On graphs with equal dominating and c-dominating energy

S. M. Hosamani, V. B. Awati and R. M. Honmore

Abstract

Graph energy and domination in graphs are most studied areas of graph theory. In this paper we try to connect these two areas of graph theory by introducing c-dominating energy of a graph G. First, we show the chemical applications of c-dominating energy with the help of well known statistical tools. Next, we obtain mathematical properties of c-dominating energy. Finally, we characterize trees, unicyclic graphs, cubic and block graphs with equal dominating and c-dominating energy.

1 Introduction

All graphs considered in this paper are finite, simple and undirected. In particular, these graphs do not have loops. Let G = (V, E) be a graph with the vertex set V(G) = {v1, v2, v3, ⋯, vn} and the edge set E(G) = {e1, e2, e3, ⋯, em}, that is |V(G)| = n and |E(G)| = m. The vertex u and v are adjacent if uvE(G). The open(closed) neighborhood of a vertex vV(G) is N(v) = {u : uvE(G)} and N[v] = N(v) ∪ {v} respectively. The degree of a vertex vV(G) is denoted by dG(v) and is defined as dG(v) = |N(v)|. A vertex vV(G) is pendant if |N(v)| = 1 and is called supporting vertex if it is adjacent to pendant vertex. Any vertex vV(G) with |N(v)| > 1 is called internal vertex. If dG(v) = r for every vertex vV(G), where r ∈ ℤ+ then G is called r-regular. If r = 2 then it is called cycle graph Cn and for r = 3 it is called the cubic graph. A graph G is unicyclic if |V| = |E|. A graph G is called a block graph, if every block in G is a complete graph. For undefined terminologies we refer the reader to [16].

A subset DV(G) is called dominating set if N[D] = V(G). The minimum cardinality of such a set D is called the domination number γ(G) of G. A dominating set D is connected if the subgraph induced by D is connected. The minimum cardinality of connected dominating set D is called the connected dominating number γc(G) of G [27].

The energy E(G) of a graph G is equal to the sum of the absolute values of the eigenvalues of the adjacency matrix of G. This quantity, introduced almost 30 years ago [13] and having a clear connection to chemical problems [15], has in newer times attracted much attention of mathematicians and mathematical chemists [3, 8, 9, 10, 11, 12, 20,22, 23, 24, 28, 30, 31].

In connection with energy (that is defined in terms of the eigenvalues of the adjacency matrix), energy-like quantities were considered also for the other matrices: Laplacian [15], distance [17], incidence [18], minimum covering energy [1] etc. Recall that a great variety of matrices has so far been associated with graphs [4, 5, 10, 29].

Recently in [25] the authors have studied the dominating matrix which is defined as:

Let G = (V, E) be a graph with V(G) = {v1, v2, ⋯, vn} and let DV(G) be a minimum dominating set of G. The minimum dominating matrix of G is the n × n matrix defined by AD(G) = (aij), where aij = 1 if vivjE(G) or vi = vjD, and aij = 0 if vivjE(G).

The characteristic polynomial of AD(G) is denoted by fn(G, μ) := det(μIAD(G)).

The minimum dominating eigenvalues of a graph G are the eigenvalues of AD(G). Since AD(G) is real and symmetric, its eigenvalues are real numbers and we label them in non-increasing order μ1μ2 ≥ ⋯ ≥ μn. The minimum dominating energy of G is then defined as

ED(G)=i=1n|μi|.

Motivated by dominating matrix, here we define the minimum connected dominating matrix abbreviated as (c-dominating matrix). The c-dominating matrix of G is the n × n matrix defined by ADc(G) = (aij), where

aij=1,ifvivjE;1,ifi=jand viDc;0,otherwise.

The characteristic polynomial of ADc(G) is denoted by fn(G, λ) := det(λ IADc(G)).

The c-dominating eigenvalues of a graph G are the eigenvalues of ADc(G). Since ADc(G) is real and symmetric, its eigenvalues are real numbers and we label them in non-increasing order λ1λ2 ≥ ⋯ ≥ λn. The c-dominating energy of G is then defined as

EDc(G)=i=1n|λi|.

To illustrate this, consider the following examples:

Example 1

Let G be the 5-vertex path P5, with vertices v1, v2, v3, v4, v5 and let its minimum connected dominating set be Dc = {v2, v3, v4}. Then

ADc(G)=0100011100011100011100010

The characteristic polynomial of ADc(G) is λ5 – 3λ4λ3 + 5λ2 + λ – 1 = 0. The minimum connected dominating eigenvalues are λ1 = 2.618, λ2 = 1.618, λ3 = 0.382, λ4 = –1.000 and λ5 = –0.618.

Therefore, the minimum connected dominating energy is EDc(P5) = 6.236.

Example 2

Consider the following graph

Let G be a tree T as shown above and let its minimum connected dominating set be Dc = {b, d, e, f, g, h}. Then

ADc(G)=0100000000111100000001000000000101100000000111000000001110000000011101000000111000000001000000001000

By direct calculation, we get the minimum connected dominating eigenvalues are λ1 = 2.945, λ2 = 2.596, λ3 = 1.896, λ4 = 1.183, λ5 = –1.263, λ6 = –1.152, λ7 = 0.579, λ8 = 0.000, λ9 = –0.268 and λ10 = –0.516.

Therefore, the minimum connected dominating energy is EDc(T) = 12.398.

Example 3

The c-dominating energy of the following graphs can be calculated easily:

  1. EDc(Kn) = (n – 2) + n(n2)+5, where Kn is the complete graph of order n.
  2. EDc(K1,n–1) = 4n3 where K1,n–1 is the star graph.
  3. EDc(Kn×2) = (2n – 3) + 4n(n1)9, where Kn×2 is the coctail party graph.

In this paper, we are interested in studying the mathematical aspects of the c-dominating energy of a graph. This paper has organized as follows: The section 1, contains the basic definitions and background of the current topic. In section 2, we show the chemical applicability of c-dominating energy for molecular graphs G. The section 3, contains the mathematical properties of c-dominating energy. In the last section, we have characterized, trees, unicyclic graphs and cubic graphs and block graphs with equal minimum dominating energy and c-dominating energy. Finally, we conclude this paper by posing an open problem.

2 Chemical Applicability of EDc(G)

We have used the c-dominating energy for modeling eight representative physical properties like boiling points(bp), molar volumes(mv) at 20C, molar refractions(mr) at 20C, heats of vaporization (hv) at 25C, critical temperatures(ct), critical pressure(cp) and surface tension (st) at 20C of the 74 alkanes from ethane to nonanes. Values for these properties were taken from http://www.moleculardescriptors.eu/dataset.htm. The c-dominating energy EDc(G) was correlated with each of these properties and surprisingly, we can see that the EDc has a good correlation with the heats of vaporization of alkanes with correlation coefficient r = 0.995.

The following structure-property relationship model has been developed for the c-dominating energy EDc(G).

hv=10EDc(G)±5.

Figure 3
Figure 3

Correlation of EDc(G) with heats of vaporization of alkanes.

Citation: Applied Mathematics and Nonlinear Sciences 4, 2; 10.2478/AMNS.2019.2.00047

3 Mathematical Properties of c-Dominating Energy of Graph

We begin with the following straightforward observations.

Observation 1

Note that the trace ofADc(G) = γc(G).

Observation 2

LetG = (V, E) be a graph withγc-setDc. Letfn(G, λ) = c0λn + c1λn–1 + ⋯ + cnbe the characteristic polynomial ofG. Then

  1. c0 = 1,
  2. c1 = –|Dc| = –γc(G).

Theorem 3

Ifλ1, λ2, ⋯, λnare the eigenvalues ofADc(G), then

  1. i=1nλi=γc(G)
  2. i=1nλi2=2m+γc(G).

Proof

  1. Follows from Observation 1.
  2. The sum of squares of the eigenvalues of ADc(G) is just the trace of ADc(G)2. Therefore
    i=1nλi2=i=1nj=1naijaji=2i<j(aij)2+i=1n(aii)2=2m+γc(G).

We now obtain bounds for EDc(G) of G, similar to McClelland’s inequalities [21] for graph energy.

Theorem 4

LetGbe a graph of ordernand sizemwithγc(G) = k. Then

EDc(G)n(2m+k).

Proof

Let λ1λ2 ≥ ⋯ ≥ λn be the eigenvalues of ADc(G). Bearing in mind the Cauchy-Schwarz inequality,

(i=1naibi)2(i=1nai)2(i=1nbi)2

we choose ai = 1 and bi = |λi|, which by Theorem 3 implies

EDc2=(i=1n|λi|)2n(i=1n|λi|2)=ni=1nλi2=2(2m+k).

Theorem 5

LetGbe a graph of ordernand sizemwithγc(G) = k. Letλ1λ2 ≥ ⋯ ≥ λnbe a non-increasing arrangement of eigenvalues ofADc(G). Then

EDc(G)2mn+nkα(n)(|λ1||λn|)2

whereα(n)=n[n2](11n[n2]), where [x] denotes the integer part of a real numberk.

Proof

Let a1, a2, ⋯, an and b1, b2, ⋯, bn be real numbers for which there exist real constants a, b, A and B, so that for each i, i = 1, 2, ⋯, n, aaiA and bbiB. Then the following inequality is valid (see [6]).

ni=1naibii=1naii=1nbiα(n)(Aa)(Bb),

where α(n)=n[n2](11n[n2]). Equality holds if and only if a1 = a2 = ⋯ = an and b1 = b2 = ⋯ = bn.

We choose ai := |λi|, bi := |λi|, a = b := |λn| and A = B := |λ1|, i = 1, 2, ⋯, n, inequality (4) becomes

|ni=1n|λi|2(i=1n|λi|)2|α(n)(|λ1||λn|)2.

Since EGc(G)=i=1n|λi|,i=1n|λi|2=i=1n|λi|2=2m+k and EDc(G)n(2m+k), the inequality (5) becomes

n(2m+k)(EDc)2α(n)(|λ1||λn|)2(EDc)22mn+nkα(n)(|λ1||λn|)2.

Hence equality holds if and only if λ1 = λ2 = ⋯ = λn.□

Corollary 6

LetGbe a graph of ordernand sizemwithγc(G) = k. Letλ1λ2 ≥ ⋯ ≥ λnbe a non-increasing arrangement of eigenvalues ofADc(G). Then

EDc(G)2mn+nkn24(|λ1||λn|)2.

Proof

Since α(n)=n[n2](11n[n2])n24, therefore by (3), result follows.□

Theorem 7

LetGbe a graph of ordernand sizemwithγc(G) = k. Letλ1λ2 ≥ ⋯ ≥ λnbe a non-increasing arrangement of eigenvalues ofADc(G). Then

EGc(G)|λ1||λ2|n+2m+k|λ1|+|λn|.

Proof

Let a1, a2, ⋯, an and b1, b2, ⋯, bn be real numbers for which there exist real constants r and R so that for each i, i = 1, 2, ⋯, n holds raibiRai. Then the following inequality is valid (see [11]).

i=1nbi2+rRi=1nai2(r+R)i=1naibi.

Equality of (8) holds if and only if, for at least one i, 1 ≤ in holds rai = bi = Rai.

For bi := |λi|, ai := 1 r := |λn| and R := |λ1|, i = 1, 2, ⋯, n inequality (8) becomes

i=nn|λi|2+|λ1||λn|i=1n1(|λ1|+|λn|)i=1n|λi|.

Since i=1n|λi|2=i=1nλi2=2m+k,i=1n|λi|=EDc(G), from inequality (9),

2m+k+|λ1||λn|n(λ1+λn)EDc(G)

Hence the result.□

Theorem 8

LetGbe a graph of ordernand sizemwithγc(G) = k. Ifξ = |detADc(G)|, then

EDc(G)2m+k+n(n1)ξ2n.

Proof

(EDc(G))2=(i=1n|λi|)2=i=1n|λi|2+ij|λi||λj|.

Employing the inequality between the arithmetic and geometric means, we obtain

1n(n1)ij|λi||λj|(ij|λi||λj|)1n(n1).

Thus,

(EDG)2i=1n|λi|2+n(n1)(ij|λi||λj|)1n(n1)i=1n|λi|2+n(n1)(ij|λi|2(n1))1n(n1)=2m+k+n(n1)ξ2n.

Lemma 9

Ifλ1(G) is the largest minimum connected dominating eigenvalue ofADc(G), thenλ12m+γc(G)n.

Proof

Let X be any non-zero vector. Then we have λ1(A)=maxX0{XAXXX}, see [16]. Therefore, λ1(ADc(G)) ≥ JAJJJ=2m+γc(G)n.

Next, we obtain Koolen and Moulton’s [19] type inequality for EDc(G).

Theorem 10

IfGis a graph of ordernand sizemand 2m + γC(G) ≥ n, then

EDc(G)2m+γc(G)n+(n1)[(2m+γc(G))(2m+γc(G)n)2].

Proof

Bearing in mind the Cauchy-Schwarz inequality,

(i=1naibi)2(i=1nai)2(i=1nbi)2.

Put ai = 1 and bi = |λi| then

(i=2naibi)2(n1)(i=2nbi)2(EDc(G)λ1)2(n1)(2m+γc(G)λ12)EDc(G)λ1+(n1)(2m+γc(G)λ12).

Let

f(x)=x+(n1)(2m+γc(G)x2).

For decreasing function

f(x)01x(n1)(n1)(2m+γc(G)x2)0x2m+γc(G)n.

Since (2m + k) ≥ n, we have 2m+γc(G)n2m+γc(G)nλ1. Also f(λ1)f(2m+γc(G)n).

i.eEDc(G)f(λ1)f(2m+γc(G)n).i.eEDc(G)f(2m+γc(G)n)

Hence by (12), the result follows.□

4 Graphs with equal Dominating and c-Dominating Energy

Its a natural question to ask that for which graphs the dominating energy and c-dominating energy are equal. To answer this question, we characterize graphs with equal dominating energy and c-dominating energy. The graphs considered in this section are trees, cubic graphs, unicyclic graphs, block graphs and cactus graphs.

Theorem 11

LetG = Tbe a tree with at least three vertices, thenED(G) = EDc(G) if and only if every internal vertex ofTis a support vertex.

Proof

Let G = T be a tree of order at least 3. Let F = {u1, u2, ⋯, uk} be the set of internal vertices of T. Then clearly F is the minimal dominating set of G. Therefore in AD(G) the values of ui = 1 in the diagonal entries. Further, observe that 〈F〉 is connected. Hence F is the minimal connected dominating set. Therefore, AD(G) = ADc(G). In general, AD(G) = ADc(G) is true if every minimum dominating set is connected. In other words, AD(G) = ADc(G) if γ(G) = γc(G). Therefore, the result follows from Theorem 2.1 in [2].□

In the next three theorems we characterize unicyclic graphs with AD(G) = ADc(G). Since, AD(G) = ADc(G) if γ(G) = γc(G). Therefore, the proof of our next three results follows from Theorem 2.2, Theorem 2.4 and Theorem 2.5 in [2].

Theorem 12

LetGbe a unicyclic graph with cycleC = u1u2 ⋯, unu1n ≥ 5 and letX = {vC : dG(v) ≥ 2}. ThenED(G) = EDc(G) if the following conditions hold:

  1. (a). EveryvVN[X] withdG(V) ≥ 2 is a support vertex.
  2. (b). 〈Xis connected and |X| ≤ 3.
  3. (c). IfX〉 = P1orP3, both vertices inN(X) of degree at least 3 are supports and ifX〉 = P2, at least one vertex inN(X) of degree at least three is a support.

Theorem 13

LetGbe unicyclic graph with |V(G)| ≥ 4 containing a cycleC = C3, and letX = {vC : dG(v) = 2}. ThenED(G) = EDc(G) if the following conditions hold:

  1. (a). EveryvVN[X] withdG(V) ≥ 2 is a support vertex.
  2. (b). There exists some uniquevCwithdG(v) ≥ 3 or for everyvCofdG(v) ≥ 3 is a support.

Theorem 14

LetGbe unicyclic graph with |V(G)| ≥ 5 containing a cycleC = C4, and letX = {vC : dG(v) = 2}. ThenED(G) = EDc(G) if the following conditions hold:

  1. (a). EveryvVN[X] withdG(V) ≥ 2 is a support vertex.
  2. (b). If |X| = 1, all the three remaining vertices ofCare supports and if |X| ≥ 2, Ccontains at least one support.

Theorem 15

LetGbe a connected cubic graph of ordern, ThenED(G) = EDc(G) ifGK4, C6,K3,3, G1orG2whereG1andG2are given in Fig. 4.

Theorem 16

LetGbe a block graph of withl ≥ 2. ThenED(G) = EDc(G) if every cutvertex ofGis an end block cutvertex.

Proof

Since ED(G) = EDc(G) if γ(G) = γc(G). Therefore, the result follows from Theorem 2 in [7].□

We conclude this paper by posing the following open problem for the researchers:

Open Problem: Construct non- cospectral graphs with unequal domination and connected domination numbers having equal dominating energy and c-dominating energy.

Communicated by Juan Luis García Guirao

References

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    C. Adiga, A. Bayad, I. Gutman and S. Srinivas, The minimum covering energy of a graph, Kragujevac Journal of Science 34 (2012), 39–56.

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    S.Arumugam, J. Paulraj Joseph, On graphs with equal domination and connected domination numbers, Discrete Math. 206 (1999) 45-49.

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    X. Chen, L. Sun and H. Xing, Characterization of graphs with equal domination and connected domination numbers, Discrete Math. 289(2004) 129–135.

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    C. Coulson and G. Rushbrooke, Note on the method of molecular orbitals, Mathematical Proceedings of the Cambridge Philosophical Society 36 (1940), 193–200.

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    V. Consonni and R. Todeschini, New spectral index for molecule description, MATCH Communications in Mathematical and in Computer Chemistry 60 (2008), 3–14.

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    D. Cvetković, P. Rowlinson and S. Simi, Eigenspaces of Graphs, Cambridge University Press (1997).

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    J. Diaz and F. Metcalf, Stronger forms of a class of inequalities of G. Pólya-G.Szegö and L. V. Kantorovich, Bulletin of the American Mathematical Society 69 (1963), 415–418.

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    W. Gao, Z. Iqbal, M. Ishaq, A. Aslam, R. Sarfraz, Topological aspects of dendrimers via distance based descriptors, IEEE Access, 2019, 7(1), 35619-35630.

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    I. Gutman, The energy of a graph, Ber. Math.-Statist. Sekt. Forschungszentrum Graz 103 (1978), 1–22.

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    I. Gutman and O. Polansky, Mathematical Concepts in Organic Chemistry, Springer–Verlag, Berlin, 1986.

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    I. Gutman and B. Zhou, Laplacian energy of a graph, Linear Algebra and its Applications 414 (2006), 29–37.

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    F. Harary, Graph Theory, Addison–Wesely, Reading, 1969.

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    G. Indulal, I. Gutman and A. Vijayakumar, On distance energy of graphs, MATCH Communications in Mathematical and in Computer Chemistry 60 (2008), 461–472.

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    M. Jooyandeh, D. Kiani and M. Mirzakhah, Incidence energy of a graph, MATCH Communications in Mathematical and in Computer Chemistry 62 (2009), 561–572.

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    J. Koolen and V. Moulton, Maximal energy graphs, Advances in Applied Mathematics 26 (2001), 47–52.

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    J. Liu and B. Liu, A Laplacian-energy like invariant of a graph, MATCH Communications in Mathematical and in Computer Chemistry 59 (2008), 355–372.

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    B. McClelland, Properties of the latent roots of a matrix: The estimation of π-electron energies, The Journal of Chemical Physics 54 (1971), 640–643.

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    I. Milovanovć, E. Milovanovć and A. Zakić, A short note on graph energy, MATCH Communications in Mathematical and in Computer Chemistry 72 (2014), 179–182.

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    J. Rada, Energy ordering of catacondensed hexagonal systems, Discrete Applied Mathematics 145 (2005), 437–443.

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    M. Rajesh Kanna, B. Dharmendra and G. Sridhara, The minimum dominating energy of a graph, International Journal of Pure and Applied Mathematics 85 (2013), 707–718.

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    H. Sachs, beziehungen zwischen den in einem graphen enthaltenen kreisen und seinem charakteristischen polynom, Ibid. 11 (1963), 119–134.

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    E. Sampathkumar and H.Walikar, The connected domination number of a graph, Journal of Mathematical and Physical Sciences 13 (1979), 607–613.

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    I. Shparlinski, On the energy of some circulant graphs, Linear Algebra and its Applications 414 (2006), 378–382.

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    N. Trinajstić, Chemical graph theory, CRC Press (1992).

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    A. R. Virk, M, Quraish, Some invariants of flower graph, Appl. Math. Nonl. Sc., 2018, 3, 427-432

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    B. Zhou, Energy of a graph, MATCH Communications in Mathematical and in Computer Chemistry 51 (2004), 111– 118.

If the inline PDF is not rendering correctly, you can download the PDF file here.

  • [1]

    C. Adiga, A. Bayad, I. Gutman and S. Srinivas, The minimum covering energy of a graph, Kragujevac Journal of Science 34 (2012), 39–56.

  • [2]

    S.Arumugam, J. Paulraj Joseph, On graphs with equal domination and connected domination numbers, Discrete Math. 206 (1999) 45-49.

  • [3]

    A. Aslam, S. Ahmad, M. A. Binyamin, W. Gao, Calculating topological indices of certain OTIS Interconnection networks, Open Chemistry, 2019, 17, 220-228.

  • [4]

    A. Balban, Chemical applications of graph theory, Academic Press (1976).

  • [5]

    R. Bapat, Graphs and Matrices, Hindustan Book Agency (2011).

  • [6]

    M. Biernacki, H. Pidek and C. Ryll-Nardzewski, Sur une inégalité entre des intégrales déefinies, Univ. Marie Curie- Sktoodowska A4 (1950), 1–4.

  • [7]

    X. Chen, L. Sun and H. Xing, Characterization of graphs with equal domination and connected domination numbers, Discrete Math. 289(2004) 129–135.

  • [8]

    C. Coulson and G. Rushbrooke, Note on the method of molecular orbitals, Mathematical Proceedings of the Cambridge Philosophical Society 36 (1940), 193–200.

  • [9]

    V. Consonni and R. Todeschini, New spectral index for molecule description, MATCH Communications in Mathematical and in Computer Chemistry 60 (2008), 3–14.

  • [10]

    D. Cvetković, P. Rowlinson and S. Simi, Eigenspaces of Graphs, Cambridge University Press (1997).

  • [11]

    J. Diaz and F. Metcalf, Stronger forms of a class of inequalities of G. Pólya-G.Szegö and L. V. Kantorovich, Bulletin of the American Mathematical Society 69 (1963), 415–418.

  • [12]

    W. Gao, Z. Iqbal, M. Ishaq, A. Aslam, R. Sarfraz, Topological aspects of dendrimers via distance based descriptors, IEEE Access, 2019, 7(1), 35619-35630.

  • [13]

    I. Gutman, The energy of a graph, Ber. Math.-Statist. Sekt. Forschungszentrum Graz 103 (1978), 1–22.

  • [14]

    I. Gutman and O. Polansky, Mathematical Concepts in Organic Chemistry, Springer–Verlag, Berlin, 1986.

  • [15]

    I. Gutman and B. Zhou, Laplacian energy of a graph, Linear Algebra and its Applications 414 (2006), 29–37.

  • [16]

    F. Harary, Graph Theory, Addison–Wesely, Reading, 1969.

  • [17]

    G. Indulal, I. Gutman and A. Vijayakumar, On distance energy of graphs, MATCH Communications in Mathematical and in Computer Chemistry 60 (2008), 461–472.

  • [18]

    M. Jooyandeh, D. Kiani and M. Mirzakhah, Incidence energy of a graph, MATCH Communications in Mathematical and in Computer Chemistry 62 (2009), 561–572.

  • [19]

    J. Koolen and V. Moulton, Maximal energy graphs, Advances in Applied Mathematics 26 (2001), 47–52.

  • [20]

    J. Liu and B. Liu, A Laplacian-energy like invariant of a graph, MATCH Communications in Mathematical and in Computer Chemistry 59 (2008), 355–372.

  • [21]

    B. McClelland, Properties of the latent roots of a matrix: The estimation of π-electron energies, The Journal of Chemical Physics 54 (1971), 640–643.

  • [22]

    I. Milovanovć, E. Milovanovć and A. Zakić, A short note on graph energy, MATCH Communications in Mathematical and in Computer Chemistry 72 (2014), 179–182.

  • [23]

    M. Naeem, M. K. Siddiqui, J. L. G. Guirao, W. Gao, New and modified eccentric indices of octagonal grid Om n, 2018, 3, 209-228.

  • [24]

    J. Rada, Energy ordering of catacondensed hexagonal systems, Discrete Applied Mathematics 145 (2005), 437–443.

  • [25]

    M. Rajesh Kanna, B. Dharmendra and G. Sridhara, The minimum dominating energy of a graph, International Journal of Pure and Applied Mathematics 85 (2013), 707–718.

  • [26]

    H. Sachs, beziehungen zwischen den in einem graphen enthaltenen kreisen und seinem charakteristischen polynom, Ibid. 11 (1963), 119–134.

  • [27]

    E. Sampathkumar and H.Walikar, The connected domination number of a graph, Journal of Mathematical and Physical Sciences 13 (1979), 607–613.

  • [28]

    I. Shparlinski, On the energy of some circulant graphs, Linear Algebra and its Applications 414 (2006), 378–382.

  • [29]

    N. Trinajstić, Chemical graph theory, CRC Press (1992).

  • [30]

    A. R. Virk, M, Quraish, Some invariants of flower graph, Appl. Math. Nonl. Sc., 2018, 3, 427-432

  • [31]

    B. Zhou, Energy of a graph, MATCH Communications in Mathematical and in Computer Chemistry 51 (2004), 111– 118.

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