Ratio Mathematica Volume 44, 2022 Some Graph Parameters of Clique graph of Cyclic Subgroup graph on certain Non- Abelian Groups S. Ragha* R. Rajeswari† Abstract The aim of this paper is to examine various graph parameters of clique graph of cyclic subgroup graph on certain non-abelian groups and also we obtain some theorems and results in detail. Keywords: Cyclic Subgroup graph, Clique graph, Hub number, Topological Indices 2010 AMS subject classification: 05C09,05C25,05C12,05C50‡ *Research Scholar (Reg. No. 20212012092008), PG & Research Department of Mathematics, A.P.C Mahalaxmi College for Women, Thoothukudi-628002, Tamil Nadu, India. Affiliated to Manonmaniam Sundaranar University, Abishekapatti, Tirunelveli-627012, Tamil Nadu, India.raghasankar810@gmail.com †Assistant Professor, PG &Research Department of Mathematics, A.P.C Mahalaxmi College for Women, Thoothukudi-628002, Tamil Nadu, India. rajimuthuram@gmail.com ‡Received on June 17th, 2022. Accepted on Sep 1st, 2022. Published on Nov 30th, 2022. doi: 10.23755/rm.v44i0.927. ISSN: 1592-7415. eISSN: 2282-8214. ©The Authors.This paper is published under the CC-BY licence agreement. 371 S. Ragha and R. Rajeswari 1. Introduction Algebraic Graph theory is a branch of mathematics in which graphs are constructed from the algebraic structures such as groups, rings etc. J. John Arul Singh and S. Devi [3] have introduced the notion of Cyclic Subgroup Graph of a finite group. The concept of Clique graphs was discussed at least as early as 1968 by Hamelink and Ronald. C. After that, Roberts and Spencer have given the concept of A Characterization of Clique Graphs [2] in 1971. Gutman introduced the concept of Energy in 1978. A topological index of a graph G is a numerical parameter mathematically derived from the graph structure. It is a graph invariant which does not depend on the labeling or pictorial representation of the graph and it is the graph invariant number calculated from a graph representing a molecule. Our present work is provoked by the above study. In section 2, we discuss some graph parameters for the clique graph of cyclic subgroup graph on certain non-abelian groups. In section 3, we examine some energies on clique graph of cyclic subgroup graph for dihedral group and generalised quaternion group. In section 4, we give some topological indices for the clique graph of cyclic subgroup graph on certain non-abelian groups. In this paper, 𝑝 represents prime number and 𝑝𝑞 represents distinct primes where 𝑞 > 𝑝. Before entering, let us look into some necessary definitions and notations. The cyclic subgroup graphΓ𝑧 (𝐺) for a finite group 𝐺 is a simple undirected graph in which the cyclic subgroups are vertices and two distinct subgroups are adjacent if one of them is a subset of the other. The clique graph𝒦(𝐺) of an undirected simple graph 𝐺, is a graph with a node for each maximal cliques in 𝐺. Two vertices in 𝒦(𝐺) are adjacent when their corresponding maximal cliques in 𝐺 share at least one vertex in common. For an integer 𝑛 ≥ 3, the dihedral group𝐷2𝑛 of order 2𝑛 is 𝐷2𝑛 =< 𝑟, 𝑓: 𝑟 𝑛 = 𝑓 2 = 1, 𝑓𝑟𝑓 = 𝑟−1 >. The generalized quaternion group 𝑄4𝑛 =< 𝑎, 𝑏: 𝑏 2 = 𝑎𝑛, 𝑎2𝑛 = 𝑒, 𝑏𝑎𝑏−1 = 𝑎−1 >, where 𝑒 is the identity element.A hub set in a graph 𝐺 is a set 𝐻 of vertices in 𝐺, such that any two vertices outside 𝐻 are connected by a path whose all internal vertices lie in 𝐻. The hub number of 𝐺, denoted by ℎ(𝐺), is the minimum cardinality of a hub set in 𝐺. Let G be a simple undirected graph, note that 𝑢𝑖 ~𝑢𝑗 denotes that 𝑢𝑖 is adjacent to 𝑢𝑗 for 1 ≤ 𝑖 ≠ 𝑗 ≤ 𝑛. The adjacency matrix of G denoted by 𝐴(𝐺) = (𝑎𝑖𝑗 ) is an 𝑛 × 𝑛 matrix defined as 𝑎𝑖𝑗 = 1 when 𝑢𝑖 ~𝑢𝑗 and 0 otherwise. The sum of the absolute values of the eigen values of its adjacency matrix is called the energy i.e., 𝐸(𝐺) = ∑ |𝜆𝑖 | 𝑛 𝑖=1 . The closed neighborhood matrix𝑁 = [𝑛𝑖,𝑗 ] = 𝐴 + 𝐼𝑛 has 𝑛𝑖,𝑗 = 1 if and only if 𝑢𝑖 ∈ 𝑁[𝑢𝑗 ]. The sum of the absolute values of the eigen values of its closed neighborhood matrix is called the closed neighborhood energy. The Laplacian matrix and Signless Laplacian matrix is defined as 𝐿(𝐺) = 𝐷(𝐺) − 𝐴(𝐺) and 𝑆𝐿(𝐺) = 𝐷(𝐺) + 𝐴(𝐺) where 𝐴(𝐺) is the adjacency matrix and 𝐷(𝐺) is the diagonal matrix of vertex degrees. 372 Some Graph Parameters of Clique graph of Cyclic Subgroup graph on certain Non-Abelian Groups 2. Some Graph Parameters of Clique graph of Cyclic Subgroup graph on certain Non-Abelian Groups Theorem 2.1: If 𝒦(Γ𝑧(𝐺)) be a clique graph of cyclic subgroup graph on a non-abelian group, then 𝒦(Γ𝑧 (𝐺)) is biconnected. Proof: Let 𝒦(Γ𝑧(𝐺)) be the clique graph of cyclic subgroup graph on a non-abelian group. Now, 𝑊1, 𝑊2, … , 𝑊𝑛 be the maximal cliques of Γ𝑧 (𝐺). By the definition of clique graph, 𝑉(𝑈) = 𝑊1, 𝑊2, … , 𝑊𝑛 and (𝑊𝑖 , 𝑊𝑗 ) ∈ 𝐸(𝑈) if only if 𝑖 ≠ 𝑗 and 𝑊𝑖 ∩ 𝑊𝑗 ≠ ∅ and take 𝑈 = 𝒦(Γ𝑧(𝐺)). For Γ𝑧 (𝐺), there exists a universal vertex which is adjacent to rest of its vertices. Now, any two vertices in 𝒦(Γ𝑧(𝐺)) are adjacent only when their corresponding maximal cliques in Γ𝑧 (𝐺) have atleast one vertex in common. It is clear that, there is a path in between every starting vertex and ending vertex. Even after removing any vertex, the graph remains connected. Now concluding that 𝐺 is connected and it does not contain any articulation point which results to a biconnected graph. Theorem 2.2: For a clique graph of cyclic subgroup graph on any non-abelian group, the hub number is 0. Proof: Based on the proof of 2.1, for 𝒦(Γ𝑧 (𝐺)), every pair of vertices are adjacent. Hence, there does not exist an intermediate vertex lies in the hub set. In this case, the minimum hub set is a null set. Hence, ℎ (𝒦(Γ𝑧 (𝐺))) = 0. Theorem 2.3: Let 𝒢 = 𝒦(Γ𝑧(𝐺)) be a clique graph of cyclic subgroup graph on any non-abelian group and |𝑉(𝒢)| = 𝑚. Then ℊ(𝒢) = 𝓂(𝒢) = 𝑚. Proof: Let 𝒢 = 𝒦(Γ𝑧(𝐺)) be a clique graph of cyclic subgroup graph on any non- abelian group. The geodetic closure of a vertex set 𝑆 ⊂ 𝒱 is the set of all vertices 𝑦 ∈ 𝒱which lies in some geodesic in 𝒢 joining two vertices 𝑢 and 𝑣 of 𝑆. Clearly for 𝒦(Γ𝑧 (𝐺)), there exists 𝑚 maximal cliques which is connected by at least one vertex in common. Now, the resulting graph consists of 𝑚 independent vertices in it. Hence ℊ(𝒢) = 𝑚. Consider, a set 𝐷 of vertices of 𝒦(Γ𝑧(𝐺)) is a monophonic set of 𝒦(Γ𝑧(𝐺)), if each vertex 𝑣 ∈ 𝒦(Γ𝑧(𝐺)) lies on an 𝑢 − 𝑤 monophonic path in 𝒦(Γ𝑧 (𝐺)) for some 𝑢, 𝑤 ∈ 𝐷 and the minimum cardinality of a monophonic set of 𝒦(Γ𝑧 (𝐺)), 𝓂(𝒢) = 𝑚. Theorem 2.4: For any non-abelian group, 𝒦(Γ𝑧(𝐺)) is non-planar. Proof directly follows from theorem 2.1. Theorem 2.5: Let 𝒢 = 𝒦(Γ𝑧(𝐺)) be a clique graph of cyclic subgroup graph on any non-abelian group and |𝑉(𝒢)| = 𝑚. Then 𝜅(𝒢) = 𝑚 − 1. Proof: For 𝒦(Γ𝑧 (𝐺)), by removing 𝑚 − 1 vertices which makes the graph disconnected. 373 S. Ragha and R. Rajeswari Theorem 2.6: The independence number of 𝒦(Γ𝑧 (𝐺)) is 1. Proof follows from direct computation. Theorem 2.7: For any non-abelian group, ℎ (𝒦(Γ𝑧 (𝐺))) ≠ 𝛾 (𝒦(Γ𝑧(𝐺))). Proof: Let 𝒦(Γ𝑧(𝐺)) be a clique graph of cyclic subgroup graph on any non-abelian group. Here, every pair of vertices are adjacent. Choose any one vertex 𝑢1 ∈ 𝒦(Γ𝑧(𝐺)) as a dominating set, which is adjacent to all other vertices. Hence, the domination number is 1. By theorem 2.2, this theorem can be proved. 3.Various Graph Energies on 𝓚(𝚪𝒛(𝑫𝟐𝒏)) and 𝓚(𝚪𝒛(𝑸𝟒𝒏)) Theorem 3.1: The adjacency energy on clique graph of cyclic subgroup graph for a dihedral group of order 2𝑛, 𝑛 ∈ ℕ and 𝑛 > 2 is 𝐸 (𝒦(Γ𝑧 (𝐷2𝑛))) = { 2𝑛 𝑖𝑓 𝑛 = 𝑝, 𝑝2 2(𝑛 + 1) 𝑖𝑓 𝑛 = 𝑝𝑞 Proof: Case i: For 𝑛 = 𝑝, 𝑝2 The vertex set of 𝒦(Γ𝑧(𝐷2𝑛)) = {𝑢1, 𝑢2, … , 𝑢𝑛+1}. The Adjacency matrix can be written as 𝐴 = 𝐽𝑛+1 − 𝐼𝑛+1. The obtained characteristic polynomial is (𝑥 − 𝑛)(𝑥 + 1)𝑛 . The spectrum of 𝒦(Γ𝑧 (𝐷2𝑛))will be written as, 𝑠𝑝𝑒𝑐 (𝒦(Γ𝑧(𝐷2𝑛))) = { 𝑛 1 −1 𝑛 }. For 𝑛 = 𝑝, 𝑝2, 𝐸 (𝒦(Γ𝑧(𝐷2𝑛))) = 2𝑛 Case ii: For 𝑛 = 𝑝𝑞 The vertex set of 𝒦(Γ𝑧(𝐷2𝑛)) = {𝑢1, 𝑢2, … , 𝑢𝑛+2}. The Adjacency matrix can be written as 𝐴 = 𝐽𝑛+2 − 𝐼𝑛+2. The obtained characteristic polynomial is (𝑥 − (𝑛 + 1))(𝑥 + 1)𝑛+1. The spectrum of 𝒦(Γ𝑧 (𝐷2𝑛))will be written as, 𝑠𝑝𝑒𝑐 (𝒦(Γ𝑧(𝐷2𝑛))) = { 𝑛+1 1 −1 𝑛+1 }. For 𝑛 = 𝑝𝑞, 𝐸 (𝒦(Γ𝑧 (𝐷2𝑛))) = 2(𝑛 + 1). Theorem 3.2: If 𝑛 = 𝑝, then the eigen values of 𝐴 (𝒦(Γ𝑧 (𝑄4𝑛))) are 𝑛 + 1 with multiplicity 1 &−1 with multiplicity 𝑛 + 1 and 𝐸 (𝒦(Γ𝑧 (𝑄4𝑛))) = 2(𝑛 + 1). Proof: The vertex set of 𝒦(Γ𝑧(𝑄4𝑛)) = {𝑢1, 𝑢2, … , 𝑢𝑛+2}. The Adjacency matrix can be written as 𝐴 = 𝐽𝑛+2 − 𝐼𝑛+2. The obtained characteristic polynomial is −(𝑥 − (𝑛 + 1))(𝑥 + 1)𝑛+1. The spectrum of 𝒦(Γ𝑧 (𝑄4𝑛))will be written as, 𝑠𝑝𝑒𝑐 (𝒦(Γ𝑧(𝑄4𝑛))) = { 𝑛+1 1 −1 𝑛+1 }. For 𝑛 = 𝑝, 𝐸 (𝒦(Γ𝑧 (𝑄4𝑛))) = 2(𝑛 + 1). 374 Some Graph Parameters of Clique graph of Cyclic Subgroup graph on certain Non-Abelian Groups Theorem 3.3: The closed neighborhood energy on clique graph of cyclic subgroup graph for a dihedral group of order 2𝑛, 𝑛 ∈ ℕ and 𝑛 > 2 is 𝐸𝑁 (𝒦(Γ𝑧 (𝐷2𝑛))) = { 𝑛 + 1 𝑖𝑓 𝑛 = 𝑝, 𝑝2 𝑛 + 2 𝑖𝑓 𝑛 = 𝑝𝑞 Proof: Case i: For 𝑛 = 𝑝, 𝑝2 The vertex set of 𝒦(Γ𝑧(𝐷2𝑛)) = {𝑢1, 𝑢2, … , 𝑢𝑛+1}. The Closed Neighborhood matrix can be written as 𝑁 = 𝐽𝑛+1. The obtained characteristic polynomial is (𝑥 − (𝑛 + 1))𝑥𝑛. The Closed Neighborhood spectrum of 𝒦(Γ𝑧(𝐷2𝑛))will be written as, 𝑠𝑝𝑒𝑐 (𝒦(Γ𝑧 (𝐷2𝑛))) = { 𝑛+1 1 0 𝑛 }. For 𝑛 = 𝑝, 𝑝2, 𝐸𝑁 (𝒦(Γ𝑧(𝐷2𝑛))) = 𝑛 + 1. Case ii: For 𝑛 = 𝑝𝑞 The vertex set of 𝒦(Γ𝑧(𝐷2𝑛)) = {𝑢1, 𝑢2, … , 𝑢𝑛+2}. The Closed Neighborhood matrix can be written as 𝑁 = 𝐽𝑛+2. The obtained characteristic polynomial is (𝑥 − (𝑛 + 2))𝑥𝑛+1. The Closed Neighborhood spectrum of 𝒦(Γ𝑧(𝐷2𝑛))will be written as, 𝑠𝑝𝑒𝑐 (𝒦(Γ𝑧 (𝐷2𝑛))) = { 𝑛+2 1 0 𝑛+1 }. For 𝑛 = 𝑝𝑞, 𝐸𝑁 (𝒦(Γ𝑧 (𝐷2𝑛))) = 𝑛 + 2. Theorem 3.4: If 𝑛 = 𝑝, then the eigen values of 𝑁 (𝒦(Γ𝑧 (𝑄4𝑛))) are 𝑛 + 2 with multiplicity 1 &0 with multiplicity 𝑛 + 1 and 𝐸𝑁 (𝒦(Γ𝑧 (𝑄4𝑛))) = 𝑛 + 2. Proof: The vertex set of 𝒦(Γ𝑧(𝑄4𝑛)) = {𝑢1, 𝑢2, … , 𝑢𝑛+2}. The Closed Neighbourhood matrix can be written as 𝑁 = 𝐽𝑛+2. The obtained characteristic polynomial is −(𝑥 − (𝑛 + 2))𝑥𝑛+1. The Closed Neighborhood spectrum of 𝒦(Γ𝑧(𝑄4𝑛))will be written as, 𝑠𝑝𝑒𝑐 (𝒦(Γ𝑧 (𝑄4𝑛))) = { 𝑛+2 1 0 𝑛+1 }. For 𝑛 = 𝑝, 𝐸𝑁 (𝒦(Γ𝑧 (𝑄4𝑛))) = 𝑛 + 2. Theorem 3.5: The Laplacian spectrum on clique graph of cyclic subgroup graph for a dihedral group of order 2𝑛, 𝑛 ∈ ℕ and 𝑛 > 2 is (i) For 𝑛 = 𝑝, 𝑝2, 𝑠𝑝𝑒𝑐 (𝒦(Γ𝑧 (𝐷2𝑛))) = { 𝑛+1 𝑛 0 1 } (ii) For 𝑛 = 𝑝𝑞, 𝑠𝑝𝑒𝑐 (𝒦(Γ𝑧(𝐷2𝑛))) = { 𝑛+2 𝑛+1 0 1 }. Proof: Case i: For 𝑛 = 𝑝, 𝑝2 The vertex set of 𝒦(Γ𝑧(𝐷2𝑛)) = {𝑢1, 𝑢2, … , 𝑢𝑛+1}. The Laplacian matrix can be written as 𝐿 = (𝑛 + 1)𝐼𝑛+1 − 𝐽𝑛+1. The obtained characteristic polynomial is (𝑥 − (𝑛 + 1))𝑛𝑥. 375 S. Ragha and R. Rajeswari The Laplacian spectrum of 𝒦(Γ𝑧 (𝐷2𝑛))will be written as, 𝑠𝑝𝑒𝑐 (𝒦(Γ𝑧(𝐷2𝑛))) = { 𝑛+1 𝑛 0 1 }. Case ii: For 𝑛 = 𝑝𝑞 The vertex set of 𝒦(Γ𝑧(𝐷2𝑛)) = {𝑢1, 𝑢2, … , 𝑢𝑛+2}. The Laplacian matrix can be written as 𝐿 = (𝑛 + 2)𝐼𝑛+2 − 𝐽𝑛+2. The obtained characteristic polynomial is(𝑥 − (𝑛 + 2)) 𝑛+1 𝑥. The Laplacian spectrum of 𝒦(Γ𝑧 (𝐷2𝑛))will be written as, 𝑠𝑝𝑒𝑐 (𝒦(Γ𝑧(𝐷2𝑛))) = { 𝑛+2 𝑛+1 0 1 }. Theorem 3.6: If 𝑛 = 𝑝, then the eigen values of 𝐿 (𝒦(Γ𝑧 (𝑄4𝑛))) are 𝑛 + 2 with multiplicity 𝑛 + 1 &0 with multiplicity 1. Proof: The vertex set of 𝒦(Γ𝑧(𝑄4𝑛)) = {𝑢1, 𝑢2, … , 𝑢𝑛+2}. The Laplacian matrix can be written as 𝐿 = (𝑛 + 2)𝐼𝑛+2 − 𝐽𝑛+2. The obtained characteristic polynomial is −(𝑥 − (𝑛 + 2)) 𝑛+1 𝑥. The Laplacian spectrum of 𝒦(Γ𝑧 (𝑄4𝑛))will be written as, 𝑠𝑝𝑒𝑐 (𝒦(Γ𝑧(𝑄4𝑛))) = { 𝑛+2 𝑛+1 0 1 }. Theorem 3.7: The signless Laplacian spectrum on clique graph of cyclic subgroup graph for a dihedral group of order 2𝑛, 𝑛 ∈ ℕ and 𝑛 > 2 is (𝑖)𝐹𝑜𝑟 𝑛 = 𝑝, 𝑝2, 𝑠𝑝𝑒𝑐 (𝒦(Γ𝑧 (𝐷2𝑛))) = { 2𝑛 1 𝑛 − 1 𝑛 } (𝑖𝑖)𝐹𝑜𝑟 𝑛 = 𝑝𝑞, 𝑠𝑝𝑒𝑐 (𝒦(Γ𝑧 (𝐷2𝑛))) = { 2𝑛 + 2 1 𝑛 𝑛 + 1 } Proof: Case i: For 𝑛 = 𝑝, 𝑝2 The vertex set of 𝒦(Γ𝑧(𝐷2𝑛)) = {𝑢1, 𝑢2, … , 𝑢𝑛+1}. The Signless Laplacian matrix can be written as 𝑆𝐿 = 𝐽𝑛+1 + (𝑛 − 1)𝐼𝑛+1. The obtained characteristic polynomial is (𝑥 − 2𝑛)(𝑥 − (𝑛 − 1))𝑛. The Signless Laplacian spectrum of 𝒦(Γ𝑧(𝐷2𝑛))will be written as, 𝑠𝑝𝑒𝑐 (𝒦(Γ𝑧 (𝐷2𝑛))) = { 2𝑛 1 𝑛−1 𝑛 }. Case ii: For 𝑛 = 𝑝𝑞 The vertex set of 𝒦(Γ𝑧(𝐷2𝑛)) = {𝑢1, 𝑢2, … , 𝑢𝑛+2}. The Signless Laplacian matrix can be written as 𝑆𝐿 = 𝐽𝑛+2 + 𝑛𝐼𝑛+2. The obtained characteristic polynomial is (𝑥 − (2𝑛 + 2))(𝑥 − 𝑛)𝑛+1. The Signless Laplacian spectrum of 𝒦(Γ𝑧(𝐷2𝑛))will be written as, 𝑠𝑝𝑒𝑐 (𝒦(Γ𝑧 (𝐷2𝑛))) = { 2𝑛+2 1 𝑛 𝑛+1 }. 376 Some Graph Parameters of Clique graph of Cyclic Subgroup graph on certain Non-Abelian Groups Theorem 3.8: If 𝑛 = 𝑝, then the eigen values of 𝑆𝐿 (𝒦(Γ𝑧 (𝑄4𝑛))) are 2𝑛 + 2 with multiplicity 1 &𝑛 with multiplicity 𝑛 + 1. Proof: The vertex set of 𝒦(Γ𝑧(𝑄4𝑛)) = {𝑢1, 𝑢2, … , 𝑢𝑛+2}. The Signless Laplacian matrix can be written as 𝑆𝐿 = 𝐽𝑛+2 + 𝑛𝐼𝑛+2. The obtained characteristic polynomial is (𝑥 − (2𝑛 + 2))(𝑥 − 𝑛)𝑛+1. The Signless Laplacian spectrum of 𝒦(Γ𝑧(𝑄4𝑛))will be written as, 𝑠𝑝𝑒𝑐 (𝒦(Γ𝑧 (𝑄4𝑛))) = { 2𝑛+2 1 𝑛 𝑛+1 }. 4.Some Topological Indices on Clique graph of Cyclic Subgroup graph for certain Non-Abelian Groups Theorem 4.1: If 𝒦(Γ𝑧(𝐺)) be a clique graph of cyclic subgroup graph on a non- abelian group and |𝑉 (𝒦(Γ𝑧(𝐺)))| = 𝑚, then the balaban index is 𝐽 (𝒦(Γ𝑧 (𝐺))) = 𝑚3−𝑚2 2(𝑚2−3𝑚+4) Proof:𝐽 (𝒦(Γ𝑧 (𝐺))) = 𝑦 𝑦−𝑥+2 ∑ 1 √𝑤(𝑢).𝑤(𝑣)𝑢𝑣∈𝐸(𝒦(Γ𝑧(𝐺))) , where the sum is taken over all edges of a connected graph 𝐺, 𝑥 and 𝑦 are the cardinalities of the vertex and the edge set of 𝐺, 𝑤(𝑢) and 𝑤(𝑣) denoted the sum of distances from u (resp.v) to all other vertices of G. 𝐽 (𝒦(Γ𝑧 (𝐺))) = 𝑚2−𝑚 𝑚2−𝑚−2𝑚+4 ( 𝑚(𝑚−1) 2(𝑚−1) ) = 𝑚2−𝑚 𝑚2−3𝑚+4 ( 𝑚 2 ) = 𝑚3−𝑚2 2(𝑚2−3𝑚+4) Theorem 4.2: If 𝒦(Γ𝑧(𝐺)) be a clique graph of cyclic subgroup graph on a non- abelian group and |𝑉 (𝒦(Γ𝑧(𝐺)))| = 𝑚, then atom bond connectivity status index is 𝐴𝐵𝐶𝑆 (𝒦(Γ𝑧 (𝐺))) = 1 √2 𝑚√𝑚 − 2. Proof: 𝐴𝐵𝐶𝑆 (𝒦(Γ𝑧 (𝐺))) = ∑ √ 𝜎(𝑢)+𝜎(𝑣)−2 𝜎(𝑢)𝜎(𝑣)𝑢𝑣∈𝐸(𝒦(Γ𝑧(𝐺))) = (√ (𝑚 − 1) + (𝑚 − 1) − 2 (𝑚 − 1)(𝑚 − 1) ) 𝑚(𝑚 − 1) 2 = 1 √2 𝑚√𝑚 − 2. Theorem 4.3: If 𝒦(Γ𝑧(𝐺)) be a clique graph of cyclic subgroup graph on a non- abelian group and |𝑉 (𝒦(Γ𝑧(𝐺)))| = 𝑚, then the arithmetic-geometric status index is 𝐴𝐺𝑆 (𝒦(Γ𝑧(𝐺))) = 𝑚(𝑚−1) 2 . 377 S. Ragha and R. Rajeswari Proof: 𝐴𝐺𝑆 (𝒦(Γ𝑧 (𝐺))) = ∑ 𝜎(𝑢)+𝜎(𝑣) 2√𝜎(𝑢)𝜎(𝑣)𝑢𝑣∈𝐸(𝒦(Γ𝑧(𝐺))) = 𝑚 − 1 + 𝑚 − 1 2√(𝑚 − 1)(𝑚 − 1) × 𝑚(𝑚 − 1) 2 = 𝑚(𝑚 − 1) 2 Theorem 4.4: If 𝒦(Γ𝑧(𝐺)) be a clique graph of cyclic subgroup graph on a non- abelian group and |𝑉 (𝒦(Γ𝑧(𝐺)))| = 𝑚, then (𝑖)The First Zagrebdegree eccentricityindex, 𝐷𝐸1 (𝒦(Γ𝑧 (𝐺))) = 𝑚 3 (𝑖𝑖)The Second Zagreb degree eccentricity index, 𝐷𝐸2 (𝒦(Γ𝑧 (𝐺))) = 𝑚3(𝑚−1) 2 . Proof: (i) 𝐷𝐸1 (𝒦(Γ𝑧 (𝐺))) = ∑ (𝑒𝑖 + 𝑑𝑖 ) 2 𝑣𝑖∈𝑉(𝒦(Γ𝑧(𝐺))) (where 𝑒𝑖 be the eccentricity and 𝑑𝑖 be the degree) = 𝑚(1 + 𝑚 − 1)2 = 𝑚3 (𝑖𝑖)𝐷𝐸2 (𝒦(Γ𝑧(𝐺))) = ∑ (𝑒𝑖 + 𝑑𝑖 )(𝑒𝑗 + 𝑑𝑗 ) 𝑣𝑖𝑣𝑗∈𝐸(𝒦(Γ𝑧(𝐺))) = 𝑚(𝑚 − 1) 2 (1 + 𝑚)(1 + 𝑚) = 𝑚3 (𝑚−1) 2 . References [1] S. Arumugam, Ramachandran, Invitation to Graph theory, SciTech Publications Pvt. Ltd, India, 2006. [2] Fred S. Roberts and Joel H. Spencer, A Characterizations of clique graphs, Journal of Combinatorial theory 10,102-108(1971). [3] J. John Arul Singh, S. Devi, Cyclic Subgroup Graph of a Finite Group, International Journal of Pure and Mathematics, Vol. 111 No.3 2016, 403-408. [4] Kulli V.R, Computation of Status Neighborhood Indices of Graphs, International Journal of Recent Scientific Research, Vol 11, Issue 04(B), pp. 38079-38085, April 2020. [5] Subarsha Banerjee, Laplacian Spectra of Comaximal Graph of ℤ𝑛, arXiv:2005.02316v2[math.CO] 23 Nov 2020. [6] Subarsha Banerjee, Prime Coprime Graph of a Finite Group, arXiv:1911.02763v2 [math.CO] 3Feb 2021. [7] Veena Mathad and Sultan Senan Mahde, The Minimum Hub Energy of a graph, Palestine Journal of Mathematics, Vol.6(1) (2017), 247 – 256. 378