index BΠ×(G) = BΠ2(G) =

Q ue[dG(u)dG(e)].

Recently many other indices were

studied, for example, in [4] and [9]. The Line graph L(G) is the graph with vertex set V (L) = E(G) and whose vertices correspond to the edges of G with two vertices being adjacent if and only if the corresponding edges in G have a vertex in common two. For more details, we refer to [5]. 2 Results

Theorem 2.1 Let G be a r- regular graph with n ≥ 2 vertices. Then (i) BΠ1(L(G)) = [2(3r − 4)]nr(r−1), (ii) BΠ1(L(G)) = [4(r − 1)(2r − 3)]nr(r−1). Proof. Let G be a r- regular graph with n ≥ 2 vertices. By algebraic method, we have and graph is (2r − 2) - regular and BΠ∗(L(G)) =

1). Since line graph of a r - regular Q ue[dL(G)(u)

∗ dL(G)(e)]. Hence, we have

the following cases: Case 1. BΠ+(L(G)) = BΠ1(L(G)) = Que[dL(G)(u) + dL(G)(e)]

Case 2. BΠ×(L(G)) = BΠ2(L(G)) = Que[dL(G)(u) × dL(G)(e)]

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Theorem 2.2 Let G be a r- regular graph with n ≥ 2 vertices. Then

. Corollary 2.3 Let Cn be a cycle with n ≥ 3 vertices. Then BΠ1(L(Cn)) = BΠ2(L(Cn)) = 42n. Corollary 2.4 Let Kn be a complete graph with n ≥ 3 vertices. Then (i) BΠ1(L(Kn)) = (6n − 14)n(n−1)(n−2), (ii) BΠ1(L(Kn)) = [4(n − 2)(2n − 5)]n(n−1)(n−2). Theorem 2.5 Let Pn be a path with n ≥ 4 vertices. Then (i) BΠ1(L(Pn)) = 9 × 24n−14, (ii) BΠ1(L(Pn)) = 24n−14. Proof. Let Pn be a path with n ≥ 4 vertices. Since L(Pn) ∼= Pn−1. By algebraic method, we have |V (L(Pn))| = n − 1 and |E(L(Pn))| = n − 2. We have two partitions of the vertex set V (L(Pn)) as follows: V1 = {v ∈ V (L(Pn)) : dL(Pn)(v) = 1};|V1| = 2, and V2 = {v ∈ V (L(Pn)) : dL(Pn)(v) = 2};|V2| = n − 3. Also we have two partitions of the edge set E(L(Pn)) as follows: E1 = {uv ∈ E(L(Pn)) : dL(Pn)(u) = 1,dL(Pn)(v) = 2};|E1| = 2, and E2 = {uv ∈ E(L(Pn)) : dL(Pn)(u) = dL(Pn)(v) = 2};|E2| = n − 4. Then BΠ∗(L(Pn)) = Que[dL(Pn)(u) ∗ dL(Pn)(e)] Y [dL(Pn)(u) ∗ dL(Pn)(e)] + Y [dL(Pn)(u) ∗ dL(Pn)(e)]

uv∈E1

uv∈E2

We have the following two cases are arise: Case 1. BΠ+(L(Pn)) = BΠ1(L(Pn)) Y

[(1 + 1) × (2 + 1)] × Y [(2 + 2) × (2 + 2)]

uv∈E1

uv∈E2

= (2 × 3)2 × (4 × 4)n−4 = 9 × 24n−14. Case 2. BΠ×(L(Pn)) = BΠ2(L(Pn)) =

uv∈E1

[(1 × 1) × (2 × 1)] × Y [(2 × 2) × (2 × 2)] uv∈E2

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= (1 × 2)2 × (4 × 4)n−4 = 24n−14. Thus the result follows. By above Theorem, we have the following result without proof. Theorem 2.6 Let Pn be a path with n ≥ 4 vertices. Then BΠ1(L(Pn)) = 9 BΠ2(L(Pn)). Corollary 2.7 Let Cn be a cycle and Pn be a path with n ≥ 4 vertices. Then (i) BΠ1(L(Pn)) = 9 × 2−14BΠ1(L(Cn)), (ii) BΠ2(L(Pn)) = 2−14 BΠ2(L(Cn)). Theorem 2.8 Let Kr,s be a complete bipartite graph with 1 ≤ r ≤ s vertices. Then (i) BΠ1(L(Kr,s)) = [3r + 3s − 8]rs(r+s−2), (ii) BΠ2(L(Kr,s)) = [(r + s − 2)(2r + 2s − 6)]rs(r+s−2). Proof. Let Kr,s be a complete bipartite graph with 1 ≤ r ≤ s vertices. By algebraic method, we have |V (L(Kr,s))| = rs, and |E(L(Kr,s))| =. Since line graph of complete bipartite graph Kr,s is a (r+s−2)-regular graph and BΠ∗(L(Kr,s)) = Que[dL(Kr,s)(u) ∗ dL(Kr,s)(e)]. We have (i)

BΠ+(L(Kr,s)) = BΠ1(S(Kr,s))

(ii) BΠ×(L(Kr,s)) = BΠ2(L(Kr,s))

The following results are immediate from above theorem. Corollary 2.9 Let K1,s be a star graph with s ≥ 1 vertices. Then (i) BΠ∗(L(K1,s)) = BΠ∗(Ks), (ii) BΠ1(L(K1,s)) = (3s − 5)s(s−1), (iii) BΠ2(L(K1,s)) = [2(s − 1)(s − 2)]s(s−1), Corollary 2.10 Let Kr,r be a regular complete bipartite graph with r ≥ 2 vertices. Then Copyright © 2021, Scholarly Research Journal for Interdisciplinary Studies

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