On q-distance energy of a graph

M. R. Rajesh Kanna 1 , R. Pradeep Kumar 2 , Soner Nandappa 2 ,  and Ismail Naci Cangul 3
  • 1 Department of Mathematics, Sri D Devaraj Urs Government First Grade College, Hunsur, India
  • 2 Department of Studies in Mathematics, GSSS Institute of Engineering and Technology for Women, KRS Road, Metagalli, Mysuru, India
  • 3 Bursa Uludag University, Faculty of Arts and Science, Department of Mathematics, Gorukle, Turkey
M. R. Rajesh Kanna, R. Pradeep Kumar
  • Department of Studies in Mathematics, GSSS Institute of Engineering and Technology for Women, KRS Road, Metagalli, Mysuru, 570 016, India
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, Soner Nandappa
  • Department of Studies in Mathematics, GSSS Institute of Engineering and Technology for Women, KRS Road, Metagalli, Mysuru, 570 016, India
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and Ismail Naci Cangul
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  • Bursa Uludag University, Faculty of Arts and Science, Department of Mathematics, Gorukle, 16059, Bursa, Turkey
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Abstract

Three main tools to study graphs mathematically are to make use of the vertex degrees, distances and matrices. The classical graph energy was defined by means of the adjacency matrix in 1978 by Gutman and has a large number of applications in chemistry, physics and related areas. As a result of its importance and numerous applications, several modifications of the notion of energy have been introduced since then. Most of them are defined by means of graph matrices constructed by vertex degrees. In this paper we define another type of energy called q-distance energy by means of distances and matrices. We study some fundamental properties and also establish some upper and lower bounds for this new energy type.

1 Introduction and preliminaries

Let G be a graph with n vertices and m edges and let A = (aij) be the adjacency matrix of G. The eigenvalues λ1,λ2,...,λn of A in non-increasing order are called the eigenvalues of the graph G. As A is real symmetric, the eigenvalues of G are real with sum equal to zero. The energy E(G) of G is defined by I. Gutman, [7], to be the sum of the absolute values of the eigenvalues of G, i.e.

E(G)=i=1n|λi|.
For details on the mathematical aspects of the theory of graph energy, see the review [9], papers [4, 5, 8] and the references cited therein. The basic properties including various upper and lower bounds for the energy of a graph have been established in [16, 18], and the notion of graph energy has been found to have remarkable chemical applications in the molecular orbital theory of conjugated molecules, [6, 10]. In [11], a QSPR study is made for the energy of certain graph theoretical matrices. In [17], some graph operations are realized.

The distance matrix of G is the square matrix of order n whose (i, j)-th entry is the distance between the vertices vi and vj which is defined as the length of the shortest path between these two vertices. Let μ1, μ2, ..., μn be the eigenvalues of the distance matrix of G. The distance energy DE is defined by

DE=DE(G):=i=1n|μi|.
Detailed information on distance energy can be found in [3, 13, 14, 21]. The distance energy of the join of two given graphs can be found in [20]. In [19], a generalization of the distance notion is given.

Recently R. B. Bapat et al., [1], defined a new distance matrix, called as the q-distance matrix denoted by

Aq(G)=(qij).
For an indeterminate q, the entries qij of this new matrix are defined by
qij={1+q+q2++qk1,ifk=dij,0,ifi=j,
where k = dij is the distance between the vertices vi and vj. Each entry of Aq(G) is a polynomial in q. Observe that Aq(G) is an entry-wise non-negative matrix for all q ≥ −1.

The characteristic polynomial of Aq(G) is defined by

fn(G,μ)=det(μIAq(G)).
The q-distance eigenvalues of the graph G are similarly the eigenvalues of Aq(G). Since Aq(G) is real and symmetric, its eigenvalues are also real numbers and we label them in non-increasing order μ1μ2···μn. The q-distance energy of G is denoted by Eq(G) and is defined by
Eq(G)=i=1n|μi|.
Note that the trace of Aq(G) = 0 and also if q = 1, then the q-distance energy coincides with distance energy of a graph.

Example 1

Consider a crown graphS60as in Fig. 1.1.

Figure 1.1
Figure 1.1

The crown graphS06

Citation: Applied Mathematics and Nonlinear Sciences 5, 2; 10.2478/amns.2020.2.00017

As

Aq(S60)=(01+q1+q1+q+q2111+q01+q11+q+q211+q1+q0111+q+q21+q+q21101+q1+q11+q+q211+q01+q111+q+q21+q1+q0),
the characteristic polynomial ofS06is
(μ+q2q+1)(μq23q5)(μ+q2+2q+1)2(μq2+1)2.
Then the q-distance spectrum ofS60would be
(q2+q1q2+3q+5q22q1q211122)
and therefore the q-distance energy ofS60is found as
Eq(S60)=|(q2q+1)|+|q2+3q+5|+2|(q2+2q+1)|+2|q21|=q2q+1+q2+3q+5+2q2+4q+2+2q22=6q2+6q+6.

2 Properties of the q-distance eigenvalues

Here we study some fundamental properties of the q-distance eigenvalues. We start with the following well-known lemmas:

Lemma 2

Let G be a graph with the adjacency matrix A and the spectrum spec(G) = {μ1, μ2,..., μn}. Then it is well-known that

detA=i=1nμi.
In addition, for any polynomial P(x), the value P(μ) is an eigenvalue of P(A) and hence
detP(A)=i=1nP(μi).

Lemma 3

LetB=(B0B1B1B0)be a symmetric 2 × 2 block matrix. Then the spectrum of B is the union of the spectra of B0 + B1and B0B1.

We can now prove the following results on q-distance eigenvalues:

Theorem 4

Let G be an (n,m) graph of diameter 2 with the q-distance eigenvalues μ1, μ2,..., μn. Then

i=1nμi2=2m+(n2n2m)(1+q)2.

Proof

In a q-distance adjacency matrix Aq(G), there are 2m elements equal to 1 and n2n − 2m elements equal to (1 + q). Therefore

i=1nμi2=i=1nj=1nqijqji=i=1nj=1n(qij)2=(2m)(1)2+(n2n2m)(1+q)2=2m+(n2n2m)(1+q)2.

Theorem 5

Let G be an (n,m) graph of diameter 2 and let μ1be its greatest q-distance eigenvalue. Then

μ1n(n1)(1+q)2mqn.

Proof

Let G be a connected graph of diameter 2 with its vertices labeled as v1,v2,...,vn and let di denote the degree of vi. As G is of diameter 2, it is easy to observe that the ith row of Aq consists of di times 1s and ndi − 1 times 2s. Let X = [1,1,1,...,1] be a vector containing only 1s. Then by the Rayleigh principle, we have

μ1XAqXTXXT=i=1n(di(1)+(ndi1)(1+q))n=(2m+(n1)n(1+q)2m(1+q))n=n(n1)(1+q)2mqn.

Theorem 6

Let G be an r-regular graph of diameter 2 with r, μ2,..., μn as its eigenvalues. Then the q-distance eigenvalues of G arerq + (n − 1)(1 + q),−2 − (1 + q), −3 − (1 + q),...,−n − (1 + q).

Proof

Let G be an r-regular graph with diameter 2 and adjacency matrix A. A¯ is the adjacency matrix of G¯ . Then the q-distance adjacency matrix of G will be

Aq=A+(1+q)A¯.
If r, μ2,..., μn are the eigenvalues of A with rμ2···μn, then n − 1 − r,μ2 − 1, −μ3 − 1,...,−μn − 1 are the eigenvalues of A¯ . By Eqn. (1), the theorem is proved.

Theorem 7

Let G be a connected r-regular graph of diameter one or two with the adjacency matrix A and spec(G) = {r, μ2, μ3,..., μn}. Then the product graph H = G × K2is (r + 1)-regular and of diameter 2 or 3 with spec(H) = {−rq(1 + q) + 2n(1 + q) + q2(n − 1) − 1,−i(1 + q) − (1 + 2q + q2),−rq(1 − q) + q2(1 − n) − 1,−i(1 − q) − (1 − q2)} for i = 1,2,3,...,n.

Proof

Since G is of diameter 1 or 2, its q-distance matrix is A+(1+q)A¯ . Then the q-distance matrix of H is of the form

(A+(1+q)A¯J+qA+(q+q2)A¯J+qA+(q+q2)A¯A+(1+q)A¯).
By Lemma 3, the spectrum of H is the union of the spectra of (1+q)A+(1+2q+q2)A¯+J and (1q)A+(1q2)A¯J . If r, μ2,..., μn are the eigenvalues of A with rμ2 ≥ ··· ≥ μn then nr − 1,−μ2 − 1,−μ3 − 1,...,−μn − 1 are the eigenvalues of A¯ . Also we know that n,0,0,...,0 are the eigenvalues of J. Therefore, the theorem follows.

3 Bounds for the q-distance energy

In this section, we find several bounds for the q-distance energy Eq(G). The first one is a sequel of the work of McClelland's, [18].

Theorem 8

Let G be a simple (n,m) graph with diameter 2. If P = |detAq(G)|, then

2m+(n2n2m)(1+q)2+n(n1)P2nEq(G)n(2m+(n2n2m)(1+q)2).

Proof

Recall that the Cauchy-Schwarz inequality states that

(i=1naibi)2(i=1nai2)(i=1nbi2).
If we substitute ai = 1 and bi = | μi |, then we obtain
(i=1n|μi|)2(i=1n1)(i=1nμi2).
Hence by Theorem 4, we have
Eq2(G)n(2m+(n2n2m)(1+q)2)
and therefore we obtain
Eq(G)n(2m+(n2n2m)(1+q)2).
Since the arithmetic mean is not smaller than the geometric mean, we have
1n(n1)ij|μiμj|[ij|μiμj|]1n(n1)=[i=1n|μi|2(n1)]1n(n1)=[i=1n|μi|]2n=[i=1nμi]2n=|detAq(G)|2n=P2n.
Therefore
ij|μiμj|n(n1)P2n.
Now consider
Eq2(G)=(i=1n|μi|)2=i=1n|μi|2+ij|μi||μj|.
Therefore, by Theorem 4, we obtain
Eq2(G)2m+(n2n2m)(1+q)2+n(n1)P2n
and hence
Eq(G)2m+(n2n2m)(1+q)2+n(n1)P2n.

Now, we find another bound for Eq(G) which is a sequel to the work of Koolen and Moulton's, [12].

Theorem 9

If G is an (n,m) graph with diameter 2 so that

n(n1)(1+q)2mqn1,
then
Eq(G)n(n1)(1+q)2mqn+(n1)[(2m+(n2n2m)(1+q)2(n(n1)(1+q)2mqn)2].

Proof

By substituting ai = 1 and bi =| μi | in Cauchy-Schwarz inequality, we have

(i=2n|μi|)2i=2n1i=2nμi2[Eq(G)μ1]2(n1)(2m+(n2n2m)(1+q)2).
Hence
Eq(G)μ1+(n1)(2m+(n2n2m)(1+q)2μ12).
Let
f(x)=x+(n1)(2m+(n2n2m)(1+q)2x2).
Then for a decreasing function f (x), the fact f(x) ≤ 0 implies that
1x(n1)(n1)(2m+(n2n2m)(1+q)2x2)0.
From this we obtain
x2m+(n2n2m)(1+q)2n.
Therefore the function f (x) is decreasing in the interval
(2m+(n2n2m)(1+q)2n,2m+(n2n2m)(1+q)2).
Clearly the number (n(n−1)(1+q)−2mq)/n belongs to that interval and since μ1 ≥ (n(n−1)(1+q)−2mq)/n, we have (n(n1)(1+q)2mq)/nμ12m+(n2n2m)(1+q)2 . By Lemma 5.3, we can write f (μ1) ≤ f((n(n − 1)(1 + q) − 2mq)/n). Hence
Eq(G)f(μ1)f((n(n1)(1+q)2mq)/n)
implying the result.

Bapat and Pati, [2], proved that if the graph energy is a rational number, then it is an even integer. A similar result for q-distance energy can be given as follows:

Lemma 10

Let G be an (n,m) graph. If the q-distance energy Eq(G) of G is a rational number, then

Eq(G)|0|(mod2).

Proof

Proof is similar to Theorem 5.4 of [15].

4 Join of two graphs

One of the ways of studying graphs is to make use of smaller graphs usually those subgraphs whose own are the components of the given graph. Similarly to this idea, many graph operations, sometimes called graph products, are defined to make the necessary calculations on some given graphs by means of similar calculations on some smaller graphs. In this section, we shall study one of the most practical of these products, called the join, of two graphs and calculate the q-distance energy of it. Other operations can be applied similarly to obtain some other properties.

Definition 1

The join of two graphs G1 and G2 denoted by G1G2 is a larger graph obtained from G1 and G2 by joining each vertex of G1 to all the vertices of G2.

Figure 4.1
Figure 4.1

Join of two graphs

Citation: Applied Mathematics and Nonlinear Sciences 5, 2; 10.2478/amns.2020.2.00017

Theorem 11

Let G1be an r1-regular graph on n1vertices having diam(G1) ≤ 2 and G2be an r2-regular graph on n2vertices having diam(G2) ≤ 2. Let further φ(G1 : μ) and φ(G2 : μ) be the q-distance characteristic polynomials of G1and G2, respectively. Then the q-distance characteristic polynomial of the q-distance matrix of G1G2is

(μ(1+q)(n11)+r1q)(μ(1+q)(n21)+r2q)n1n2(μ(1+q)(n11)+r1q)(μ(1+q)(n21)+r2q)ϕ(G1:μ)ϕ(G2:μ).

Proof

Let us assume that v1, v2, ..., vn1 be the vertices of the graph G1 and u1, u2, ..., un2 be the vertices of the graph G2. Let qij denote the q-distance between the vertices vi and vj in G1 and qij denote the q-distance between the vertices ui and uj in G2. In G1, every vertex is at distance 1 from r1 vertices and at distance 1 + q from the remaining n1 − 1 − r1 vertices. Therefore for i = 1, 2, 3, ..., n1, we can write

j=1n1qij=1(r1)+(1+q)(n11r1)=r1+(1+q)(n11)r1r1q=(1+q)(n11)r1q
and similarly, for i = 1,2,3,...,n2, we have
j=1n2qij=(1+q)(n21)r2q.

Let Eq(G1G2) be the q-distance adjacency matrix of the join graph G1G2. Then this matrix Aq(G1G2) has the form

v1   v2  v3   vn1  u1  u2  u3   un2v1v2v3vn1u1u2u3un1(0q12q13q1n11111q210q23q2n11111q31q320q3n11111qn11qn12qn130111111110q12q13q1n21111q210q23q2n21111q31q320q3n21111qn21qn22qn230)(n1+n2)×(n1+n2)
Let φ(G1G2 : μ) denote the q-distance characteristic polynomial of G1G2; i.e.,
ϕ(G1G2:μ)=|μEq(G1G2)|.

This polynomial is equal to the following determinant

|μq12q13q1n11111q21μq23q2n11111q31q32μq3n11111qn11qn12qn13μ11111111μq12q13q1n21111q21μq23q2n21111q31q32μq3n21111qn21qn22qn23μ|(n1+n2)×(n1+n2).

Applying the row operations Rn1+2=Rn1+2Rn1+1 ; Rn1+3=Rn1+3Rn1+1 ;··· ; Rn1+n2=Rn1+n2Rn1+1 to the above determinant, we see that the determinant becomes

|μq12q13q1n11111q21μq23q2n11111q31q32μq3n11111qn11qn12qn13μ11111111μq12q13q1n20000q21μμ+q12q23+q13q2n2+q1n20000q31μq32+q12μ+q13q3n2+q1n20000qn21μqn22+q12qn23+q13μ+q1n2|.

Applying the column operation Cn1+1=Cn1+1+Cn1+2++Cn1+n2 , using the fact that qij=qji and the equations above, the same determinant becomes

|μq12q1n1n2111q21μq2n1n2111q31q32q3n1n2111qn11qn12μn2111111μ(1+q)(n21)r2qq12q13q1n20000μ+q12q23+q13q2n2+q1n20000q32+q12μ+q13q3n2+q1n20000qn22+q12qn23+q13μ+q1n2|=|μq12q13q1n1n2q21μq23q2n1n2q31q32μq3n1n2qn11qn12qn13μn21111μ(1+q)(n21)+r2q||B|,
where
|B|=|μ+q12q23+q13q2n2+q1n2q32+q12μ+q13q3n2+q1n2qn22+q12qn23+q12μ+q1n2|(n21)×(n21).
Applying the row operations R2=R2R1 , R3=R3R1 , ···, Rn1=Rn1R1 , the above determinant transforms to
|μq12q13q1n1n2q21μμ+q12q23+q13q2n1+q1n10q31μq32+q12μ+q13q3n1+q1n10qn11μqn12+q12qn13+q13μ+q1n101111μ(1+q)(n21)+r2q||B|.
Applying the column operation C1=C1+C2++Cn1 and using the above equations, the determinant becomes
|μ(1+q)(n11)+r1qq12q13q1n1n20μ+q12q23+q13q2n1+q1n100q32+q12μ+q13q3n1+q1n100qn12+q12qn13+q13μ+q1n10n1111μ(1+q)(n21)+r2q||B|.
Expanding it along the first column C1, we obtain
ϕ(G1G2:μ)={(μ(1+q)(n11)+r1q)Δ1(1)n1n1Δ2}|B|,
where
Δ1=|μ+q12q23+q13q2n1+q1n10q32+q12μ+q13q3n1+q1n10qn12+q12qn13+q13μ+q1n10111μ(1+q)(n11)+r2q|=(μ(1+q)(n11)+r2q)|A|(1)n1+n2=(μ(1+q)(n11)+r2q)|A|
and
Δ2=|q12q13q1n1n2μ+q12q23+q13q2n1+q1n10q32+q12μ+q13q3n1+q1n10qn12+q12qn13+q13μ+q1n10|=(1)n1+1(n2)|A|=n2(1)n1|A|.
Therefore we have
ϕ(G1G2:μ)=((μ(1+q)(n11)+r1q)(μ(1+q)(n21)+r2q)|A|)(1)n1n1n2(1)n1|A|)|B|,
i.e.
ϕ(G1G2:μ)=|A||B|[(μ(1+q)(n11)+r1q)(μ(1+q)(n21)+r2q)n1n2]
where
|A|=|μ+q12q23+q13q2n1+q1n1q32+q12μ+q13q3n1+q1n1qn12+q12qn13+q13μ+q1n1|.
Clearly
|A|=1(μ(1+q)(n11)+r1q)×|μ(1+q)(n11)+r1qq12q13q1n10μ+q12q23+q13q2n1+q1n10q32+q12μ+q13q3n1+q1n10qn12+q12qn13+q13μ+q1n1|.
Applying the operation C1=C1(C2+C3++Cn1) , the determinant becomes
|A|=1μ(1+q)(n11)+r1q×|μq12q13q1n1μq21μ+q12q23+q13q2n1+q1n1μqn31q32+q12μ+q13q3n1+q1n1μqn11qn12+q12qn13+q13μ+q1n1|.
Applying the operations R = R2 + R1; R3=R3+R1 ;··· ; Rn=Rn1+R1 , we have
|A|=1μ(1+q)(n11)+r1q×|μq12q13q1n1q21μq23q2n1q31q32μq3n1qn11qn12qn13μ|=1μ(1+q)(n11)+r1qϕ(G1;μ).
Similarly,
|B|=1μ(1+q)(n21)+r2qϕ(G2;μ).
By substituting these values in the above equation, we have the required result.

Theorem 12

Let G1and G2be r1and r2regular graphs with n1and n2vertices, respectively. If diam(G1) ≤ 2 and diam(G2) ≤ 2, then

Eq(G1G2)={Eq(G1)+Eq(G2),ifRKn1n2,Eq(G1)+Eq(G2)(R+K)+(R+K)24(RKn1n2),ifRK<n1n2,
where R = (1 + q)(n1 − 1) − r1q and K = (1 + q)(n2 − 1) − r2q.

Proof

From Theorem 2, we have

ϕ(G1G2;μ)=((μ(1+q)(n11)+r1q)(μ(1+q)(n21)+r2q)n1n2))(μ(1+q)(n11)+r1q)(μ(1+q)(n21)+r2q)ϕ(G1;μ)ϕ(G2;μ)
implying that
ϕ(G1G2;μ)=((μR)(μK)n1n2)ϕ(G1;μ)ϕ(G2;μ)(μR)(μK),
where R = (1 + q)(n1 − 1) − r1q and K = (1 + q)(n2 − 1) − r2q; i.e.,
(μR)(μK)ϕ(G1G2;μ)=((μR)(μK)n1n2)ϕ(G1;μ)ϕ(G2;μ).
That is,
(μR)(μK)ϕ(G1G2;μ)=(μ2(R+K)μ+RKn1n2)ϕ(G1;μ)ϕ(G2;μ).

Let

P1(μ)=(μR)(μK)ϕ(G1G2;μ)
and
P2(μ)=(μ2(R+K)μ+RKn1n2)ϕ(G1;μ)ϕ(G2;μ).
The roots of the equation P1(μ) = 0 are R,K and q–distance eigenvalues of G1G2. Therefore, the sum of the absolute values of the roots of P1(μ) = 0 is
R+K+Eq(G1G2)
and similarly the roots of the equation P2(μ) = 0 are q–distance eigenvalues of G1 and G2 and hence
R+K+(R+K)24(RKn1n2)2,R+K(R+K)24(RKn1n2)2.
Therefore, the sum of the absolute values of the roots of P2(μ) = 0 is
Eq(G1)+Eq(G2)+|R+K+(R+K)24(RKn1n2)2|+|R+K(R+K)24(RKn1n2)2|.
Since P1(μ) = P2(μ), from above equations, we get,
R+K+Eq(G1G2)=Eq(G1)+Eq(G2)+|R+K+(R+K)24(RKn1n2)2|+|R+K(R+K)24(RKn1n2)2|.

Case 1: If RK ≥ n1n2, the last equation reduces to

R+K+Eq(G1G2)=Eq(G1)+Eq(G2)+R+K+(R+K)24(RKn1n2)2+R+K(R+K)24(RKn1n2)2,
and therefore
R+K+Eq(G1G2)=Eq(G1)+Eq(G2)+R+K
implying that
Eq(G1G2)=Eq(G1)+Eq(G2).

Case 2: If RK < n1n2, this time the last equation above reduces to

R+K+Eq(G1G2)=Eq(G1)+Eq(G2)+R+K+(R+K)24(RKn1n2)2+R+K(R+K)24(RKn1n2)2.
Hence we get
R+K+Eq(G1G2)=Eq(G1)+Eq(G2)+(R+K)24(RKn1n2)
and finally
Eq(G1G2)=Eq(G1)+Eq(G2)(R+K)+(R+K)24(RKn1n2).

5 Brief summary and conclusion

Energy is a very important subject of graph theory with many applications in physics and chemistry. Similarly to the classical graph energy, there are a few other types of energy in graphs which are similarly defined by means of some other matrices. In this paper, we have defined a new type of energy called q-distance energy. As the distances are calculated between the vertices of the graph representing the atoms in the corresponding molecule, the q-distance energy is expected to have applications in chemistry due to its effect on the intermolcecular forces which affect the graph energy. The q-distance energy has been obtained for the join of two graphs. Similar studies can be made for other graph operations. Also, we have established lower and upper bounds for this new energy.

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    S. M. Hosamani, B. B. Kulkarni, R. G. Boli, V. M. Gadag, (2017), QSPR Analysis of Certain Graph Theoretical Matrices and Their Corresponding Energy, Appl. Maths and Nonlin. Sciences, 2 (1), 131–150

    • Crossref
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    J. H. Koolen, V. Moulton, (2001), Maximal Energy Graphs, Adv. Appl. Math., 26, 47–52

    • Crossref
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    G. Indulal, (2009), Sharp Bounds on the Distance Spectral Radius and the Distance Energy of Graphs, Lin. Algebra Appl., 430, 106–113

    • Crossref
    • Export Citation
  • [14]

    G. Indulal, I. Gutman, A. Vijayakumar, (2008), On Distance Energy of Graphs, MATCH Commun. Math. Comput. Chem., 60, 461–472

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    M. R. R. Kanna, B. N. Dharmendra, G. Sridhara, (2013), Minimum dominating energy of a graph, International Journal of Pure and Applied Mathematics, 85 (4), 707–718

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    H. Liu, M. Lu, F. Tian, (2007), Some Upper Bounds for the Energy of Graphs, Journal of Mathematical Chemistry, 41 (1), 45–57

    • Crossref
    • Export Citation
  • [17]

    V. Lokesha, T. Deepika, P. S. Ranjini, I. N. Cangul, (2017), Operations of Nanostructures via SDD, ABC4 and GA5 indices, Appl. Maths and Nonlin. Sciences, 2 (1), 173–180

    • Crossref
    • Export Citation
  • [18]

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

    • Crossref
    • Export Citation
  • [19]

    A. R. Nizami, A. Perveen, W. Nazeer, M. Baqir, (2018), WALK POLYNOMIAL: A New Graph Invariant, Appl. Maths and Nonlin. Sciences, 3 (1), 321–330

    • Crossref
    • Export Citation
  • [20]

    H. S. Ramane, I. Gutman, D. S. Revankar, (2008), Distance equienergetic graphs, MATCH Commun. Math. Comput. Chem., 60, 473–484

  • [21]

    H. S. Ramane, D. S. Revankar, I. Gutman, S. B. Rao, B. D. Acharya, H. B. Walikar, (2008), Bounds for the distance energy of a graph, Kragujevac J. Math., 31, 59–68

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    R. B. Bapat, A. K. Lal, S. Pati, (2006), A q-analogue of the distance matrix of a tree, Linear Algebra and its Applications, 416 (2–3), 799–814

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    R. B. Bapat, S. Pati, (2011), Energy of a graph is never an odd integer, Bull. Kerala Math. Assoc., 1, 129–132

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    G. Caporossi, E. Chasset, B. Furtula, (2009), Some conjectures and properties on distance energy, Les Cahiers du GERAD, G-2009-64, 1–7

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    D. Cvetković, I. Gutman (eds.), (2009), Applications of Graph Spectra, Beograd: Matematicki institut SANU

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    D. Cvetković, I. Gutman (eds.), (2011), Selected Topics on Applications of Graph Spectra, Beograd: Matematicki institut SANU

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    A. Graovac, I. Gutman, N. Trinajstić, (1977), Topological Approach to the Chemistry of Conjugated Molecules, Springer, Berlin

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    I. Gutman, (1978), The Energy of a Graph, Ber. Math-Statist. Sekt. Forschungsz. Graz, 103, 1–22

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    I. Gutman, (2001), The Energy of a Graph: Old and New Results, ed. by A. Betten, A. Kohnert, R. Laue, A. Wassermann, Algebraic Combinatorics and Applications, Springer, Berlin, 196–211

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    I. Gutman, X. Li, J. Zhang, (2009), Graph Energy, ed. by M. Dehmer, F. Emmert-Streib, Analysis of Complex Networks. From Biology to Linguistics, Wiley - VCH, Weinheim, 154–174

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    I. Gutman, O. E. Polansky, (1986), Mathematical Concepts in Organic Chemistry, Springer-Verlag, Berlin

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    S. M. Hosamani, B. B. Kulkarni, R. G. Boli, V. M. Gadag, (2017), QSPR Analysis of Certain Graph Theoretical Matrices and Their Corresponding Energy, Appl. Maths and Nonlin. Sciences, 2 (1), 131–150

    • Crossref
    • Export Citation
  • [12]

    J. H. Koolen, V. Moulton, (2001), Maximal Energy Graphs, Adv. Appl. Math., 26, 47–52

    • Crossref
    • Export Citation
  • [13]

    G. Indulal, (2009), Sharp Bounds on the Distance Spectral Radius and the Distance Energy of Graphs, Lin. Algebra Appl., 430, 106–113

    • Crossref
    • Export Citation
  • [14]

    G. Indulal, I. Gutman, A. Vijayakumar, (2008), On Distance Energy of Graphs, MATCH Commun. Math. Comput. Chem., 60, 461–472

  • [15]

    M. R. R. Kanna, B. N. Dharmendra, G. Sridhara, (2013), Minimum dominating energy of a graph, International Journal of Pure and Applied Mathematics, 85 (4), 707–718

  • [16]

    H. Liu, M. Lu, F. Tian, (2007), Some Upper Bounds for the Energy of Graphs, Journal of Mathematical Chemistry, 41 (1), 45–57

    • Crossref
    • Export Citation
  • [17]

    V. Lokesha, T. Deepika, P. S. Ranjini, I. N. Cangul, (2017), Operations of Nanostructures via SDD, ABC4 and GA5 indices, Appl. Maths and Nonlin. Sciences, 2 (1), 173–180

    • Crossref
    • Export Citation
  • [18]

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

    • Crossref
    • Export Citation
  • [19]

    A. R. Nizami, A. Perveen, W. Nazeer, M. Baqir, (2018), WALK POLYNOMIAL: A New Graph Invariant, Appl. Maths and Nonlin. Sciences, 3 (1), 321–330

    • Crossref
    • Export Citation
  • [20]

    H. S. Ramane, I. Gutman, D. S. Revankar, (2008), Distance equienergetic graphs, MATCH Commun. Math. Comput. Chem., 60, 473–484

  • [21]

    H. S. Ramane, D. S. Revankar, I. Gutman, S. B. Rao, B. D. Acharya, H. B. Walikar, (2008), Bounds for the distance energy of a graph, Kragujevac J. Math., 31, 59–68

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