For a simple graph $G$ with $n$ vertices and $m$ edges having adjacency eigenvalues $\lambda_1,\lambda_2, \dots,\lambda_n$, the energy $E(G)$ of $G$ is defined as $E(G)=\sum_{i=1}^{n} |\lambda_i|$. We obtain the upper bounds for $E(G)$ in terms of the vertex covering number $\tau$, the number of edges $m$, maximum vertex degree $d_1$ and second maximum vertex degree $d_2$ of the connected graph $G$. These upper bounds improve some recently known upper bounds for $E(G)$. Further, these upper bounds for $E(G)$ imply a natural extension to other energies like distance energy and Randi\'{c} energy associated to a connected graph $G$

Ganie, Hilal A.

Pirzada, Shariefuddin

Samee, Uma Tul

Le Matematiche (Dipartimento di Matematica e Informatica, Università degli Studi di Catania)

LE MATEMATICHEVol. LXXIV (2019) – Issue I, pp. 163–172doi: 10.4418/2019.74.1.11ON GRAPH ENERGY, MAXIMUM DEGREE AND VERTEXCOVER NUMBERHILAL A. GANIE - U. SAMEE - S. PIRZADAFor a simple graph G with n vertices and m edges having adjacencyeigenvalues λ1,λ2, . . . ,λn, the energy E (G) of G is defined as E (G) =∑ni=1 |λi|. We obtain the upper bounds for E (G) in terms of the vertexcovering number τ , the number of edges m, maximum vertex degree d1and second maximum vertex degree d2 of the connected graph G. Theseupper bounds improve some recently known upper bounds for E (G). Fur-ther, these upper bounds for E (G) imply a natural extension to other en-ergies like distance energy and Randic´ energy associated to a connectedgraph G.1. IntroductionLet G be a finite simple graph with n vertices and m edges having vertex setV (G) = {v1,v2, . . . ,vn} and edge set E(G) = {e1,e2, . . . ,em}. Throughout thispaper by a graph G, we mean the graph G = (V,E) having n vertices and medges, unless otherwise stated. The adjacency matrix A = (ai j) of G is a (0,1)-square matrix of order n and the spectrum of the adjacency matrix is the spec-trum of the graph G.Submission received : 26 December 2018AMS 2010 Subject Classification: 05C30, 05C50Keywords: Adjacency matrix, singular values, energy, vertex covering number.Corresponding author: S. Pirzada164 HILAL A. GANIE - U. SAMEE - S. PIRZADAIf λ1,λ2, . . . ,λn are the adjacency eigenvalues of G, the energy of G is de-fined asE (G) =n∑i=1|λi|.This concept introduced by Gutman [12] is intensively studied in Chemistry,since it can be used to approximate the total pi-electron energy of a molecule,see [15] and the references therein. This spectrum-based graph invariant hasbeen much studied in both chemical and mathematical literature. Among thepioneering results on graph energy are the lower and upper bounds for energy,see [1, 10, 14, 15, 19] and the references therein. For more information aboutthe energy of a graph, we refer to the most recent papers [1, 14, 19].A subset S of the vertex set V (G) is said to be a covering set of G if everyedge of G is incident to at least one vertex in S. A covering set with minimumcardinality among all covering sets is called the minimum covering set of G andits cardinality, denoted by τ = τ(G), is called vertex covering number of thegraph G. If H is a subgraph of the graph G, we denote the graph obtained byremoving the edges in H from G by G \H (that is, only the edges of H areremoved from G).The neighbourhood of a vertex vi ∈ V (G), denoted by N(vi) is defined asthe set of all the vertices of G which are adjacent to vi, that is, N(vi) = {v j :viv j ∈ E(G)}. A graph G of order n with m edges is said to be c-cyclic, wherec≥ 0 is an integer, if m = n+ c−1. For the eigenvalue λ of G, λ [k] denotes itsmultiplicity.Further, as usual Pn, Cn, Kn and Ks,t , respectively, denote the path on n ver-tices, the cycle on n vertices, the complete graph on n vertices and the completebipartite graph on s+ t vertices. For other undefined notations and terminologyfrom spectral graph theory, the readers are referred to [4, 18].The rest of the paper is organized as follows. In Section 2, we obtain someupper bounds for E (G) in terms of the vertex covering number τ , the numberof edges m, maximum vertex degree d1 and second maximum vertex degree d2of the connected graph G. These upper bounds improve some recently knownupper bounds for the energy E (G) of a connected graph. In Section 3, we extendthe upper bound obtained for E (G) in Section 2, to other energies like distanceenergy and Randic´ energy associated to a connected graph G.2. Bounds for the energy of a graphGiven a complex m× n matrix M, its energy, denoted by E (M), is the sum ofits singular values (the positive square roots of the eigenvalues of the matrixON GRAPH ENERGY, MAXIMUM DEGREE AND VERTEX COVER NUMBER 165M∗M), see [16]. If M is a real symmetric matrix of order n and if si(M),xi(M)denote the singular values and the eigenvalues of M, respectively, then si(M) =|xi(M)|, for all i = 1,2, . . . ,n. In the light of this definition, if λ1,λ2, . . . ,λn arethe eigenvalues of the graph G, then the energy E (G) = E (A(G)), [16], isE (G) =n∑i=1|λi|, (1)where λi are the eigenvalues of the graph G.The following lemma can be seen in [6].Lemma 2.1. Let X,Y and Z be square matrices of order n such that Z = X +Y .Thenn∑i=1si(Z)≤n∑i=1si(X)+n∑i=1si(Y ).Moreover, equality holds if and only if there exists an orthogonal matrix P suchthat PX and PY are both positive semidefinite matrices.Wang and Ma [19] obtained the following upper bound for the energy E (G)in terms of the vertex covering number τ and the maximum vertex degree ∆:E (G)≤ 2τ√∆, (2)with equality if and only if G is the disjoint union of τ copies of K1,∆ togetherwith some isolated vertices.Now, we obtain an upper bound for the energy E (G) in terms of the vertexcovering number τ and the number of edges m of the graph G.Theorem 2.2. ALet G be a connected graph of order n≥ 2 and m edges havingvertex covering number τ . ThenE (G)≤ 2√mτ, (3)with equality if and only if τ = 1 and G∼= K1,n−1.Proof. Let G be a connected graph with vertex set V (G) = {v1,v2, . . . ,vn} andedge set E(G) = {e1,e2, . . . ,em}. Let τ be the vertex covering number and Cbe the minimum vertex covering set of G. Without loss of generality, let C ={v1,v2, . . . ,vτ}.Let G1,G2, . . . ,Gτ be the spanning subgraphs of G corresponding to the166 HILAL A. GANIE - U. SAMEE - S. PIRZADAvertices v1,v2, . . . ,vτ of C, having vertex set same as G and edge sets defined asfollows.E(Gi) = {vivt : vt ∈ N(vi)\{v1,v2, . . . ,vi−1}}, i = 1,2, . . . ,τ.For i = 1,2, . . . ,τ, let mi = |E(Gi)|. It is clear that E(G) = E(G1)∪E(G2)∪·· · ∪E(Gτ) and Gi = K1,mi ∪ (n−mi− 1)K1, for all i = 1,2, . . . ,τ . Also, it iseasy to see that the adjacency matrix A(G) now can be decomposed asA(G) = A(G1)+A(G2)+ · · ·+A(Gτ). (4)The adjacency spectrum of Gi = K1,mi ∪ (n−mi−1)K1 is {±√mi,0[n−2]}.Therefore,E (Gi) = E (K1,mi ∪ (n−mi−1)K1) = 2√mi, for all i = 1,2, . . . ,τ. (5)Now, applying Lemma 2.1 to equation (4) and using (1), (5) and Cauchy-Schwarz’s inequality, we haveE (G)≤ E (G1)+E (G2)+ · · ·+E (Gτ)= 2√m1+2√m2+ · · ·+2√mτ= 2τ∑i=1√mi ≤ 2√ττ∑i=1mi = 2√mτ,as E(G) = E(G1)∪E(G2)∪ ·· ·∪E(Gτ) gives m =τ∑i=1mi. This proves the firstpart of the proof.Assume that equality holds in (3), so all the inequalities above occur asequalities. Since G is connected, it is clear that equality occurs in (4) if and onlyif G ∼= K1,n−1. Also, since equality occurs in Cauchy-Schwarz’s inequality ifand only if m1 = m2 = · · ·= mτ , it follows that the equality occurs in (3) if andonly if τ = 1 and G∼= K1,n−1.Conversely, if τ = 1 and G∼=K1,n−1, then it is easy to see that equality holdsin (3).Remark 2.3. For a connected graph G of order n≥ 2 having m edges, it can beseen that the upper bound (3) is better than the upper bound (2) for all graphs Gwith m ≤ τ∆. In particular, if G ∼= Kn or Kr,s, r ≤ s, then m = n(n−1)2 , τ = n−1, ∆= n−1 or m= rs, τ = r, ∆= s. It can be seen that m≤ τ∆ always hold. If Gis a c-cyclic graph, c≥ 0, then m= n+c−1 and so m≤ τ∆ gives c≤ τ∆−n+1.If G is a path Pn, then c = 0, τ = bn2c, ∆= 2 and so c≤ τ∆−n+1 always hold.Similarly, it can be seen that for a cycle Cn, we always have c≤ τ∆−n+1. WeON GRAPH ENERGY, MAXIMUM DEGREE AND VERTEX COVER NUMBER 167now give a construction of an infinite family of graphs for which the conditionm ≤ τ∆ holds. Let Sω(a,a, . . . ,a, . . . ,a), a ≥ 1 be the family of connectedgraphs of order n = ω+aω with m edges having a pendent vertices attached toeach of the ω vertices of the clique Kω (see Figure 1). For this graph it is clearthat τ = ω , ∆ = ω − 1+ a and m = ω(ω−1)2 + n−ω . We have ω ≥ 1, whichgives ω(ω − 1) ≥ 0, further implies that 2ω2− 4ω + 2n ≥ ω2− 3ω + 2n, or2ω2− 2(n−ω)− 2ω ≥ ω2− 3ω + 2n, or 2ω(ω − a+ 1) ≥ ω2− 3ω + 2n, orm≤ τ∆, as n−ω = aω .We now obtain an upper bound for the energy E (G) in terms of the vertexcovering number τ , the maximum vertex degree d1 and the second maximumvertex degree d2 of the graph G.Theorem 2.4. Let G be a connected graph of order n ≥ 2 and m edges hav-ing vertex covering number τ . If d1 and d2 are the maximum and the secondmaximum vertex degrees in G, thenE (G)≤ 2√d1+2(τ−1)√d2, (6)with equality if and only if τ = 1 and G∼= K1,n−1.Proof. Let G be a connected graph with vertex set V (G) = {v1,v2, . . . ,vn} andedge set E(G) = {e1,e2, . . . ,em}. Let τ be the vertex covering number andC be the minimum vertex covering set of G. Without loss of generality letC = {v1,v2, . . . ,vτ}. Let G1,G2, . . . ,Gτ be the spanning subgraphs of G cor-responding to the vertices v1,v2, . . . ,vτ of C as defined in Theorem 2.2. Now,proceeding similarly as in Theorem 2.2, we arrive atE (G)≤ E (G1)+E (G2)+ · · ·+E (Gτ)= 2√m1+2√m2+ · · ·+2√mτ .Let d1 ≥ d2 ≥ d3 ≥ ·· · ≥ dn be the degree sequence of the graph G, wheredi = d(vi), for all i is the degree of the vertex vi. Since C is a covering set withminimum cardinality, we can pick the vertices in C as follows.If v1 has maximum degree in the graph G, we pick v1 as the first vertexin C. If all the edges of the graph G are incident to v1, then C = {v1} is theminimum vertex covering set, otherwise, if v2 has the maximum degree in thegraph G−{v1}, where G−{v1} is the graph obtained from G by removing thevertex v1 and all the edges incident to v1, we pick v2 as the second vertex in C.If all the edges of the graph G−{v1} are incident to v2, then C = {v1,v2} isthe minimum vertex covering set, otherwise, we proceed similarly, to obtain theother elements v3,v4, . . . ,vτ of the minimum vertex covering set C.Let C = {v1,v2 . . . ,vτ} be the minimum vertex covering set obtained in this168 HILAL A. GANIE - U. SAMEE - S. PIRZADAway. It is clear that the degree of v1 in G1 = Km1,1∪ (n−m1−1)K1 is d1, givingm1 = d1. Also, the degree of v2 in G1 = Km2,1∪ (n−m2− 1)K1 is either d2 ord2−1, depending on whether v1 and v2 are non-adjacent or adjacent in G, givingm2 ≤ d2. Similarly, it can be seen that mi ≤ d2, for all i = 3,4, . . . ,τ . Therefore,it follows thatE (G)≤ 2√m1+2√m2+ · · ·+2√mτ = 2√d1+2(τ−1)√d2,proving the first part. Equality case can be discussed similar to that as seen inTheorem 2.2.Remark 2.5. For a connected graph G of order n ≥ 2 with m edges havingmaximum and second maximum vertex degrees d1 and d2 respectively, the upperbound (6) is always better than the upper bound (2). If τ = 1, then G ∼= Kn−1,1and so the equality occurs in both the upper bounds. For τ ≥ 2, we have 2√d1+2(τ−1)√d2 ≤ 2τ√d1, giving d1 ≥ d2, which is always true.In a graph G, for the adjacent vertices v1 and v2 with degrees respectivelyd1 and d2, we have the following observation, the proof of which follows on thesimilar lines as the proof of Theorem 2.4.Theorem 2.6. Let G be a connected graph of order n ≥ 2 and m edges havingvertex covering number τ . If v1 and v2 are adjacent vertices in G having degreesrespectively d1 and d2, thenE (G)≤ 2√d1+2(τ−1)√d2−1.Equality holds if and only if τ = 1 and G∼= K1,n−1.Remark 2.7. For a connected graph G of order n≥ 2 with m edges, let v1 and v2be the vertices of G having maximum and second maximum vertex degrees d1and d2, respectively. If v1 and v2 are adjacent, it can seen that the upper boundgiven by Theorem 2.6 is always better than the upper bound given by (2).ON GRAPH ENERGY, MAXIMUM DEGREE AND VERTEX COVER NUMBER 169uu uu uuHHHHHuuuuuu uuuu uuuuuuuuuuuuuuuuuu uuaaaaaaFigure 1: The graph Sω(a,a, . . . ,a), a≥ 1, for ω = 6.3. Bounds for distance and Randic´ energyIt is well known that by using different structural properties one can associatevarious matrices like Laplacian matrix, signless Laplacian matrix, distance ma-trix, normalized Laplacian matrix, Randic´ matrix etc, to a graph G. It is naturalto define an energy like quantity for these matrices. The energy of distance ma-trix called distance energy is defined in [13], the energy of Randic´ matrix calledRandic´ energy is defined in [11], the energy of normalized Laplacian matrixcalled normalized Laplacian energy is defined in [3]. In [11] it is shown thatfor a connected graph G the Randic´ energy and normalized Laplacian energyare same. For more on energy-like definitions of a graph with respect to varioustypes of matrices and related results, we refer to [5, 7–9, 17] and the referencestherein.The following theorem gives an upper for the distance energy DE (G) interms of the vertex covering number τ , the maximum degree d1 and the secondmaximum degree d2 of the graph G.Theorem 3.1. Let G be a connected graph of order n ≥ 2 and m edges havingvertex covering number τ . If d1 and d2 are the maximum and second maximumvertex degrees in G, thenDE (G)≤ 2(d1−1)+2√d21 −d1+1+2(τ−1)[d2−1+√d22 −d2+1],with equality if and only if τ = 1 and G∼= K1,n−1.170 HILAL A. GANIE - U. SAMEE - S. PIRZADAProof. Using the fact that the distance spectrum of K1,n−1 is{n−2+√n2−3n+3,n−2−√n2−3n+3, −2[n−2]}and proceeding similarly as in Theorem 2.4, the result follows.The following upper bound for the distance energy DE (G) of a connectedgraph G was obtained in [2]:DE (G)≤√2(n−1)∑i< jd2i j +n|det(D(G))|2n , (7)where di j is the distance between ith and jth vertex of G.Remark 3.2. It is easy to see that for several families of connected graphs G oforder n ≥ 2 having vertex covering number τ , maximum degree d1 and secondmaximum degree d2, the upper bound given by Theorem 3.1 is better than theupper bound given by (7). If G∼= K1,n−1, then τ = 1, d1 = n−1,d2 = 1 and sothe upper bound given by Theorem 3.1 is better than the upper bound given by(7). This is because equality occurs for G∼= K1,n−1 in Theorem 3.1, while as theupper bound (7) is strict for G∼= K1,n−1.The following theorem gives an upper for the Randic´ energy RE (G), interms of the vertex covering number τ of the graph G.Theorem 3.3. Let G be a connected graph of order n ≥ 2 and m edges havingvertex covering number τ . ThenRE (G)≤ 2τ,with equality if and only if τ = 1 and G∼= K1,n−1.Proof. Using the fact that the Randic´ spectrum of K1,n−1 is{1,0[n−2],−1}and proceeding similarly as in Theorem 2.4, the result follows.The following upper bound for the Randic´ energy RE (G) of a connectedgraph G was obtained in [3]:RE (G)≤√1528(n+1). (8)ON GRAPH ENERGY, MAXIMUM DEGREE AND VERTEX COVER NUMBER 171Remark 3.4. It can be seen that for the connected graphs G of order n ≥ 2having vertex covering number τ , the upper bound given by Theorem 3.3 isbetter than the upper bound given by (8), for all τ ≤√1528n+12 . In particular, ifG ∼= Ka,b, with a ≤ b and a ≤√1528n+12 , then the upper bound Theorem 3.3 isalways better than the upper bound given by (8).Acknowledgements. The research of S. Pirzada is supported by SERB-DST,New Delhi under the research project number MTR/2017/000084.REFERENCES[1] E. Andrade, M. Robbiano and B. San Martin, A lower bound for the energy ofsymmetric matrices and graphs, Linear Algebra Appl. 513 (2017) 264-275.[2] S. B. Bozkurt, A. D. Gu¨ngo¨ra and B. Zhou, Note on the distance energy of graphs,MATCH Commun. Math. Comput. Chem. 64 (2010) 129-134.[3] M. Cavers, S. Fallat and S. Kirkland, On the normalized Laplacian energy andgeneral Randic´ index R−1 of graphs, Linear Algebra Appl. 433 (2010) 172-190.[4] D. Cvetkovic´, M. Doob and H. Sachs, Spectra of graphs-Theory and Application,Academic Press, New York, 1980.[5] K. C. Das, S. Sorgun and I. Gutman, On Randic´ energy, MATCH Comm.Math.Comput. Chem. 73 (2015) 81-92.[6] K. Fan, Maximum properties and inequalities for the eigenvalues of completelycontinuous operators, Proc. Natl. Acad. Sci. USA 37 (1951) 760-766.[7] H. A. Ganie and S. Pirzada, On the bounds for signless Laplacian energy of agraph, Discrete Appl. Math. 228 (2017) 3-13.[8] H. A. Ganie, A. M. Alghamdi and S. Pirzada, On the sum of the Laplacian eigen-values of a graph and Brouwer’s conjecture, Linear Algebra Appl. 501 (2016)376-389.[9] H. A. Ganie, B. A. Chat and S. Pirzada, On the signless Laplacian energy of agraph and energy of line graph, Linear Algebra Appl. 544 (2018) 306-324.[10] H. A. Ganie, S. Pirzada and E. T. Baskoro, On energy, Laplacian energy and p-fold graphs, Elec. J. Graph Theory Appl. 3 (1) (2015) 94-107.[11] I. Gutman, B. Furtula, and S. B. Bozkurt, On Randic´ energy, Linear Algebra Appl.442 (2014) 50-57.[12] I. Gutman, The energy of a graph: old and new results, in: A. Betten, A. Kohn-ert, R. Laue, A. Wassermann (Eds.), Algebraic Combinatorics and Applications,Springer-Verlag, Berlin, (2001) 196-211.172 HILAL A. GANIE - U. SAMEE - S. PIRZADA[13] G. Indulal, I. Gutman and A. Vijaykumar, On the distance energy of a graph,MATCH Commun. Math. Comput. Chem. 60 (2008) 461-472.[14] A. Jahanbani, Some new lower bounds for energy of graphs, Appl. Math.Comp.296 (2017) 233-238.[15] X. Li, Y. Shi and I. Gutman, Graph Energy, Springer, New York, 2012.[16] V. Nikiforov, The energy of graphs and matrices, J. Math. Anal. Appl. 326 (2007)1472-1475.[17] S. Pirzada and H. A. Ganie, On the Laplacian eigenvalues of a graph and Laplacianenergy, Linear Algebra Appl. 486 (2015) 454–468.[18] S. Pirzada, An Introduction to Graph Theory, Universities Press, Orient Black-Swan, Hyderabad (2012).[19] L. Wang and X. Ma, Bounds of graph energy in terms of vertex cover number,Linear Algebra Appl. 517 (2017) 207-216.HILAL A. GANIEDepartment of Mathematics, University of Kashmir,Srinagar, Kashmir, Indiae-mail: hilahmad1119kt@gmail.comU. SAMEEDepartment of Mathematics, Islamia College of Science and Commerce,Srinagar, Kashmir, Indiae-mail: drumatulsamee@gmail.comS. PIRZADADepartment of Mathematics, University of Kashmir,Srinagar, Kashmir, Indiae-mail: pirzadasd@kashmiruniversity.ac.in

On graph energy, maximum degree and vertex cover number

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On graph energy, maximum degree and vertex cover number

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Le Matematiche (Dipartimento di Matematica e Informatica, Università degli Studi di Catania)