Let vi and vj be two vertices of a graph G. The maximum degree matrix of G is given in [2] by dij = {max {di, dj} if vi and vj are adjacent 0 otherwise. Similarly the (i, j)-th entry of the minimum degree matrix is defined by taking the minimum degree instead of the maximum degree above, [1]. In this paper, we have elucidated a relation between maximum degree energy of p−shadow graphs with the maximum degree energy of its underlying graph. Similarly, a relation has been derived for minimum degree energy also. We disprove the results EM(S0 (G)) = 2EM(G) and Em(S0 (G)) = 2Em(G) given by Zheng-Qing Chu et al. [3] by giving some counterexamples.Publisher's Versio

Cangül, İsmail Naci

Prakasha, Kunkunadu Nanjundappa

Saravanan, Kadirvel

Srinivasa Rao, K

TWMS Journal of Applied and Engineering Mathematics

English

Isik University Academic Open Access

TWMS J. App. and Eng. Math. V.12, N.1, 2022, pp. 1-10MAXIMUM AND MINIMUM DEGREE ENERGIES OF p-SPLITTINGAND p-SHADOW GRAPHSK. S. RAO1, K. SARAVANAN1, K. N. PRAKASHA2, I. N. CANGUL3, §Abstract. Let vi and vj be two vertices of a graph G. The maximum degree matrix ofG is given in [2] bydij ={max {di, dj} if vi and vj are adjacent0 otherwise.Similarly the (i, j)-th entry of the minimum degree matrix is defined by taking theminimum degree instead of the maximum degree above, [1]. In this paper, we haveelucidated a relation between maximum degree energy of p−shadow graphs with themaximum degree energy of its underlying graph. Similarly, a relation has been derivedfor minimum degree energy also. We disprove the results EM (S′(G)) = 2EM (G) andEm(S′(G)) = 2Em(G) given by Zheng-Qing Chu et al. [3] by giving some counterexam-ples.Keywords and Phrases: maximum degree energy, minimum degree energy, splittinggraph, shadow graph.Mathematics Subject Classification: 05C50, 05C351. IntroductionThe adjacency matrix of a graph G is known to be a {0, 1} matrix with the (i, j)-thentry 1 if vi and vj are adjacent, and 0 if vi and vj are non-adjacent. Gutman introducedthe notion of energy of a graph contingent on adjacency matrix which was emanated bythe motivation of Hückel molecular orbital approximation, [5]. He defined the energy of agraph as the sum of the absolute values of eigenvalues of the adjacency matrix. Due to theintensive use of the adjacency matrix, many other graph matrices have been introducedwhich are related to different properties of graph, see e.g. [4, 7, 9]. Distance matrix,1 Department of Mathematics, Sri Chandrasekharendra Saraswathi Viswa Mahavidyalaya,Kanchipuram, Tamilnadu, India.e-mail: raokonda@yahoo.com; ORCID: https://orcid.org/0000-0002-2357-0933.e-mail: kadirvelsaravanan@gmail.com; ORCID: https://orcid.org/0000-0001-9373-19932 Department of Mathematics, Vidyavardhaka College of Engineering, Mysuru, India.e-mail: prakashamaths@gmail.com; ORCID: https://orcid.org/0000-0002-6908-4076.3 Department of Mathematics, Faculty of Arts and Science, Bursa Uludag University, 16059,Bursa, Turkey.e-mail: cangul@uludag.edu.tr; ORCID: https://orcid.org/0000-0002-0700-5774; corresponding author.§ Manuscript received: March 19, 2021; accepted: August 2, 2021.TWMS Journal of Applied and Engineering Mathematics, Vol.12, No.1 © Işık University, Departmentof Mathematics, 2022; all rights reserved.12 TWMS J. APP. AND ENG. MATH. V.12, N.1, 2022Randic matrix, Laplacian matrix, partition matrix, sum connectivity matrix, minimumcovering matrix etc. are some of such matrices.Let G be a simple graph with n vertices {v1, v2, · · · , vn} and let di be the degree ofvi for i = 1, 2, 3, · · · , n. The maximum degree matrix is defined by Adiga and Smithain [2] asdij ={max {di, dj} if vi and vj are adjacent0 otherwiseLet µ1, µ2, · · · , µn be the maximum degree eigenvalues of M(G). As the maximumdegree matrix is a real symmetric matrix with zero trace, these maximum degree eigenval-ues are real with sum equal to zero. The maximum degree energy of a graph G is definedsimilarly to the classical adjacency energy asEM (G) =n∑i=1|µi| .It is shown that if the maximum degree energy of a graph is rational, then it must bean even integer [2]. K.Srinivasa Rao, et al. [8] obtained bounds on the maximum degreeeigenvalues for a general graph and as well as some frequently used graph classes. Severalunicyclic graph classes are defined and their maximum degree eigenvalues and energy arecalculated [8].The minimum degree matrix, [1], is defined similarly to the maximum degree matrixwith the change in (i, j)-th entry. Here the (i, j)-th entry is the minimum of the degreesof two adjacent vertices vi and vj . Let ρ1, ρ2, · · · , ρn be the minimum degree eigenvaluesof the minimum degree matrix. The minimum degree energy is defined asEm(G) =n∑i=1|ρi| .Proposition 1.1. [6] Let A ∈Mm, B ∈Mn. Let λ be an eigenvalue of A correspondingto an eigenvector x and µ be an eigenvalue of B corresponding to an eigenvector y. Thenλµ is an eigenvalue of A⊗B corresponding to the eigenvector x⊗ y.2. Maximum and minimum degree energy of splitting graphsA derived graph is a graph which is obtained from a given graph according to some setof rules. One of the derived graphs is called splitting graph. The splitting graph S′(G) ofa graph G is obtained by adding a new vertex u′to each vertex u such that u′is adjacentto every vertex that is adjacent to u in G. See Fig. 1 as an example:Figure 1RAO ET AL.: MAXIMUM AND MINIMUM DEGREE ENERGIES OF P -SPLITTING AND ... 32.1. Errors. Zheng-Qing Chu et al., [3], gave the following relations between EM (S′(G))and EM (G) and also between Em(S′(G)) and Em(G). These two results are proven in thispaper to be incorrect. We provide some counter-examples which disprove these results.First we recall the erroneous results:Theorem 2.1. [3] For a graph G,EM (S′(G)) = 2EM (G).Theorem 2.2. [3] For a graph G,Em(S′(G)) = 2Em(G).Maximum and minimum degree matrices of G and S′(G) given in Fig. 1 areM(G) = 0 2 22 0 02 0 0andm(G) = 0 1 11 0 01 0 0and alsoM(S′(G)) =0 4 4 0 4 44 0 0 2 0 04 0 0 2 0 00 2 2 0 0 04 0 0 0 0 04 0 0 0 0 0 and m(S′(G)) =0 2 2 0 1 12 0 0 2 0 02 0 0 2 0 00 2 2 0 0 01 0 0 0 0 01 0 0 0 0 0 .Here EM (G) = 5.6569, Em(G) = 2.8284, EM (S′(G)) = 20.3961 and Em(S′(G)) = 10.198.We can easily observe that EM (S′(G)) 6= 2EM (G) and Em(S′(G)) 6= 2Em(G). With thisexample, we can conclude that Theorem 1 and Theorem 2 of [3] are not correct. This isdue to the wrong construction of M(S′(G)) and m(S′(G)) in the proof of Theorem 1 andTheorem 2.2.2. Maximum degree energy of p-splitting graphs. The p-splitting graph S′p(G) ofa graph G is obtained by adding p new vertices, say {u1, u2, · · · , up}, to each vertex u ofG such that for 1≤ i ≤ p, ui is adjacent to each vertex that is adjacent to u in G.In this section, we consider an r-regular graph G and obtain the maximum degree energyof a p-splitting graph S′p(G) in terms of the maximum degree energy of the graph G. Alsowe obtain the maximum degree energy of p-splitting graphs of some classes of r-regulargraphs.Theorem 2.3. If G is an r-regular graph, thenEM (Sp′(G)) = (p+ 1)(√1 + 4p)EM (G).Proof. Let {u1, u2, · · · , un} be the vertices of an r-regular graph G. Then the maximumdegree matrix M(G) is of order n where (i, j)-th entry is given byM(G)(i, j) ={max(di, dj) if ui and uj are adjacent,0 otherwise.4 TWMS J. APP. AND ENG. MATH. V.12, N.1, 2022Let {u1i , u2i , · · · , upi } be the vertices corresponding to each vi which are added to G toobtain S′p(G) such that N(u1i ) = N(u2i ) = · · · = N(vpi ) = N(vi) for i = 1, 2, · · · , n. Themaximum degree matrix of S′p(G) is a block matrix of the formM(S′p(G)) =(p+ 1)M(G) (p+ 1)M(G) · · · (p+ 1)M(G)(p+ 1)M(G) 0 · · · 0(p+ 1)M(G) 0 · · · 0....... . ....(p+ 1)M(G) 0 · · · 0=p+ 1 p+ 1 · · · p+ 1p+ 1 0 · · · 0p+ 1 0 · · · 0....... . ....p+ 1 0 · · · 0⊗M(G) = A⊗M(G)where O is a null matrix and A=p+ 1 p+ 1 · · · p+ 1p+ 1 0 · · · 0p+ 1 0 · · · 0· · · · · ·p+ 1 0 · · · 0.The spectrum of A is 0 (p+ 1)(1 +√1 + 4p)2 (p+ 1)(1−√1 + 4p)2p− 1 1 1.Hence the spectrum of M(S′p(G)) is(0µ1 · · · 0µn Xµ1 · · · Xµn Y µ1 · · · Y µnp− 1 · · · p− 1 1 · · · 1 1 · · · 1)where X =(p+ 1)(1 +√1 + 4p)2and Y =(p+ 1)(1−√1 + 4p)2. HenceEM (S′p(G)) =n∑i=1∣∣∣∣(p+ 1)(1±√1 + 4p2)µi∣∣∣∣= (p+ 1)n∑i=1|µi|(1 +√1 + 4p2+√1 + 4p− 12)= (p+ 1)(√1 + 4p)EM (G).Corollary 2.1. If G is a cycle graph of order n ≥ 3, thenEM (S′p(Cn)) = 4(p+ 1)√1 + 4pn−1∑k=0∣∣∣∣cos 2kπn∣∣∣∣ .Proof. If G is a cycle graph Cn (n ≥ 3), then EM (Cn) = 4∑n−1k=0∣∣∣∣cos 2kπn∣∣∣∣, [8].Since Cn is a 2-regular graph, we have the required result using Theorem 2.3. RAO ET AL.: MAXIMUM AND MINIMUM DEGREE ENERGIES OF P -SPLITTING AND ... 5Corollary 2.2. If G is complete graph of order n, thenEM (S′p(Kn)) = 2(p+ 1)√1 + 4p(n− 1)2.Proof. If G is the complete graph Kn, then EM (Kn) = 2(n−1)2, [2]. Since Kn is an n−1regular graph, using Theorem 2.3, we have the required result. Corollary 2.3. If Kn,n is a complete bipartite graph, thenEM (S′p(Kn,n)) = 2n2(p+ 1)√1 + 4p.Proof. If G is a complete bipartitie graph, then EM (Km,n) = 2√mn3 for m ≥ n, [8].Therefore EM (Kn,n) = 2n2. Hence by Theorem 2.3, we have the required result. Corollary 2.4. If G is a crown graph on 2n vertices, thenEM (S′p(G)) = 4(p+ 1)√1 + 4p(n− 1)2.Proof. If G is a crown graph on 2n vertices then EM (G) = 4(n − 1)2, [8]. Hence byTheorem 2.3, we have the required result. 2.3. Minimum Degree Energy of p-Splitting Graphs. If G is an r-regular graph,then m(G) = M(G). Hence we have the following result:Theorem 2.4. If G is an r-regular graph, thenEm(G) = EM (G).Theorem 2.5. If G is an r-regular graph, thenEm(Sp′(G)) =√(p+ 1)2 + 4pEm(G).Proof. Let {u1, u2, · · · , un} be the vertices of an r-regular graph G and m(G) be theminimum degree matrix. Let {u1i , u2i , · · · , upi } be the vertices corresponding to each viwhich are added in G to obtain S′p(G) such that N(u1i ) = N(u2i ) = · · · = N(vpi ) = N(vi),i = 1, 2, · · · , n. Then the minimum degree matrix of S′p(G) is a block matrix of the formm(S′p(G)) =(p+ 1)m(G) m(G) · · · m(G)m(G) O · · · Om(G) O · · · O....... . ....m(G) O · · · O=p+ 1 1 · · · 11 0 · · · 01 0 · · · 0....... . ....1 0 · · · 0p+1⊗m(G)=A⊗m(G)6 TWMS J. APP. AND ENG. MATH. V.12, N.1, 2022where O is a null matrix and A=p+ 1 1 · · · 11 0 · · · 01 0 · · · 0· · · · · ·1 0 · · · 0. Hence the spectrum of A is 0 p+ 1 +√(p+ 1)2 + 4p2p+ 1−√(p+ 1)2 + 4p2p− 1 1 1 .Now the spectrum of m(S′p(G)) is(0µ1 · · · 0µn Pµ1 · · · Pµn Qµ1 · · · Qµnp− 1 · · · p− 1 1 · · · 1 1 · · · 1)where P =p+ 1 +√(p+ 1)2 + 4p2and Q =p+ 1−√(p+ 1)2 + 4p2. HenceEm(S′p(G)) =n∑i=1∣∣∣∣∣p+ 1±√(p+ 1)2 + 4p2µi∣∣∣∣∣=n∑i=1|µi|(p+ 1 +√(p+ 1)2 + 4p2+√(p+ 1)2 + 4p− (p+ 1)2)=√(p+ 1)2 + 4pEm(G).Corollary 2.5. If Cn is the cycle graph of order n (n ≥ 3), thenEm(S′p(Cn)) = 4√(p+ 1)2 + 4pn−1∑k=0∣∣∣∣cos 2kπn∣∣∣∣ .Corollary 2.6. If Kn is the complete graph of order n, thenEm(S′p(Kn)) = 2(n− 1)2√(p+ 1)2 + 4p.Corollary 2.7. If Kn,n is complete bipartite graph, thenEp(S′(Kn,n)) = 2n2√(p+ 1)2 + 4p.Corollary 2.8. If G is crown graph with 2n vertices thenEp(S′(G)) = 4(n− 1)2√(p+ 1)2 + 4p.3. Maximum and minimum degree energies of p-shadow graphsThe shadow graph D2(G) of a connected graph G is constructed by taking two copiesof G say G′and G′′and joining each vertex u′in G′to the neighbors of the correspondingu′′in G′′. For example The p-shadow graph Dp(G) of a connected graph G is similarlyconstructed by taking p copies of G, say G1, G2, · · · , Gp and then joining each vertex uof Gi to the neighbors of the corresponding vertex v in Gj , for 1 ≤ i, j ≤ p. For exampleRAO ET AL.: MAXIMUM AND MINIMUM DEGREE ENERGIES OF P -SPLITTING AND ... 7Figure 2. The shadow graph of P33.1. Maximum degree energy of p-shadow graphs.Theorem 3.1. For a graph G, we haveEM (Dp(G)) = p2EM (G).Proof. Let {u1, u2, · · · , un} be the set of vertices of a graph G. Then the maximumdegree matrix of G is the same with the one in the proof of Theorem 2.3. Consider p-copies of graph G, say G1, G2, · · · , Gp with vertices u11, u22, · · · , upn. To obtain Dp(G),each vertex v in Gj is joined to the neighbors of the corresponding vertex v in Gj , for 1≤ i, j ≤ p. Then M(Dp(G)) is a block matrix of order np and it is of the formM(Dp(G)) =pM(G) pM(G) · · · pM(G)pM(G) pM(G) · · · pM(G)pM(G) pM(G) · · · pM(G)....... . ....pM(G) pM(G) · · · pM(G)=p p · · · pp p · · · pp p · · · p....... . ....p p · · · p⊗M(G)= pJp ⊗M(G).8 TWMS J. APP. AND ENG. MATH. V.12, N.1, 2022Since the spectrum of pJp is(0 p2p− 1 1), the spectrum of M(Dp(G)) is(0µ1 · · · 0µn p2µ1 · · · p2µn0 · · · 0 1 · · · 1).Hence we deduceEM (Dp(G)) =n∑i=1|p2µi|= p2EM (G).In the following corollaries, we obtain the maximum degree energies of shadow graphsof some classes of graphs:Corollary 3.1. If G is a cycle graph of order n ≥ 3, thenEM (Dp(Cn)) = 4p2n−1∑k=0∣∣∣∣cos(2kπn)∣∣∣∣ .Corollary 3.2. If G is a complete graph of order n, thenEM (Dp(Kn)) = 2p2(n− 1)2.Corollary 3.3. If Km,n is the complete bipartite graph with m ≥ n, thenEM (Dp(Km,n)) = 2p2√mn3.Proof. Since EM (Km,n) = 2√mn3 for m ≥ n [8], EM (Dp(Km,n)) = 2p2√mn3. Corollary 3.4. If G is a path graph on n vertices, thenEM (Dp(Pn)) = 4p2n∑k=1∣∣∣∣cos kπn+ 1∣∣∣∣ .Proof. If G is a path graph on n vertices, then EM (Pn) = 4∑nk=1∣∣∣∣cos kπn+ 1∣∣∣∣, [8]. There-fore EM (Dp(Pn)) = 4p2∑nk=1∣∣∣∣cos kπn+ 1∣∣∣∣ . Corollary 3.5. If G is a crown graph with 2n vertices, thenEM (Dp(G)) = 4p2(n− 1)2.3.2. Minimum degree energy of p-shadow graphs. In this section, we obtain theminimum degee energy of a p-shadow graph in terms of minimum degree energy. Alsoobtained the minimum degree energy of p-shadow graphs of some classes of regular graphs.Theorem 3.2. For a graph G, Em(Dp(G)) = p2Em(G).Proof. Let m(G) be the minimum degree matrix of G. Then, mimimum degree matrix ofp-shadow graph of G is block matrix of order pn and it is of the formm(Dp(G)) =p ·m(G) p ·m(G) · · · ·m(G)p ·m(G) p ·m(G) · · · p ·m(G)....... . ....p ·m(G) p ·m(G) · · · p ·m(G) = Jp ⊗m(G).RAO ET AL.: MAXIMUM AND MINIMUM DEGREE ENERGIES OF P -SPLITTING AND ... 9Since the spectrum of pJp is (0 p2p− 1 1),the spectrum of m(Dp(G)) is(0ρ1 · · · 0ρn m2ρ1 · · · m2ρn0 · · · 0 1 · · · 1).HenceEm(Dp(G)) =n∑i=1|p2ρi| = p2Em(G).We now obtain the minimum degree energy of shadow graph of some classes of graphsin the following corollaries:Corollary 3.6. If G is a cycle graph of order n ≥ 3, thenEm(Dp(Cn)) = 4p2n−1∑k=0∣∣∣∣cos 2kπn∣∣∣∣ .Corollary 3.7. If G is a complete graph of order n, thenEm(Dp(Kn)) = 2p2(n− 1)2.Corollary 3.8. If G is crown graph with 2n vertices, thenEm(Dp(G)) = 4p2(n− 1)2.Corollary 3.9. If Kr,s is the complete bipartite graph with r ≥ s, thenEm(Dp(Kr,s)) = 2p2√rs3.References[1] C. Adiga, C. S. Shivakumar Swamy, Bounds on the Largest of Minimum Degree Eigenvalues of Graphs,International Mathematical Forum, 5, (37), (2010), 1823-1831.[2] C. Adiga, M. Smitha, On Maximum Degree Energy of a Graph, Int. J. Contemp. Math. Sciences, 4,(8) (2009), 385-396.[3] Z.-Q. Chu, S. Nazeer, T. J. Zia, I. Ahmed, S. Shahid, Some New Results on Various Graph Energiesof the Splitting Graph, Journal of Chemistry, 2019, Article ID 7214047, 12 pages.[4] G. K. Gok, Some bounds on the Seidel energy of graphs, TWMS J. App. Eng. Math., 9, (4) (2019),949-956.[5] I. Gutman, The energy of a graph, Ber. Math.-Statist. Sekt. Forsch. Graz, (100-105): 103, (22), (1978),10. Steiermaerkisches Mathematisches Symposium (Stift Rein, Graz), (1978).[6] R. A. Horn, C. R. Johnson, Topics in Matrix Analysis, Cambridge Univ. Press, Cambridge, (1991).[7] M. H. Nezhaad, M. Ghorbani, Seidel borderenergetic graphs, TWMS J. App. Eng. Math., 10, (2),(2020), 389-399.[8] K. S. Rao, K. N. Prakasha, K. Saravanan, I. N. Cangul, Maximum Degree Energy, Advanced Studiesin Contemporary Mathematics, 31, (1), (2021), 49-66.[9] E. Sampathkumar, S. V. Roopa, K. A. Vidya, M. A. Sriraj, Partition energy of some trees and theirgeneralized complements, TWMS J. App. and Eng. Math., 10, (2), (2020), 521-531.10 TWMS J. APP. AND ENG. MATH. V.12, N.1, 2022K. Srinivasa Rao completed his Ph.D in mathematics at Aacharya Nagarjuna Uni-versity, M.Phil. at Madurai Kamaraj University, M.Sc. at JNT university, andPGDCS at the University of Hyderbad. His primary research focuses on ChemicalGraph Theory and Algebra. Currently, he is working as a professor of mathematicsat Sri Chandrasekharendra Saraswathi Viswa Mahavidyalaya, Kanchipuram, India.K. Saravanan is a research scholar at Sri Chandrasekharendra Saraswathi ViswaMahavidyalaya (Deemed to be University), Kanchipuram, India. He obtained hisbachelor degree and master degree in mathematics at the University of Madras, India.Presently, he is working in the Department of Mathematics, Sri ChandrasekharendraSaraswathi Viswa Mahavidyalaya (Deemed to be University), Kanchipuram, India.K. N. Prakasha for the photography and short autobiography, see TWMS J. App. and Eng. Math.V.9, N.4, 2019.Ismail Naci Cangul for the photography and short autobiography, see TWMS J. App. and Eng.Math., V.6, N.2, 2016.

Maximum and minimum degree energies of p-splitting and p-shadow graphs

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Maximum and minimum degree energies of p-splitting and p-shadow graphs

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