EUROPEAN JOURNAL OF PURE AND APPLIED MATHEMATICS 2025, Vol. 18, Issue 4, Article Number 5360 ISSN 1307-5543 – ejpam.com Published by New York Business Global Bounds on Spectral Radius and Signless Laplacian Spectral Radius for Generalized Core-Satellite Graphs Malathy V.1, Kalyani Desikan1,∗ 1 Department of Mathematics, SAS, Vellore Institute of Technology, Chennai, India Abstract. A Generalized core-satellite graph Θ(c,S, η∗) belongs to the family of graphs of diam- eter two. It has a central core of nodes connected to a few satellites, where all satellite cliques are not identical and might be of different sizes. These graphs can be used to model any real-world complex network. Using core-satellite graphs, properties like hierarchical structure can be conve- niently modeled for large complex networks. In this paper, we obtain the lower and upper bounds for the spectral radius and signless Laplacian spectral radius of the Generalized core-satellite graph, in terms of number of vertices, number of edges, and the graph parameters associated with the structure of the graph in both satellites and the core. 2020 Mathematics Subject Classifications: 05C50 Key Words and Phrases: Spectral radius, signless Laplacian spectral radius, generalized core- satellite graphs 1. Introduction Generalized core-satellite graphs can be used to model any complex real-world network. Using core-satellite graphs, properties like hierarchical structure can be conveniently mod- eled for large, complex networks. A hierarchical design model provides a reference topology that separates a network into distinct layers. Each of these layers has a series of functions that define its role in the network. This model facilitates making the network scalable, stable, deterministic, and reliable, provides better security, is effortless to manage and design, provides enhanced performance, and is also cost-efficient. Factors like these com- bine to make core-satellite graphs a dynamic model design for certain types of real-world networks [1, 2]. Generalized core-satellite graphs are denoted as Θ(c,S, η∗) and belong to the family of graphs of diameter two. It has a central core of vertices connected to a few satellites, where all satellite cliques are not identical and might be of different sizes. There is no restriction on having the same number of vertices or nodes. ∗Corresponding author. DOI: https://doi.org/10.29020/nybg.ejpam.v18i4.5360 Email addresses: malathy.viswanathan2015@vit.ac.in (M. V.), kalyanidesikan@vit.ac.in (K. Desikan) https://www.ejpam.com 1 Copyright: © 2025 The Author(s). (CC BY-NC 4.0) M. V., K. Desikan / Eur. J. Pure Appl. Math, 18 (4) (2025), 5360 2 of 20 Estrada and Benzi [1] introduced the Generalized core-satellite graphs, where they gen- eralized both the windmill and complete split graphs and analyzed certain features like clustering, assortativity, and spectral properties. The authors characterized the eigen- structure of these graphs’ adjacency and Laplacian matrices, observed that their Laplacian eigenvalues are integral, and commented on the asymptotic behavior of quantities such as the synchronizability index and infection threshold. Also, they determined the general and spectral properties of core-satellite graphs and provided the bounds of the spectral radius of the Generalized core-satellite graph. Generalized core-satellite class of graphs is found to resolve most of the social network issues where the network’s graphs are analyzed. Due to the expanding usage of social networks such as LinkedIn, Facebook, Instagram, Twitter, and Google+, malicious users seek to impinge on the privacy of other users and exploit their credentials by creating fraudulent accounts. This has become a cause of concern for users. Hence, social network providers are attempting to identify these users and their fake accounts to remove them from social networking environments using graph analysis. Graph similarity measures are used in graph analysis. Some significant similarity measures namely Jaccard, cosine, and L1 norm are a few measures where Friendship graphs (core-satellite graph) are utilized to detect suspicious accounts [3]. Many results on bounds for the spectral radius and signless Laplacian spectral radius have been given as functions of graph parameters such as the number of vertices, edges, degree sequence, average 2-degree, diameter, covering number, domination number, inde- pendence number, and others [4–7]. In [8], Nair Abreu et al. mentioned that this class of graphs is equivalent to chordal graphs having only one minimal vertex separator and the subclass of a quasi-threshold graph. They demonstrated that this class of graphs belongs to the hierarchical structure of chordal graphs. Moreover, in [9], Das has proved a con- jecture on the complete split-like graph. In [10], Liu et al. presented several upper and lower bounds on the k-th largest eigenvalue of Aα - matrix and characterized the extremal graphs corresponding to the bounds obtained. We find many recent articles on spectral radius and signless Laplacian spectral radius. In [11], Wang and Guo investigated the upper bounds of the spectral radius of the coalescense of two graphs generalizing some results by Passbani and Salemi in 2019 and as an appli- cation provided a new sharp upper bound on the spectral radius of a tree. Ghorbani and Amraei [12], investigated the spectra of certain classes of vertex or edge-transitive graphs for extended adjacency matrix and obtained some new bounds for both the smallest and the largest eigenvalues examining the behaviour of Aex-energy of a graph. In [13], authors characterized irregular bipartite graphs with maximum spectral radius and presented an upper bound on the spectral radius in terms of the order and maximum degree. Das and Liu in [14] proved complete split graph CS(n, α), the graph on n vertices con- sisting of a clique on (n − α) vertices and an independent set on the remaining α where (1 ≤ α ≤ n − 1) vertices in which each vertex of the clique is adjacent to each vertex of M. V., K. Desikan / Eur. J. Pure Appl. Math, 18 (4) (2025), 5360 3 of 20 the independent set. They determined its Laplacian spectrum when 1 ≤ α ≤ n − 1 and the signless Laplacian spectrum when 1 ≤ α ≤ n− 1, α ̸= 3. In [15], the authors obtained sharp upper bounds on the Q-index of (minimally) 2-connected graphs with given size, and characterize the corresponding extremal graphs. In this article, we have considered Generalized core-satellite graph Θ(c,S, η∗) and ob- tained results on the upper and lower bounds for the spectral radius and signless Laplacian spectral radius in terms of the parameters related to the structure of the graph through a different approach. 2. Preliminaries Figure 1: Generalized core-satellite graph (η1Kα1)▽Kϕ ∪ (η2Kα2)▽Kϕ ∪ ... ∪ (ηkKαk )▽Kϕ In this section, we discuss some preliminary findings that will be required to support our main results. Let G = (V (G), E(G)) be a simple, connected, undirected, and finite graph. Let the order, |V (G)|, be n and let the size of the graph, |E(G)|, be m, respectively. Let A(G) denote the (0,1) - adjacency matrix and D(G) the diagonal matrix whose diagonal entries are degree sequence of G. Let Q(G) = D(G) + A(G) be the signless Laplacian matrix of the graph G. According to Geršgorin’s Theorem, if C is an n× n real symmetric matrix then its eigen- values are non-negative real numbers. Since A(G) and Q(G) are real symmetric matrices, their eigenvalues are non-negative real numbers. The eigenvalues of A(G) and Q(G) are or- dered as ρ(G) = ρ1(G) ≥ ρ2(G) ≥ ... ≥ ρn(G) and µ(G) = µ1(G) ≥ µ2(G) ≥ ... ≥ µn(G), respectively. The largest eigenvalue of the adjacency matrix of the graph (known as spec- tral radius) and signless Laplacian matrix (known as signless Laplacian spectral radius) M. V., K. Desikan / Eur. J. Pure Appl. Math, 18 (4) (2025), 5360 4 of 20 are denoted as ρ1(G) and µ1(Q(G)), respectively. Definition 1. [1] The join (or complete product) Gψ1 ▽ Gψ2 of graphs Gψ1 and Gψ2 is obtained from Gψ1 ∪Gψ2 by joining every vertex of Gψ1 with every vertex of Gψ2. The Generalized core-satellite graph consists of core clique c = Kϕ with ϕ nodes and satellites S1, S2, S3, ..., Sk with Kα1 ,Kα2 , ...,Kαk cliques. Let η1, η2, η3,...,ηk be the number of copies of the cliques Kα1 ,Kα2 ,...,Kαk having degrees dα1 , dα2 , ..., dαk , respectively. Let dα1 < dα2 < .... < dαk . Let S = (S1, S2, ..., Sk) where each Si is the ith satellites in the graph having ηi copies of Kαi cliques and let η∗ = (η1, η2, ..., ηk) . Hence, the Generalized core-satellite graph is denoted as Θ(c,S, η∗). Theorem 1. [1] The spectral radius (Perron eigenvalue) ρ1(G) is given by the largest root of (λ− ϕ+ 1) k∏ i=1 (λ− αi + 1) = ϕ k∑ i=1 ηiαi ∏ j ̸=i (λ− αj + 1) and satisfies the bounds (ϕ− 1) + max 1≤i≤k (αi) < ρ1(G) < (ϕ− 1) + k∑ i=1 ηiαi. (1) Theorem 2. [16] Let G = (V,E) be a graph. Then min v∈V (G) ( Σ uv∈E d(u) )1/2 ≤ ρ1(G) ≤ max v∈V (G) ( Σ uv∈E d(u) )1/2 . (2) Moreover, if G is connected then either of the equalities holds iff Σ uv∈E d(u) is the same ∀ v ∈ V . Remark 1. (a) The number of edges in Si alone is ηi ( αi (αi − 1) 2 ) for i = 1, 2....k. (b) The number of edges joining the satellite graphs with vertices of the core graph is ϕηiαi M. V., K. Desikan / Eur. J. Pure Appl. Math, 18 (4) (2025), 5360 5 of 20 for i = 1, 2....k. (c) The number of edges in the core graph Kϕ alone is ϕ (ϕ− 1) 2 . Hence, the total number of edges in the core-satellite graph G is m = { k∑ i=1 ( ηiαi(αi − 1) 2 + ϕηiαi ) + ϕ(ϕ− 1) 2 } and the number of vertices of the graph G is n = ϕ+ k∑ i=1 ηiαi. Theorem 3 and Theorem 4 derived by Duan and Zhou, provide the lower bound and upper bound for the largest eigenvalue of any general non-negative matrix. Theorem 3. [17] Let A = (aij) be an n × n non-negative matrix with row sums r1, r2, ..., rn where r1 ≥ r2 ≥ .... ≥ rn. Let S and T be the smallest diagonal and the smallest non-diagonal elements of A, respectively. Let φn = (rn + S − T ) + √ (rn − S + T )2 + 4T ∑n−1 i=1 (ri − rn) 2 (3) Then ρ1(A(G)) ≥ φn. Moreover, if A is irreducible, ρ1(A(G)) = φn iff r1 = r2 = ... = rn or T > 0, and for some 2 ≤ t ≤ n, A satisfies the following conditions: (1) aii = S for 1 ≤ i ≤ t− 1. (2) aik = T for 1 ≤ i ≤ n− 1, 1 ≤ k ≤ t− 1 with k ̸= i. (3) rt = rt+1.... = rn. (4) ank = T for 1 ≤ k ≤ t− 1. Theorem 4. [17] Let A = (aij) be an n × n non-negative matrix with row sums r1,r2,...,rn where r1 ≥ r2 ≥ .... ≥ rn. Let M and N be the largest diagonal and the largest non-diagonal elements of A, respectively. Suppose N > 0. For 1 ≤ l ≤ n, let φl = (rl +M −N) + √ (rl −M +N)2 + 4N ∑l−1 i=1(ri − rl) 2 (4) Then ρ1(A(G)) ≤ φl for 1 ≤ l ≤ n. Moreover, if A is irreducible, ρ1(A(G)) = φl iff r1 = r2 = ... = rn or for some 2 ≤ t ≤ l, A satisfies the following conditions: (1) aii = M for 1 ≤ i ≤ t− 1. M. V., K. Desikan / Eur. J. Pure Appl. Math, 18 (4) (2025), 5360 6 of 20 (2) aik = N for 1 ≤ i ≤ l − 1, 1 ≤ k ≤ t− 1 with k ̸= i. (3) rt = rt+1... = rn. (4) aik = N for 1 ≤ i ≤ n, 1 ≤ k ≤ t− 1. Theorem 5. [16] Let M be a real symmetric n× n matrix, and let β be an eigenvalue of M with eigenvector x all of whose entries are non-negative. Denote the ith row sum of M by Ri(M). Then min 1≤i≤n Ri(M) ≤ β(Q(G)) ≤ max 1≤i≤n Ri(M). (5) Moreover, if all entries of x are positive then either of the equalities holds if and only if the row sums of M are all equal. Theorem 6. [16] Let M be a real symmetric n× n matrix, and let β be an eigenvalue of M with eigenvector x all of whose entries are non-negative. Denote the ith row sum of M by Ri(M). Let p be a polynomial. Then min 1≤i≤n Ri(p(M)) ≤ p(β(Q(G))) ≤ max 1≤i≤n Ri(p(M)). (6) Moreover, if all entries of x are positive then either of the equalities holds if and only if the row sums of M are all equal. Lemma 1. [5, 16] Let G = (V,E) be a simple graph. Then √ 2 min v∈V (G) √ d2 (v) + ∑ uv∈E(G) d (u) ≤ µ1(Q(G)) ≤ √ 2 max v∈V (G) √ d2(v) + ∑ uv∈E(G) d(u). Moreover, if G is connected, both the equalities hold iff 2d2 (v)+2 Σ uv∈E(G) d(u) is the same ∀v ∈ V (G). 3. Main Results 3.1. Bounds on spectral radius In this section, we derive tight upper bound and lower bound of ρ1(G) for Generalized core-satellite graphs. Theorem 7. Let G = Θ(c,S, η∗) be the Generalized core-satellite graph with n vertices and m edges. Then the lower bound and upper bound for the spectral radius of G are M. V., K. Desikan / Eur. J. Pure Appl. Math, 18 (4) (2025), 5360 7 of 20 k∑ j=1 { ϕ k∑ i=1 ( ηiαi + (ϕ− 1) ) + (αj − 1)(ϕ+ αj − 1) }1/2 ≤ kρ1(G) (7) ρ1(G) ≤ { ϕ ( k∑ i=1 ηiαi(ϕ+ αi − 1) )}1/2 . (8) Proof. We derive the lower bound given in equation (7) using the lower bound of equation (2). From Theorem 2 we observe that the lower bound is obtained by considering vertices having mimimum degree. In Si ▽Kϕ we see that vertices with minimum degree are in the satellite Si. For the Sthl clique, we have min v∈V (G) ( Σ u∼v∈E(G) d(u) )1/2 = ( Σ u∼v∈Sl d(u) )1/2 = { ϕ k∑ i=1 ( ηiαi + (ϕ− 1) ) + (αl − 1)(ϕ+ αl − 1) }1/2 ≤ ρ1(G). ( Σ u∼v∈S1 d(u) )1/2 + ( Σ u∼v∈S2 d(u) )1/2 + ...+ ( Σ u∼v∈Sk d(u) )1/2 ≤ kρ1(G). Hence, we have{ ϕ k∑ i=1 ( ηiαi + (ϕ− 1) ) + (α1 − 1)(ϕ+ α1 − 1) }1/2 + ϕ k∑ {i=1 ( ηiαi + (ϕ− 1) ) + (α2 − 1)(ϕ+ α2 − 1)  1/2 + ...+ { ϕ k∑ i=1 ( ηiαi + (ϕ− 1) ) + (αk − 1)(ϕ+ αk − 1) }1/2 ≤ kρ1(G). Hence, k∑ l=1 { ϕ k∑ i=1 ( ηiαi + (ϕ− 1) ) + (αl − 1)(ϕ+ αl − 1) }1/2 ≤ kρ1(G). (9) M. V., K. Desikan / Eur. J. Pure Appl. Math, 18 (4) (2025), 5360 8 of 20 From Theorem 2 we observe that the upper bound is obtained by considering vertices having maximum degree. In Si▽Kϕ, we see that vertices with a maximum degree are in the core Kϕ. Hence, from Theorem 2, we have, max v∈V (G) ( Σ uv∈E(G) d(u) )1/2 = ( Σ u∼v∈Kϕ d(u) )1/2 = { ϕ ( k∑ i=1 ηiαi(ϕ+ αi − 1) )}1/2 ≥ ρ1(G). (10) Hence proved. Remark 2. We observe that{ ϕ ( k∑ i=1 ηiαi(ϕ+ αi − 1) )}1/2 ≥ { ϕ (∑k i=1 ηiαi (ϕ+ αi − 1) k )}1/2 ≥ ρ1(G) (11) Theorem 8. Let G be the Generalized core-satellite graph of order n with m edges and S1, S2, S3, ...., Sk be the k satellites connected to the core graph Kϕ, then√ 2m k < ρ1(G) < √ 2m. Proof. The sum of the degrees of the vertices of satellites S1, S2, S3, ..., Sk and the sum of the degrees of the vertices of a core graph Kϕ is 2m, that is, k∑ i=1 ( Σ u∼v∈Si d(u) ) + ( Σ u∼v∈Kϕ d(u) ) = 2m. Considering the lower bound of equation (2), we have for Sl min v∈V (G) ( Σ u∼v∈E(G) d(u) )1/2 = ( Σ u∼v∈Sl d(u) )1/2 = { ϕ k∑ i=1 ( ηiαi + (ϕ− 1) ) + (αl − 1)(ϕ+ αl − 1) }1/2 ≤ ρ1(G). Hence for the Sthl clique, we have( Σ u∼v∈Sl d(u) ) = { ϕ k∑ i=1 ( ηiαi + (ϕ− 1) ) + (αl − 1)(ϕ+ αl − 1) } ≤ ρ1(G)2. M. V., K. Desikan / Eur. J. Pure Appl. Math, 18 (4) (2025), 5360 9 of 20 We have k∑ l=1 ( Σ u∼v∈Sl d(u) ) = k∑ l=1 { ϕ k∑ i=1 ( ηiαi + (ϕ− 1) ) + (αl − 1)(ϕ+ αl − 1) } ≤ kρ1(G)2. Also, we have k∑ l=1 ( Σ u∼v∈Sl d(u) ) = k∑ l=1 { ϕ k∑ i=1 ( ηiαi + (ϕ− 1) ) + (αl − 1)(ϕ+ αl − 1) } < 2m. We now show that k(ρ1(G))2 > 2m, for when ϕ ≥ 1 and k ≥ 2. We have 2m = 2 { k∑ i=1 ( ηiαi(αi − 1) 2 + ϕηiαi ) + ϕ(ϕ− 1) 2 } = k∑ i=1 ηiα 2 i + (2ϕ− 1) k∑ i=1 ηiαi + ϕ(ϕ− 1). Consider (k(ρ1(G))2 − 2m) ≥ k∑ l=1 { ϕ k∑ i=1 ( ηiαi + (ϕ− 1) ) + (αl − 1)(ϕ+ αl − 1) } − { k∑ i=1 ηiα 2 i + (2ϕ− 1) k∑ i=1 ηiαi + ϕ(ϕ− 1) } = ((k − 2)ϕ+ 1) k∑ i=1 ηiαi − k∑ i=1 ηiα 2 i + (k2 − 1)ϕ(ϕ− 1) + k∑ l=1 (αl − 1)(ϕ+ αl − 1) > 0. Therefore 2m < k(ρ1(G))2. This implies √ 2m k < ρ1(G) (12) Considering the upper bound of equation (2), we have( Σ u∼v∈Kϕ d(u) )1/2 = { ϕ k∑ i=1 (ηiαi (ϕ+ αi − 1)) }1/2 ≥ ρ1(G). M. V., K. Desikan / Eur. J. Pure Appl. Math, 18 (4) (2025), 5360 10 of 20 From the above inequality, we get 2m > { ϕ k∑ i=1 (ηiαi (ϕ+ αi − 1)) } ≥ (ρ1(G))2. Therefore √ 2m > ρ1(G). Hence √ 2m k < ρ1(G) < √ 2m. (13) Hence proved. In the following theorem, we make use of Theorem 4 to obtain the upper bound for the Generalized core-satellite graph G. In order to make use of Theorem 4 we require the row sums of the adjacency matrix. In the adjacency matrix A(G) of graph G, the vertices are arranged such that the top ϕ rows correspond to the vertices in the core Kϕ, followed by the vertices of ηk copies of the cliques Kαk ∈ Sk. The remaining rows correspond to vertices of Sk−1, Sk−2,...,S1. Let R∧ 1 , R∧ 2 ,...,R∧ k and R∧ ϕ be the row sums corresponding to the vertices of the core Kϕ and the satellites Sk, Sk−1,...,S1, i.e. R∧ ϕ,1 = R∧ ϕ,2 = ... = R∧ ϕ,ϕ = k∑ i=1 ηiαi + (ϕ− 1) = R∧ 1 R∧ k,1 = R∧ k,2 = ... = R∧ k,ηkαk = (ϕ+ αk − 1) = R∧ 2 R∧ k−1,1 = R∧ k−1,2 = ... = R∧ k−1,ηk−1αk−1 = (ϕ+ αk−1 − 1) = R∧ 3 R∧ k−2,1 = R∧ k−2,2 = ... = R∧ k−2,ηk−2αk−2 = (ϕ+ αk−2 − 1) = R∧ 4 ... ... In general, we have Rk−(j−2),1 = Rk−(j−2),2 = ... = Rk−(j−2),ηk−(j−2)αk−(j−2) = (ϕ+ αk−(j−2) − 1) = R∧ j ... R∧ 1,1 = R∧ 1,2 = ... = R∧ 1,η1α1 = (ϕ+ α1 − 1) = R∧ k+1. Theorem 9. For the Generalized core-satellite graph G of order n and size m, let S1, S2, S3, .., Sk be the k satellites connected to the core graph Kϕ. Let R∧ 1 , R ∧ 2 , ..., R ∧ k+1 M. V., K. Desikan / Eur. J. Pure Appl. Math, 18 (4) (2025), 5360 11 of 20 be the row sums corresponding to the core Kϕ and the satellites Sk, Sk−1,...,S1. The row sums are such that R∧ 1 ≥ R∧ 2 ≥ ... ≥ R∧ k+1, then ρ1(G) ≤ (ϕ+ αL − 2) 2 + √ (ϕ+ αL)2 + 4 ∑L−1 i=1 (R ∧ i −R∧ L) 2 where 1 ≤ i ≤ k and 1 ≤ L ≤ k + 1. Proof. Using Theorem 4 we have ρ1(G) ≤ (R∧ L +M −N) + √ (R∧ L −M +N)2 + 4N ∑L−1 i=1 (R ∧ i −R∧ L) 2 . From the adjacency matrix of G, we have M = 0 and N = 1. Substituting the value for N = 1, the smallest non-diagonal number in the above equation, we get ρ1(G) ≤ (R∧ L − 1) + √ (R∧ L + 1)2 + 4 ∑L−1 i=1 (R ∧ i −R∧ L) 2 . (14) Here we discuss the cases corresponding to the row sums. We have three cases depending on the choice of R∧ L, i.e. when R∧ L = R∧ 1 , R∧ L = R∧ 2 and R∧ L = R∧ j , where 3 ≤ j ≤ L. Case (i) When R∧ L = R∧ 1 , where R∧ 1 = ∑k i=1 ηiαi + (ϕ− 1) is the row sum corresponding to the vertex v ∈ Kϕ, we get ρ1(G) ≤ (R∧ 1 − 1) + √ (R∧ 1 + 1)2 2 = R∧ 1 . (15) Case (ii) When R∧ L = R∧ 2 = (ϕ+αk−1), the row sum of a vertex belonging to the satellite Sk, we get (R∧ 1 −R∧ 2 ) = ϕ { k∑ i=1 ηiαi + (ϕ− 1)− (ϕ+ αk − 1) } = ϕ ( k∑ i=1 ηiαi − αk ) = ϕ ( (n− ϕ)− αk ) . Substituting the values of R∧ 2 and (R∧ 1 −R∧ 2 ) in equation (14), we get ρ1(G) ≤ (ϕ+ αk − 1) + √ (ϕ+ αk)2 + 4(ϕ(n− ϕ)− αk) 2 . (16) Case (iii) We now discuss the case when R∧ L = R∧ j for some j where 3 ≤ j ≤ L, the row sum of a vertex belonging to the satellite Sk−(j−2) is (ϕ+ αk−(j−2) − 1) where j ̸= 2. M. V., K. Desikan / Eur. J. Pure Appl. Math, 18 (4) (2025), 5360 12 of 20 Consider j−1∑ i=1 (R∧ i −R∧ j ) = ϕ(R∧ 1 −R∧ j ) + ηkαk(αk − αk−(j−2)) + ηk−1αk−1(αk−1 − αk−(j−2)) + ... +ηk−(j−3)αk−(j−3)(αk−(j−3) − αk−(j−2)) = { ϕ ( k∑ i=1 ηiαi + (ϕ− 1)− (ϕ+ αk−(j−2) − 1) ) + j−3∑ i=0 ηk−iαk−i(αk−i − αk−(j−2)) } where 3 ≤ j ≤ L. Substituting the above expression in equation (14), we obtain ρ1(G) ≤ (ϕ+ αk−(j−2) − 2) 2 + √ (ϕ+ αk−(j−2))2 + 4 ( ϕ( ∑k i=1 ηiαi − αk−(j−2)) + ∑j−3 i=0 ηk−iαk−i(αk−i − αk−(j−2)) ) 2 . (17) Hence proved. Figure 2: S1: 3K3, S2: K5, S3: K6 join with one of the vertices of K4 (core) as illustration. M. V., K. Desikan / Eur. J. Pure Appl. Math, 18 (4) (2025), 5360 13 of 20 Example 1. Consider the Core-satellite graph given in Figure 2. This graph has S1, S2 and S3 as the satellite graphs and Kϕ = K4 as the core graph. S1 comprises of three copies of K3, S2 comprises of one copy of K5, S3 comprises of one copy K6. As an illustration, the join of the cliques in the satellite with one vertex of the core graph is shown. In fact, the remaining vertices of the core are similarly joined with cliques of the satellites. In this example, the number of vertices is 24 and the number of edges is 120. The calculated value of the spectral radius is ρ1(G) = 12.2485. We have the following observations from the bounds obtained in Theorems 7, 8 and 9. (i) From equation (7) of Theorem (7) we get the lower bound as 11.013 and from equation (8) of Theorem (7) we get the upper bound as 24.33. (ii) Using equation (11) we obtain the improved upper bound as 14.043. (iii) From equation (13) of Theorem 8, we get 8.944 < ρ1(G) < 15.49. (iv) Using Theorem 9, three cases are discussed. (a) From equation (15) of case (i) we obtain the upper bound for ρ1(G) as 23. (b) From equation (16) of case (ii) we obtain the tight upper bounds as 13 (c) From equation (17) of the case (iii) we obtain the upper bound as 15.587. From the above discussions, we observe that the tight lower and upper bounds are 11.013 and 13, respectively. 3.2. Bounds on Signless Laplacian Spectral Radius In this section, we derive the upper and lower bounds for the signless Laplacian matrix Q(G) = D(G) + A(G) for the graph G. Let µ1(Q(G)) be the signless Laplacian spectral radius. Theorem 10. Let G be a Generalized core-satellite graph with V (G) and E(G) as the set of vertices and edges of the graph G. Then the upper and lower bounds for signless Laplacian spectral radius of µ1(Q(G)) are √ 2 [ (dαk )2 + ϕ ( k∑ i=1 ηiαi + (ϕ− 1) ) + (αk − 1)(ϕ+ αk − 1) ]1/2 ≤ µ1(Q(G)) ≤ √ 2 [ k∑ i=1 ηiαi + (ϕ− 1)2 + 2 ( k∑ i=1 (ηiαiC2 + ϕηiαi) + ϕC2 ) − ( k∑ i=1 ηiαi + (ϕ− 1) )]1/2 (18) M. V., K. Desikan / Eur. J. Pure Appl. Math, 18 (4) (2025), 5360 14 of 20 Proof. Since Q(G) = D(G)+A(G), for vertex v its row sum is denoted as Rv(Q(G)) and we have Rv(Q(G)) = 2d(v). Also we have Rv(AD) = Rv(A 2) = Σ u∼v d(v) is true for the graph G. Then Rv(Q(G))2 = Rv(D(D +A) +AD +A2) = dvRv(Q) + 2( Σ u∼v d(v)) = 2(d2v + Σ u∼v d(v)) = 2(d2v + (2m− d(v)− Σ uv/∈E(G) d(u)) (19) From Lemma 1, we have √ 2 min v∈V (G) √ d2(v) + ∑ uv∈E(G) d(u) ≤ µ1(Q(G)) ≤ √ 2 max v∈V (G) √√√√d2(v) + ( 2m− d(v)− ∑ uv/∈E(G) d(u) ) (20) In this graph G, consider the vertex v taken from satellite graph Sk, since dαk = max(dαi), for i = 1, 2, ..., k. The sum of the degrees of vertices adjacent to v is the sum of the degrees of the vertices of the core graph and the remaining (αk − 1) vertices of the clique αk ∈ Sk Σ u∼v d(u)) = [ ϕ ( k∑ i=1 ηiαi + (ϕ− 1) ) + (αk − 1)(ϕ+ αk − 1) ] . We have µ1(Q(G)) ≥ √ 2 √√√√[(dαk )2 + ϕ ( k∑ i=1 ηiαi + (ϕ− 1) ) + (αk − 1)(ϕ+ αk − 1) ] (21) Similarly consider the vertex v from core Kϕ, the sum of the degrees of the vertices adjacent to v is the sum of the degrees of the vertices of the satellites, and the remaining (ϕ − 1) vertices of Kϕ. We have, for v ∈ Kϕ d(v) = k∑ i=1 ηiαi + (ϕ− 1). Also, ∑ u∼v∈Kϕ d(u) = ( 2m− d(v)− ∑ u∼v∈Kϕ d(u) ) Therefore, µ1(Q(G)) ≤ √ 2 √√√√( k∑ i=1 ηiαi + (ϕ− 1) )2 + [ 2 ( k∑ i=1 (ηiαiC2 + ϕηiαi) + ϕC2 ) − ( k∑ i=1 ηiαi + (ϕ− 1) )] . M. V., K. Desikan / Eur. J. Pure Appl. Math, 18 (4) (2025), 5360 15 of 20 (22) In the following theorem, we make use of Theorem 3 to obtain the lower and Theorem 4 to obtain the upper bounds for the Generalized core-satellite graph G. In the signless Laplacian matrix Q(G) = A(G) + D(G) of the graph G, the vertices are arranged such that the top ϕ rows correspond to the vertices in the core Kϕ, followed by the vertices of ηk copies of the cliques Kαk ∈ Sk. The remaining rows correspond to vertices of Sk−1, Sk−2,...,S1. Let R∧ 1 , R∧ 2 ,...,R∧ k and R∧ ϕ be the row sums corresponding to the vertices of the core Kϕ and the satellites Sk, Sk−1,...,S1, i.e. R∧ ϕ,1 = R∧ ϕ,2 = ... = R∧ ϕ,ϕ = 2 k∑ i=1 ηiαi + (ϕ− 1) = R∧ 1 R∧ k,1 = R∧ k,2 = ... = R∧ k,ηkαk = 2(ϕ+ αk − 1) = R∧ 2 R∧ k−1,1 = R∧ k−1,2 = ... = R∧ k−1,ηk−1αk−1 = 2(ϕ+ αk−1 − 1) = R∧ 3 R∧ k−2,1 = R∧ k−2,2 = ... = R∧ k−2,ηk−2αk−2 = 2(ϕ+ αk−2 − 1) = R∧ 4 ... ... In general for the jth term, we have Rk−(j−2),1 = Rk−(j−2),2 = ... = Rk−(j−2),ηk−(j−2)αk−(j−2) = 2(ϕ+ αk−(j−2) − 1) = R∧ j .. .. R1,1 = R1,2 = ... = R1,ηkαk = 2(ϕ+ α1 − 1) = R∧ k+1 Theorem 11. For the Generalized Core-satellite graph G of order n and size m, let S1, S2, S3, ..., Sk be the k satellites connected to the core graph Kϕ. Let R∧ 1 , R ∧ 2 , ..., R ∧ k+1 be the row sums corresponding to the core Kϕ and the satellites Sk, Sk−1,...,S1. The row sums are such that R∧ 1 ≥ R∧ 2 ≥ .... ≥ R∧ k+1, then µ1(Q(G)) ≥ (3ϕ+ 2αk−(j−2) + α1 − 3) 2 + √ (ϕ+ 2αk−(j−2) − α1)2 2 (23) where 1 ≤ i ≤ k and 3 ≤ j ≤ k + 1. Proof. To prove this theorem, applying Theorem 3 for the signless Laplacian matrix Q(G) = A(G) +D(G) of the graph G, we have µ1(Q(G)) ≥ (R∧ L + S − T ) + √ (R∧ L − S + T )2 + 4T ∑L−1 i=1 (R ∧ i −R∧ L) 2 . (24) M. V., K. Desikan / Eur. J. Pure Appl. Math, 18 (4) (2025), 5360 16 of 20 For the matrix Q(G), we observe that the smallest diagonal element S = (ϕ+α1−1), and the smallest non-diagonal element T = 0. Here we discuss the cases corresponding to the row sums R∧ L = R∧ 2 and R∧ j . Also R∧ L ̸= R∧ 1 as the lower bound is obtained by considering the row sum of any vertex vi ∈ Si. The row sums of the vertices belonging to the satellite Sk are greater compared to the row sums of the vertices belonging to the satellites Sk−1, Sk−2,...,S1. Case (i) We now consider R∧ L = R∧ 2 = 2(ϕ + αk − 1) and by substituting the values of R∧ L and T in equation (24), we get µ1(Q(G)) ≥ (R∧ 2 + S) + √ (R∧ 2 − S)2 2 Now substituting for R∧ 2 and S in the above equation, we get µ1(Q(G)) ≥ (3ϕ+ 2αk + α1 − 3) 2 + √ (ϕ+ 2αk − α1 − 1)2 2 . (25) Case (ii) We discuss the case for R∧ L = R∧ j = 2(ϕ + αk−(j−2) − 1), the row sum of the vertex belonging to the satellite Sk−(j−2), where j ̸= 2 and 3 ≤ j ≤ L. Equation (24) reduces to µ1(Q(G)) ≥ (R∧ j + S) + √ (R∧ j − S)2 2 . We see that the smallest diagonal element of Q(G) is S = (ϕ+α1−1) and the smallest non-diagonal element T = 0. Substituting for R∧ j and S in the above equation, we get µ1(Q(G)) ≥ (3ϕ+ 2αk−(j−2) + α1 − 3) 2 + √ (ϕ+ 2αk−(j−2) − α1 − 1)2 2 (26) Hence proved. Theorem 12. For the Generalized core-satellite graph G of order n and size m, let S1, S2, S3, ..., Sk be the k satellites connected to the core graph Kϕ. Let R∧ 1 , R ∧ 2 , .., R ∧ k+1 be the row sums corresponding to the core Kϕ and the satellites Sk, Sk−1,...,S1. The row sums are such that R∧ 1 ≥ R∧ 2 ≥ ... ≥ R∧ k+1, then µ1(Q(G)) ≤ (n+ 2(ϕ+ αk−(j−2) − 2)) 2 + √ 2(ϕ+ αk−(j−2) − n)2 + 8 { ϕ( ∑k i=1 ηiαi − αk−(j−2)) + ∑j−3 i=0 ηk−iαk−i(αk−i − αk−(j−2)) } 2 where 3 ≤ j ≤ k + 1. M. V., K. Desikan / Eur. J. Pure Appl. Math, 18 (4) (2025), 5360 17 of 20 Proof. We apply Theorem 4 to prove our result. We apply the values of M = ( ∑k i=1 ηiαi + (ϕ − 1)), the largest diagonal element and N = 1, the largest non-diagonal element. Consider µ1(Q(G)) ≤ (R∧ L +M −N) + √ (R∧ L −M +N)2 + 4N ∑L−1 i=1 (R ∧ i −R∧ L) 2 . Here we apply the value of N = 1, the largest non-diagonal element in the above equation, we get µ1(Q(G)) ≤ (R∧ L +M − 1) + √ (R∧ L −M + 1)2 + 4 ∑L−1 i=1 (R ∧ i −R∧ L) 2 . (27) We discuss the cases corresponding to the row sums R∧ L = R∧ 1 , R∧ 2 and R∧ j . Case (i) Consider R∧ L = R∧ 1 , where R∧ 1 = 2 (∑k i=1 ηiαi+ (ϕ− 1) ) . The vertex v ∈ Kϕ has the maximum degree. Also, the row sum of any vertex vi ∈ Si has the row sum R∧ i = 2(ϕ+ αi − 1). µ1(Q(G)) ≤ (R∧ 1 +M − 1) + √ (R∧ 1 −M + 1)2 + 4 ∑L−1 i=1 (R ∧ i −R∧ 1 ) 2 . (28) We observe that the summation L−1∑ i=1 (R∧ i −R∧ 1 ) = 0. Also (R∧ 1 +M − 1) = (3n− 4) and (R∧ 1 −M + 1) = n Substituting the above expressions in equation (28), we get µ1(Q(G)) ≤ (2n− 1). (29) Case (ii) Consider R∧ L = R∧ 2 = 2(ϕ+ αk − 1), the row sum of the vertex belonging to the satellite Sk, as the row sum of the vertex belonging to the satellite Sk is the maximum compared to the row sums of the vertices belonging to the satellite S1, S2, ..., Sk−1. µ1(Q(G)) ≤ (R∧ 2 +M − 1) + √ (R∧ 2 −M + 1)2 + 4 ∑L−1 i=1 (R ∧ i −R∧ 2 ) 2 . (30) Consider L−1∑ i=1 (R∧ i −R∧ 2 ) = ϕ(R∧ 1 −R∧ 2 ) = 2ϕ ( k∑ i=1 ηiαi + (ϕ− 1)− (ϕ+ αk − 1) ) M. V., K. Desikan / Eur. J. Pure Appl. Math, 18 (4) (2025), 5360 18 of 20 = 2ϕ ((n− ϕ)− αk) . Substituting the values of R∧ 2 , M and ∑L−1 i=1 (R ∧ i −R∧ 2 ) in equation (30), we get µ1(Q(G)) ≤ (n+ 2(ϕ+ αk − 2)) 2 + √ 2(ϕ+ 2αk − n)2 + 8 {ϕ (n− ϕ)− αk} 2 . (31) Case (iii) Consider the case R∧ L = R∧ j for some general j, where 3 ≤ j ≤ L. Here R∧ j = 2(ϕ + αk−(j−2) − 1), the row sum of the vertex belonging to the satellite Sk−(j−2) where j ̸= 2. µ1(Q(G)) ≤ (R∧ j +M − 1) + √ (R∧ j −M + 1)2 + 4 ∑L−1 i=1 (R ∧ i −R∧ j ) 2 . (32) Consider j−1∑ i=1 ( R∧ i −R∧ j ) = ( 2ϕ k∑ i=1 (ηiαi−αk−(j−2))+2ηkαk(αk−αk−(j−2))+2ηk−1αk−1(αk−1−αk−(j−2))+.... +2ηk−(j−3)αk−(j−3)(αk−(j−3) − αk−(j−2) ) = { 2ϕ ( k∑ i=1 ηiαi − αk−(j−2) ) + 2 j−3∑ i=0 ηk−iαk−i(αk−i − αk−(j−2)) } where 3 ≤ j ≤ L. Substituting for R∧ j , M and ∑L−1 i=1 ( R∧ i −R∧ j ) in equation (32), we get µ1(Q(G)) ≤ (n+ 2(ϕ+ 2αk−(j−2) − 2)) 2 . + √ 2(ϕ+ αk−(j−2) − n)2 + 8{ϕ ( (n− ϕ)− αk−(j−2) ) + ∑j−3 i=0 ηk−iαk−i(αk−i − αk−(j−2))} 2 (33) Hence proved. Remark 3. For the graph G in Example 1 the calculated value of the signless Laplacian spectral radius is µ1(Q(G)) = 30.2016. We have the following observations from Theorem 10, Theorem 11, and Theorem 12. M. V., K. Desikan / Eur. J. Pure Appl. Math, 18 (4) (2025), 5360 19 of 20 (i) We obtain the lower and upper bounds from equation (18) of Theorem 10 as 18 ≤ µ1(Q(G)) ≤ 38.626. (ii) We obtain the lower bound from Theorem 11 and to determine the tight lower bound, we discuss the following cases. (a) When R∧ L = R∧ 2 , we have from equation (25) the lower bound as µ1(Q(G)) ≥ 18. (b) When R∧ L = R∧ 3 , we have from equation (26) the lower bound as µ1(Q(G)) ≥ 16, for any j, where 3 ≤ j ≤ L we obtain this value by taking j = 3. (iii) We obtain the upper bound from Theorem 12 and to determine the tight upper bound, we discuss the following cases. (a) When R∧ L = R∧ 1 we get from equation (29), the upper bound as 46. (b) When R∧ L = R∧ 2 we get from equation (31), the upper bound as µ1(Q(G)) ≤ 30.77. (c) When R∧ L = R∧ 3 we get from equation (33), the upper bound as µ1(Q(G)) ≤ 30.79. From the above discussions, we infer that the closest lower and upper bounds are 18 and 30.77, respectively. 4. Conclusion Generalized core-satellite graphs are hierarchical network structures that are much more suitable to model real-world complex networks. This structural design has been applied as this has a wide range of benefits, for developing a network that is reliable, resilient, scalable, flexible, cost-effective, has better security, easier management design, enhanced performance, and improved cost-efficiency. In this article, we have provided varied results on bounds for the spectral radius in terms of m and k (number of satellites). We have arrived at the tight bounds from among the bounds derived. We have also derived bounds on the signless Laplacian spectral radius by applying a few conditions on the general bounds derived by Duan and Zhou, using row sums. 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