1. bookVolume 5 (2020): Issue 2 (July 2020)
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On Solutions of Fractional order Telegraph Partial Differential Equation by Crank-Nicholson Finite Difference Method

Published Online: 10 Aug 2020
Page range: 85 - 98
Received: 04 Apr 2019
Accepted: 17 Apr 2019
Journal Details
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Journal
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01 Jan 2016
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Languages
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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.

Keywords

MSC 2010

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|.E(G) = \sum\limits_{i = 1}^n |{\lambda _i}|. 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|.DE = DE(G): = \sum\limits_{i = 1}^n |{\mu _i}|. 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).{A_q}(G) = ({q_{ij}}). For an indeterminate q, the entries qij of this new matrix are defined by qij={1+q+q2++qk1,ifk=dij,0,ifi=j,{q_{ij}} = \left\{ {\matrix{ {1 + q + {q^2} + \cdots + {q^{k - 1}},} & {{\kern 1pt} \,\,\,\,{\rm if}{\kern 1pt} {\kern 1pt} {\kern 1pt} k = {d_{ij}},} \cr {0,} & {{\rm if}{\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} i = j,} \cr } } \right. 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)).{f_n}(G,\mu ) = det\,(\mu I - {A_q}(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|.{E_q}(G) = \sum\limits_{i = 1}^n |{\mu _i}|. 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 graphS60S_6^0as in Fig. 1.1.

Figure 1.1

The crown graphS06S_0^6

AsAq(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),{A_q}(S_6^0) = \left( {\matrix{ 0 & {1 + q} & {1 + q} & {1 + q + {q^2}} & 1 & 1 \cr {1 + q} & 0 & {1 + q} & 1 & {1 + q + {q^2}} & 1 \cr {1 + q} & {1 + q} & 0 & 1 & 1 & {1 + q + {q^2}} \cr {1 + q + {q^2}} & 1 & 1 & 0 & {1 + q} & {1 + q} \cr 1 & {1 + q + {q^2}} & 1 & {1 + q} & 0 & {1 + q} \cr 1 & 1 & {1 + q + {q^2}} & {1 + q} & {1 + q} & 0 \cr } } \right),the characteristic polynomial ofS06S_0^6is(μ+q2q+1)(μq23q5)(μ+q2+2q+1)2(μq2+1)2.(\mu + {q^2} - q + 1)(\mu - {q^2} - 3q - 5)(\mu + {q^2} + 2q + {1)^2}{(\mu - {q^2} + 1)^2}.Then the q-distance spectrum ofS60S_6^0would be(q2+q1q2+3q+5q22q1q211122)\left( {\matrix{ { - {q^2} + q - 1} & {{q^2} + 3q + 5} & { - {q^2} - 2q - 1} & {{q^2} - 1} \cr 1 & 1 & 2 & 2 \cr } } \right)and therefore the q-distance energy ofS60S_6^0is found asEq(S60)=|(q2q+1)|+|q2+3q+5|+2|(q2+2q+1)|+2|q21|=q2q+1+q2+3q+5+2q2+4q+2+2q22=6q2+6q+6.\matrix{ {{E_q}(S_6^0)} \hfill & { = | - ({q^2} - q + 1)| + |{q^2} + 3q + 5| + 2 \cdot | - ({q^2} + 2q + 1)| + 2 \cdot |{q^2} - 1|} \hfill \cr {} \hfill & { = {q^2} - q + 1 + {q^2} + 3q + 5 + 2{q^2} + 4q + 2 + 2{q^2} - 2} \hfill \cr {} \hfill & { = 6{q^2} + 6q + 6.} \hfill \cr }

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 thatdetA=i=1nμi.det\;A = \prod\limits_{i = 1}^n {\mu _i}.In addition, for any polynomial P(x), the value P(μ) is an eigenvalue of P(A) and hencedetP(A)=i=1nP(μi).det\;P(A) = \prod\limits_{i = 1}^n P({\mu _i}).

Lemma 3

LetB=(B0B1B1B0)B = \left( {\matrix{ {{B_0}} & {{B_1}} \cr {{B_1}} & {{B_0}} } } \right)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. Theni=1nμi2=2m+(n2n2m)(1+q)2.\sum\limits_{i = 1}^n {\mu _i}^2 = 2m + ({n^2} - n - 2m)(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.\matrix{ {\sum\limits_{i = 1}^n \mu _i^2} \hfill & { = \sum\limits_{i = 1}^n \sum\limits_{j = 1}^n {q_{ij}}{q_{ji}}} \hfill \cr {} \hfill & { = \sum\limits_{i = 1}^n \sum\limits_{j = 1}^n {{({q_{ij}})}^2}} \hfill \cr {} \hfill & { = (2m{{)(1)}^2} + ({n^2} - n - 2m)(1 + q{)^2}} \hfill \cr {} \hfill & { = 2m + ({n^2} - n - 2m)(1 + q{)^2}.} \hfill \cr }

Theorem 5

Let G be an (n,m) graph of diameter 2 and let μ1be its greatest q-distance eigenvalue. Thenμ1n(n1)(1+q)2mqn.{\mu _1} \ge {{n(n - 1)(1 + q) - 2mq} \over n}.

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.\matrix{ {{\mu _1}} \hfill & { \ge {{X{A_q}{X^T}} \over {X{X^T}}}} \hfill \cr {} \hfill & { = {{\sum\nolimits_{i = 1}^n ({d_i}(1) + (n - {d_i} - 1)(1 + q))} \over n}} \hfill \cr {} \hfill & { = {{(2m + (n - 1)n(1 + q) - 2m(1 + q))} \over n}} \hfill \cr {} \hfill & { = {{n(n - 1)(1 + q) - 2mq} \over n}.} \hfill \cr }

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¯\overline A is the adjacency matrix of G¯\overline G . Then the q-distance adjacency matrix of G will be Aq=A+(1+q)A¯.{A_q} = A + (1 + q)\overline 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¯\overline 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¯A + (1 + q)\overline 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¯).\left( {\matrix{ {A + (1 + q)\overline A } & {J + qA + (q + {q^2})\overline A } \cr {J + qA + (q + {q^2})\overline A } & {A + (1 + q)\overline A } \cr } } \right). By Lemma 3, the spectrum of H is the union of the spectra of (1+q)A+(1+2q+q2)A¯+J(1 + q)A + (1 + 2q + {q^2})\overline A + J and (1q)A+(1q2)A¯J(1 - q)A + (1 - {q^2})\overline 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¯\overline A . Also we know that n,0,0,...,0 are the eigenvalues of J. Therefore, the theorem follows.

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)|, then2m+(n2n2m)(1+q)2+n(n1)P2nEq(G)n(2m+(n2n2m)(1+q)2).\sqrt {2m + ({n^2} - n - 2m)(1 + q{)^2} + n(n - 1){P^{{2 \over n}}}} \le {E_q}(G) \le \sqrt {n(2m + ({n^2} - n - 2m)(1 + q{)^2})} .

Proof

Recall that the Cauchy-Schwarz inequality states that (i=1naibi)2(i=1nai2)(i=1nbi2).{(\sum\limits_{i = 1}^n {a_i}{b_i})^2} \le (\sum\limits_{i = 1}^n a_i^2)(\sum\limits_{i = 1}^n b_i^2). If we substitute ai = 1 and bi = | μi |, then we obtain (i=1n|μi|)2(i=1n1)(i=1nμi2).{(\sum\limits_{i = 1}^n |{\mu _i}|)^2} \le (\sum\limits_{i = 1}^n 1)(\sum\limits_{i = 1}^n \mu _i^2). Hence by Theorem 4, we have Eq2(G)n(2m+(n2n2m)(1+q)2)E_q^2(G) \le n(2m + ({n^2} - n - 2m)(1 + q{)^2}) and therefore we obtain Eq(G)n(2m+(n2n2m)(1+q)2).{E_q}(G) \le \sqrt {n(2m + ({n^2} - n - 2m)(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.\matrix{ {{1 \over {n(n - 1)}}\sum\nolimits_{i \ne j} |{\mu _i}{\mu _j}|} \hfill & { \ge {{[\prod\nolimits_{i \ne j} |{\mu _i}{\mu _j}|]}^{{1 \over {n(n - 1)}}}}} \hfill \cr {} \hfill & { = [\prod\nolimits_{i = 1}^n |{\mu _i}{|^{2(n - 1)}}{]^{{1 \over {n(n - 1)}}}}} \hfill \cr {} \hfill & { = [\prod\nolimits_{i = 1}^n |{\mu _i}|{]^{{2 \over n}}}} \hfill \cr {} \hfill & { = [\prod\nolimits_{i = 1}^n {\mu _i}{]^{{2 \over n}}}} \hfill \cr {} \hfill & { = |det{A_q}(G{{)|}^{{2 \over n}}}} \hfill \cr {} \hfill & { = {P^{{2 \over n}}}.} \hfill \cr } Therefore ij|μiμj|n(n1)P2n.\sum\limits_{i \ne j} |{\mu _i}{\mu _j}| \ge n(n - 1){P^{{2 \over n}}}. Now consider Eq2(G)=(i=1n|μi|)2=i=1n|μi|2+ij|μi||μj|.\matrix{ {E_q^2(G)} \hfill & { = {\left(\sum\nolimits_{i = 1}^n |{\mu _i}|\right)^2}} \hfill \cr {} \hfill & { = \sum\nolimits_{i = 1}^n |{\mu _i}{|^2} + \sum\nolimits_{i \ne j} |{\mu _i}||{\mu _j}|.} \hfill \cr } Therefore, by Theorem 4, we obtain Eq2(G)2m+(n2n2m)(1+q)2+n(n1)P2nE_q^2(G) \ge 2m + ({n^2} - n - 2m)(1 + q{)^2} + n(n - 1){P^{{2 \over n}}} and hence Eq(G)2m+(n2n2m)(1+q)2+n(n1)P2n.{E_q}(G) \ge \sqrt {2m + ({n^2} - n - 2m)(1 + q{)^2} + n(n - 1){P^{{2 \over n}}}} .

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 thatn(n1)(1+q)2mqn1,{{n(n - 1)(1 + q) - 2mq} \over n} \ge 1,thenEq(G)n(n1)(1+q)2mqn+(n1)[(2m+(n2n2m)(1+q)2(n(n1)(1+q)2mqn)2].{E_q}(G) \le {{n(n - 1)(1 + q) - 2mq} \over n} + \sqrt {(n - 1)[(2m + ({n^2} - n - 2m)(1 + q{)^2} - {{({{n(n - 1)(1 + q) - 2mq} \over n})}^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).{\left( {\sum\limits_{i = 2}^n |{\mu _i}|} \right)^2} \le \sum\limits_{i = 2}^n 1\sum\limits_{i = 2}^n \mu _i^2{[{E_q}(G) - {\mu _1}]^2} \le (n - 1)(2m + ({n^2} - n - 2m)(1 + q{)^2}). Hence Eq(G)μ1+(n1)(2m+(n2n2m)(1+q)2μ12).{E_q}(G) \le {\mu _1} + \sqrt {(n - 1)(2m + ({n^2} - n - 2m)(1 + q{)^2} - \mu _1^2)} . Let f(x)=x+(n1)(2m+(n2n2m)(1+q)2x2).f(x) = x + \sqrt {(n - 1)(2m + ({n^2} - n - 2m)(1 + q{)^2} - {x^2})} . Then for a decreasing function f (x), the fact f(x) ≤ 0 implies that 1x(n1)(n1)(2m+(n2n2m)(1+q)2x2)0.1 - {{x(n - 1)} \over {\sqrt {(n - 1)(2m + ({n^2} - n - 2m)(1 + q{)^2} - {x^2})} }} \le 0. From this we obtain x2m+(n2n2m)(1+q)2n.x \ge \sqrt {{{2m + ({n^2} - n - 2m)(1 + q{)^2}} \over n}} . Therefore the function f (x) is decreasing in the interval (2m+(n2n2m)(1+q)2n,2m+(n2n2m)(1+q)2).\left( {\sqrt {{{2m + ({n^2} - n - 2m)(1 + q{)^2}} \over n}} ,\sqrt {2m + ({n^2} - n - 2m)(1 + q{)^2}} } \right). 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(n(n - 1)(1 + q) - 2mq)/n \le {\mu _1} \le \sqrt {2m + ({n^2} - n - 2m)(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){E_q}(G) \le f({\mu _1}) \le f((n(n - 1)(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, thenEq(G)|0|(mod2).{E_q}(G)\;\; \equiv \;\;|0|\;\;(mod\;2).

Proof

Proof is similar to Theorem 5.4 of [15].

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

Join of two graphs

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).{{(\mu - (1 + q)({n_1} - 1) + {r_1}q)(\mu - (1 + q)({n_2} - 1) + {r_2}q) - {n_1}{n_2}} \over {(\mu - (1 + q)({n_1} - 1) + {r_1}q)(\mu - (1 + q)({n_2} - 1) + {r_2}q)}}\phi ({G_1}:\mu )\phi ({G_2}:\mu ).

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 qijq_{ij}^\prime 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\matrix{ {\sum\limits_{j = 1}^{{n_1}} {q_{ij}}} \hfill & { = 1({r_1}) + (1 + q)({n_1} - 1 - {r_1})} \hfill \cr {} \hfill & { = {r_1} + (1 + q)({n_1} - 1) - {r_1} - {r_1}q} \hfill \cr {} \hfill & { = (1 + q)({n_1} - 1) - {r_1}q} \hfill \cr } and similarly, for i = 1,2,3,...,n2, we have j=1n2qij=(1+q)(n21)r2q.\sum\limits_{j = 1}^{{n_2}} q_{ij}^\prime = (1 + q)({n_2} - 1) - {r_2}q.

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)\matrix{ {} & { {\matrix{ {\,\,\, {v_1}} & {\,\,\,\,{v_2}} & {\,\,\,{v_3}} & \,\,\cdots & {\,{v_{{n_1}}}} & {\,\,\,{u_1}} & {\,\,\,{u_2}} & {\,\,\,{u_3}} & \,\,\cdots & {\,\,{u_{{n_2}}}} \cr } } } \cr {\matrix{ {{v_1}} \cr {{v_2}} \cr {{v_3}} \cr \vdots \cr {{v_{{n_1}}}} \cr {{u_1}} \cr {{u_2}} \cr {{u_3}} \cr \vdots \cr {{u_{{n_1}}}} \cr } } & {\left( {\matrix{ 0 & {{q_{12}}} & {{q_{13}}} & \cdots & {{q_1}_{{n_1}}} & 1 & 1 & 1 & \cdots & 1 \cr {{q_{21}}} & 0 & {{q_{23}}} & \cdots & {{q_{2{n_1}}}} & 1 & 1 & 1 & \cdots & 1 \cr {{q_{31}}} & {{q_{32}}} & 0 & \cdots & {{q_{3{n_1}}}} & 1 & 1 & 1 & \cdots & 1 \cr \vdots & \vdots & \vdots & \ddots & \vdots & \vdots & \vdots & \vdots & \ddots & \vdots \cr {{q_{{n_1}}}_1} & {{q_{{n_1}}}_2} & {{q_{{n_1}}}_3} & \cdots & 0 & 1 & 1 & 1 & \cdots & 1 \cr 1 & 1 & 1 & \cdots & 1 & 0 & {q_{12}^\prime} & {q_{13}^\prime} & \cdots & {q_{1{n_2}}^\prime} \cr 1 & 1 & 1 & \cdots & 1 & {q_{21}^\prime} & 0 & {q_{23}^\prime} & \cdots & {q_{2{n_2}}^\prime} \cr 1 & 1 & 1 & \cdots & 1 & {q_{31}^\prime} & {q_{32}^\prime} & 0 & \cdots & {q_{3{n_2}}^\prime} \cr \vdots & \vdots & \vdots & \ddots & \vdots & \vdots & \vdots & \vdots & \ddots & \vdots \cr 1 & 1 & 1 & \cdots & 1 & {q_{{n_2}1}^\prime} & {q_{{n_2}2}^\prime} & {q_{{n_2}3}^\prime} & \cdots & 0 \cr } } \right)} \cr } Let φ(G1G2 : μ) denote the q-distance characteristic polynomial of G1G2; i.e., ϕ(G1G2:μ)=|μEq(G1G2)|.\phi ({G_1}\nabla {G_2}:\mu ) = |\mu - {E_q}({G_1}\nabla {G_2})|.

This polynomial is equal to the following determinant |μq12q13q1n11111q21μq23q2n11111q31q32μq3n11111qn11qn12qn13μ11111111μq12q13q1n21111q21μq23q2n21111q31q32μq3n21111qn21qn22qn23μ|(n1+n2)×(n1+n2).{\left| {\matrix{ \mu & { - {q_{12}}} & { - {q_{13}}} & \ldots & { - {q_{1{n_1}}}} & { - 1} & { - 1} & { - 1} & \ldots & { - 1}\cr { - {q_{21}}} & \mu & { - {q_{23}}} & \ldots & { - {q_{2{n_1}}}} & { - 1} & { - 1} & { - 1} & \ldots & { - 1}\cr { - {q_{31}}} & { - {q_{32}}} & \mu & \ldots & { - {q_{3{n_1}}}} & { - 1} & { - 1} & { - 1} & \ldots & { - 1}\cr \vdots & \vdots & \vdots & \ddots & \vdots & \vdots & \vdots & \vdots & \ddots & \vdots \cr { - {q_{{n_1}1}}} & { - {q_{{n_1}2}}} & { - {q_{{n_1}3}}} & \ldots & \mu & { - 1} & { - 1} & { - 1} & \ldots & { - 1}\cr { - 1} & { - 1} & { - 1} & \ldots & { - 1} & \mu & { - q_{12}^\prime} & { - q_{13}^\prime} & \ldots & { - q_{1{n_2}}^\prime}\cr { - 1} & { - 1} & { - 1} & \ldots & { - 1} & { - q_{21}^\prime} & \mu & { - q_{23}^\prime} & \ldots & { - q_{2{n_2}}^\prime}\cr { - 1} & { - 1} & { - 1} & \ldots & { - 1} & { - q_{31}^\prime} & { - q_{32}^\prime} & \mu & \ldots & { - q_{3{n_2}}^\prime}\cr \vdots & \vdots & \vdots & \ddots & \vdots & \vdots & \vdots & \vdots & \ddots & \vdots \cr { - 1} & { - 1} & { - 1} & \ldots & { - 1} & { - q_{{n_2}1}^\prime} & { - q_{{n_2}2}^\prime} & { - q_{{n_2}3}^\prime} & \ldots & \mu \cr } } \right|_{({n_1} + {n_2}) \times ({n_1} + {n_2})}}.

Applying the row operations Rn1+2=Rn1+2Rn1+1R_{{n_1} + 2}^\prime = {R_{{n_1} + 2}} - {R_{{n_1} + 1}} ; Rn1+3=Rn1+3Rn1+1R_{{n_1} + 3}^\prime = {R_{{n_1} + 3}} - {R_{{n_1} + 1}} ;··· ; Rn1+n2=Rn1+n2Rn1+1R_{{n_1} + {n_2}}^\prime = {R_{{n_1} + {n_2}}} - {R_{{n_1} + 1}} to the above determinant, we see that the determinant becomes |μq12q13q1n11111q21μq23q2n11111q31q32μq3n11111qn11qn12qn13μ11111111μq12q13q1n20000q21μμ+q12q23+q13q2n2+q1n20000q31μq32+q12μ+q13q3n2+q1n20000qn21μqn22+q12qn23+q13μ+q1n2|.\left| {\matrix{ \mu & { - {q_{12}}} & { - {q_{13}}} & \ldots & { - {q_{1{n_1}}}} & { - 1} & { - 1} & { - 1} & \ldots & { - 1}\cr { - {q_{21}}} & \mu & { - {q_{23}}} & \ldots & { - {q_{2{n_1}}}} & { - 1} & { - 1} & { - 1} & \ldots & { - 1}\cr { - {q_{31}}} & { - {q_{32}}} & \mu & \ldots & { - {q_{3{n_1}}}} & { - 1} & { - 1} & { - 1} & \ldots & { - 1}\cr \vdots & \vdots & \vdots & \ddots & \vdots & \vdots & \vdots & \vdots & \ddots & \vdots \cr { - {q_{{n_1}1}}} & { - {q_{{n_1}2}}} & { - {q_{{n_1}3}}} & \ldots & \mu & { - 1} & { - 1} & { - 1} & \ldots & { - 1}\cr { - 1} & { - 1} & { - 1} & \ldots & { - 1} & \mu & { - q_{12}^\prime} & { - q_{13}^\prime} & \ldots & { - q_{1{n_2}}^\prime}\cr 0 & 0 & 0 & \ldots & 0 & { - q_{21}^\prime - \mu } & {\mu + q_{12}^\prime} & { - q_{23}^\prime + q_{13}^\prime} & \ldots & { - q_{2{n_2}}^\prime + q_{1{n_2}}^\prime}\cr 0 & 0 & 0 & \ldots & 0 & { - q_{31}^\prime - \mu } & { - q_{32}^\prime + q_{12}^\prime} & {\mu + q_{13}^\prime} & \ldots & { - q_{3{n_2}}^\prime + q_{1{n_2}}^\prime}\cr \vdots & \vdots & \vdots & \ddots & \vdots & \vdots & \vdots & \vdots & \ddots & \vdots \cr 0 & 0 & 0 & \ldots & 0 & { - q_{{n_2}1}^\prime - \mu } & { - q_{{n_2}2}^\prime + q_{12}^\prime} & { - q_{{n_2}3}^\prime + q_{13}^\prime} & \ldots & {\mu + q_{1{n_2}}^\prime}\cr } } \right|.

Applying the column operation Cn1+1=Cn1+1+Cn1+2++Cn1+n2C_{{n_1} + 1}^\prime = {C_{{n_1} + 1}} + {C_{{n_1} + 2}} + \cdots + {C_{{n_1} + {n_2}}} , using the fact that qij=qjiq_{ij}^\prime = q_{ji}^\prime 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|,\matrix{ {\left| {\matrix{ \mu & { - {q_{12}}} & \ldots & { - {q_{1{n_1}}}} & { - {n_2}} & { - 1} & { - 1} & \ldots & { - 1}\cr { - {q_{21}}} & \mu & \ldots & { - {q_{2{n_1}}}} & { - {n_2}} & { - 1} & { - 1} & \ldots & { - 1}\cr { - {q_{31}}} & { - {q_{32}}} & \ldots & { - {q_{3{n_1}}}} & { - {n_2}} & { - 1} & { - 1} & \ldots & { - 1}\cr \vdots & \vdots & \ddots & \vdots & \vdots & \vdots & \vdots & \ddots & \vdots \cr { - {q_{{n_1}1}}} & { - {q_{{n_1}2}}} & \ldots & \mu & { - {n_2}} & { - 1} & { - 1} & \ldots & { - 1}\cr { - 1} & { - 1} & \ldots & { - 1} & {\mu - (1 + q)({n_2} - 1) - {r_2}q} & { - q_{12}^\prime} & { - q_{13}^\prime} & \ldots & { - q_{1{n_2}}^\prime}\cr 0 & 0 & \ldots & 0 & 0 & {\mu + q_{12}^\prime} & { - q_{23}^\prime + q_{13}^\prime} & \ldots & { - q_{2{n_2}}^\prime + q_{1{n_2}}^\prime}\cr 0 & 0 & \ldots & 0 & 0 & { - q_{32}^\prime + q_{12}^\prime} & {\mu + q_{13}^\prime} & \ldots & { - q_{3{n_2}}^\prime + q_{1{n_2}}^\prime}\cr \vdots & \vdots & \ddots & \vdots & \vdots & \vdots & \vdots & \ddots & \vdots \cr 0 & 0 & \ldots & 0 & 0 & { - q_{{n_2}2}^\prime + q_{12}^\prime} & { - q_{{n_2}3}^\prime + q_{13}^\prime} & \ldots & {\mu + q_{1{n_2}}^\prime}\cr } } \right|} \cr {\;\;\; = \;\;\;\left| {\matrix{ \mu & { - {q_{12}}} & { - {q_{13}}} & \cdots & { - {q_{1{n_1}}}} & { - {n_2}} \cr { - {q_{21}}} & \mu & { - {q_{23}}} & \cdots & { - {q_{2{n_1}}}} & { - {n_2}} \cr { - {q_{31}}} & { - {q_{32}}} & \mu & \cdots & { - {q_{3{n_1}}}} & { - {n_2}} \cr \vdots & \vdots & \vdots & \vdots & \vdots & \vdots \cr { - {q_{{n_1}1}}} & { - {q_{{n_1}2}}} & {{q_{{n_1}3}}} & \cdots & \mu & { - {n_2}} \cr { - 1} & { - 1} & { - 1} & \cdots & { - 1} & {\mu - (1 + q)({n_2} - 1) + {r_2}q} \cr} } \right||B|,} \cr } where |B|=|μ+q12q23+q13q2n2+q1n2q32+q12μ+q13q3n2+q1n2qn22+q12qn23+q12μ+q1n2|(n21)×(n21).|B| = {\left| {\matrix{ {\mu + q_{12}^\prime} & { - q_{23}^\prime + q_{13}^\prime} & \cdots & { - q_{2{n_2}}^\prime + q_{1{n_2}}^\prime} \cr { - q_{32}^\prime + q_{12}^\prime} & {\mu + q_{13}^\prime} & \cdots & { - q_{3{n_2}}^\prime + q_{1{n_2}}^\prime} \cr \vdots & \vdots & \vdots & \vdots \cr { - q_{{n_2}2}^\prime + q_{12}^\prime} & { - q_{{n_2}3}^\prime + q_{12}^\prime} & \cdots & {\mu + q_{1{n_2}}^\prime} \cr} } \right|_{({n_2} - 1) \times ({n_2} - 1)}}. Applying the row operations R2=R2R1R_2^\prime = {R_2} - {R_1} , R3=R3R1R_3^\prime = {R_3} - {R_1} , ···, Rn1=Rn1R1R_{{n_1}}^\prime = {R_{{n_1}}} - {R_1} , the above determinant transforms to |μq12q13q1n1n2q21μμ+q12q23+q13q2n1+q1n10q31μq32+q12μ+q13q3n1+q1n10qn11μqn12+q12qn13+q13μ+q1n101111μ(1+q)(n21)+r2q||B|.\left| {\matrix{ \mu & { - {q_{12}}} & { - {q_{13}}} & \cdots & { - {q_{1{n_1}}}} & { - {n_2}} \cr { - {q_{21}} - \mu } & {\mu + {q_{12}}} & { - {q_{23}} + {q_{13}}} & \cdots & { - {q_{2{n_1}}} + {q_{1{n_1}}}} & 0 \cr { - {q_{31}} - \mu } & { - {q_{32}} + {q_{12}}} & {\mu + {q_{13}}} & \cdots & { - {q_{3{n_1}}} + {q_{1{n_1}}}} & 0 \cr \vdots & \vdots & \vdots & \vdots & \vdots & \vdots \cr { - {q_{{n_{11}}}} - \mu } & { - {q_{{n_{12}}}} + {q_{12}}} & { - {q_{{n_{13}}}} + {q_{13}}} & \cdots & {\mu + {q_{1{n_1}}}} & 0 \cr { - 1} & { - 1} & { - 1} & \cdots & { - 1} & {\mu - (1 + q)({n_2} - 1) + {r_2}q}\cr } } \right||B|. Applying the column operation C1=C1+C2++Cn1C_1^\prime = {C_1} + {C_2} + \cdots + {C_{{n_1}}} 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|.\left| {\matrix{ {\mu - (1 + q)({n_1} - 1) + {r_1}q} & { - {q_{12}}} & { - {q_{13}}} & \cdots & { - {q_{1{n_1}}}} & { - {n_2}} \cr 0 & {\mu + {q_{12}}} & { - {q_{23}} + {q_{13}}} & \cdots & { - {q_{2{n_1}}} + {q_{1{n_1}}}} & 0 \cr 0 & { - {q_{32}} + {q_{12}}} & {\mu + {q_{13}}} & \cdots & { - {q_{3{n_1}}} + {q_{1{n_1}}}} & 0 \cr \vdots & \vdots & \vdots & \vdots & \vdots & {} \cr 0 & { - {q_{{n_{12}}}} + {q_{12}}} & { - {q_{{n_{13}}}} + {q_{13}}} & \cdots & {\mu + {q_{1{n_1}}}} & 0 \cr { - {n_1}} & { - 1} & { - 1} & \cdots & { - 1} & {\mu - (1 + q)({n_2} - 1) + {r_2}q} \cr } } \right||B|. Expanding it along the first column C1, we obtain ϕ(G1G2:μ)={(μ(1+q)(n11)+r1q)Δ1(1)n1n1Δ2}|B|,\phi ({G_1}\nabla {G_2}:\mu ) = \{ (\mu - (1 + q)({n_1} - 1) + {r_1}q){\Delta _1} - {( - 1)^{{n_1}}}{n_1}{\Delta _2}\} |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|\matrix{ {{\Delta _1}} \hfill & { = \left| {\matrix{ {\mu + {q_{12}}} & { - {q_{23}} + {q_{13}}} & \cdots & { - {q_{2{n_1}}} + {q_{1{n_1}}}} & 0 \cr { - {q_{32}} + {q_{12}}} & {\mu + {q_{13}}} & \cdots & { - {q_{3{n_1}}} + {q_{1{n_1}}}} & 0 \cr \vdots & \vdots & \vdots & \vdots & \vdots \cr { - {q_{{n_{12}}}} + {q_{12}}} & { - {q_{{n_{13}}}} + {q_{13}}} & \cdots & {\mu + {q_{1{n_1}}}} & 0 \cr { - 1} & { - 1} & \cdots & { - 1} & {\mu - (1 + q)({n_1} - 1) + {r_2}q} \cr} } \right|} \hfill \cr {} \hfill & { = (\mu - (1 + q)({n_1} - 1) + {r_2}q)|A|( - {{1)}^{{n_1} + {n_2}}}} \hfill \cr {} \hfill & { = (\mu - (1 + q)({n_1} - 1) + {r_2}q)|A|} \hfill \cr } and Δ2=|q12q13q1n1n2μ+q12q23+q13q2n1+q1n10q32+q12μ+q13q3n1+q1n10qn12+q12qn13+q13μ+q1n10|=(1)n1+1(n2)|A|=n2(1)n1|A|.\matrix{ {{\Delta _2}} \hfill & { = \left| {\matrix{ { - {q_{12}}} & { - {q_{13}}} & \cdots & { - {q_{1{n_1}}}} & { - {n_2}} \cr {\mu + {q_{12}}} & { - {q_{23}} + {q_{13}}} & \cdots & { - {q_{2{n_1}}} + {q_{1{n_1}}}} & 0 \cr { - {q_{32}} + {q_{12}}} & {\mu + {q_{13}}} & \cdots & { - {q_{3{n_1}}} + {q_{1{n_1}}}} & 0 \cr \vdots & \vdots & \vdots & \vdots & \vdots \cr { - {q_{{n_{12}}}} + {q_{12}}} & { - {q_{{n_{13}}}} + {q_{13}}} & \cdots & {\mu + {q_{1{n_1}}}} & 0 \cr} } \right|} \hfill \cr {} \hfill & { = {{( - 1)}^{ - {n_1} + 1}}( - {n_2})|A|} \hfill \cr {} \hfill & { = {n_2}{{( - 1)}^{{n_1}}}|A|.} \hfill \cr } Therefore we have ϕ(G1G2:μ)=((μ(1+q)(n11)+r1q)(μ(1+q)(n21)+r2q)|A|)(1)n1n1n2(1)n1|A|)|B|,\matrix{ {\phi ({G_1}\nabla {G_2}:\mu )} \hfill & { = ((\mu - (1 + q)({n_1} - 1) + {r_1}q)(\mu - (1 + q)({n_2} - 1) + {r_2}q)|A|)} \hfill \cr {} \hfill & { - {{( - 1)}^{{n_1}}}{n_1}{n_2}{{( - 1)}^{{n_1}}}|A|)|B|,} \hfill \cr } i.e. ϕ(G1G2:μ)=|A||B|[(μ(1+q)(n11)+r1q)(μ(1+q)(n21)+r2q)n1n2]\phi ({G_1}\nabla {G_2}:\mu ) = |A||B|[(\mu - (1 + q)({n_1} - 1) + {r_1}q) \cdot (\mu - (1 + q)({n_2} - 1) + {r_2}q) - {n_1}{n_2}] where |A|=|μ+q12q23+q13q2n1+q1n1q32+q12μ+q13q3n1+q1n1qn12+q12qn13+q13μ+q1n1|.|A| = \left| {\matrix{ {\mu + {q_{12}}} & { - {q_{23}} + {q_{13}}} & \cdots & { - {q_{2{n_1}}} + {q_{1{n_1}}}} \cr { - {q_{32}} + {q_{12}}} & {\mu + {q_{13}}} & \cdots & { - {q_{3{n_1}}} + {q_{1{n_1}}}} \cr \vdots & \vdots & \vdots & \vdots \cr { - {q_{{n_{12}}}} + {q_{12}}} & { - {q_{{n_{13}}}} + {q_{13}}} & \cdots & {\mu + {q_{1{n_1}}}} \cr } } \right|. Clearly |A|=1(μ(1+q)(n11)+r1q)×|μ(1+q)(n11)+r1qq12q13q1n10μ+q12q23+q13q2n1+q1n10q32+q12μ+q13q3n1+q1n10qn12+q12qn13+q13μ+q1n1|.\left| A \right| = {1 \over {(\mu - (1 + q)({n_1} - 1) + {r_1}q)}} \times \left| {\matrix{ {\mu - (1 + q)({n_1} - 1) + {r_1}q} & { - {q_{12}}} & { - {q_{13}}} & \cdots & { - {q_{1{n_1}}}} \cr 0 & {\mu + {q_{12}}} & { - {q_{23}} + {q_{13}}} & \cdots & { - {q_{2{n_1}}} + {q_{1{n_1}}}} \cr 0 & { - {q_{32}} + {q_{12}}} & {\mu + {q_{13}}} & \cdots & { - {q_{3{n_1}}} + {q_{1{n_1}}}} \cr \vdots & \vdots & \vdots & \vdots & \vdots \cr 0 & { - {q_{{n_{12}}}} + {q_{12}}} & { - {q_{{n_{13}}}} + {q_{13}}} & \cdots & {\mu + {q_{1{n_1}}}} \cr } } \right|. Applying the operation C1=C1(C2+C3++Cn1)C_1^\prime = {C_1} - ({C_2} + {C_3} + \cdots + {C_{{n_1}}}) , the determinant becomes |A|=1μ(1+q)(n11)+r1q×|μq12q13q1n1μq21μ+q12q23+q13q2n1+q1n1μqn31q32+q12μ+q13q3n1+q1n1μqn11qn12+q12qn13+q13μ+q1n1|.\left| A \right| = {1 \over {\mu - (1 + q)({n_1} - 1) + {r_1}q}} \times \left| {\matrix{ \mu & { - {q_{12}}} & { - {q_{13}}} & \cdots & { - {q_{1{n_1}}}} \cr { - \mu - {q_{21}}} & {\mu + {q_{12}}} & { - {q_{23}} + {q_{13}}} & \cdots & { - {q_{2{n_1}}} + {q_{1{n_1}}}} \cr { - \mu - {q_{{n_{31}}}}} & { - {q_{32}} + {q_{12}}} & {\mu + {q_{13}}} & \cdots & { - {q_{3{n_1}}} + {q_{1{n_1}}}} \cr \vdots & \vdots & \vdots & \vdots & \vdots \cr { - \mu - {q_{{n_1}1}}} & { - {q_{{n_1}2}} + {q_{12}}} & { - {q_{{n_1}3}} + {q_{13}}} & \cdots & {\mu + {q_{1{n_1}}}} \cr} } \right|. Applying the operations R = R2 + R1; R3=R3+R1R_3^\prime = {R_3} + {R_1} ;··· ; Rn=Rn1+R1R_n^\prime = {R_{{n_1}}} + {R_1} , we have |A|=1μ(1+q)(n11)+r1q×|μq12q13q1n1q21μq23q2n1q31q32μq3n1qn11qn12qn13μ|=1μ(1+q)(n11)+r1qϕ(G1;μ).\matrix{ {\left| A \right|} \hfill & { = {1 \over {\mu - (1 + q)({n_1} - 1) + {r_1}q}} \times \left| {\matrix{ \mu & { - {q_{12}}} & { - {q_{13}}} & \cdots & { - {q_{1{n_1}}}} \cr { - {q_{21}}} & \mu & { - {q_{23}}} & \cdots & { - {q_{2{n_1}}}} \cr { - {q_{31}}} & { - {q_{32}}} & \mu & \cdots & { - {q_{3{n_1}}}} \cr \vdots & \vdots & \vdots & \vdots & {} \cr { - {q_{{n_1}1}}} & { - {q_{{n_1}2}}} & { - {q_{{n_1}3}}} & \cdots & \mu \cr} } \right|} \hfill \cr {} \hfill & { = {1 \over {\mu - (1 + q)({n_1} - 1) + {r_1}q}}\phi ({G_1};\mu ).} \hfill \cr } Similarly, |B|=1μ(1+q)(n21)+r2qϕ(G2;μ).|B| = {1 \over {\mu - (1 + q)({n_2} - 1) + {r_2}q}}\phi ({G_2};\mu ). 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, thenEq(G1G2)={Eq(G1)+Eq(G2),ifRKn1n2,Eq(G1)+Eq(G2)(R+K)+(R+K)24(RKn1n2),ifRK<n1n2,{E_q}({G_1}\nabla {G_2}) = \left\{ {\matrix{ {{E_q}({G_1}) + {E_q}({G_2}),} \hfill & {{\kern 1pt} if{\kern 1pt} {\kern 1pt} {\kern 1pt} RK \ge {n_1}{n_2},} \hfill \cr {{E_q}({G_1}) + {E_q}({G_2}) - (R + K) + \sqrt {{{(R + K)}^2} - 4(RK - {n_1}{n_2})} ,} \hfill & {{\kern 1pt} if{\kern 1pt} {\kern 1pt} {\kern 1pt} RK < {n_1}{n_2},} \hfill \cr {} \hfill & {} \hfill \cr } } \right.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)\phi ({G_1}\nabla {G_2};\mu ) = {{((\mu - (1 + q)({n_1} - 1) + {r_1}q)(\mu - (1 + q)({n_2} - 1) + {r_2}q) - {n_1}{n_2}))} \over {(\mu - (1 + q)({n_1} - 1) + {r_1}q)(\mu - (1 + q)({n_2} - 1) + {r_2}q)}}\phi ({G_1};\mu )\phi ({G_2};\mu ) implying that ϕ(G1G2;μ)=((μR)(μK)n1n2)ϕ(G1;μ)ϕ(G2;μ)(μR)(μK),\phi ({G_1}\nabla {G_2};\mu ) = {{((\mu - R)(\mu - K) - {n_1}{n_2})\phi ({G_1};\mu )\phi ({G_2};\mu )} \over {(\mu - R)(\mu - K)}}, 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).(\mu - R)(\mu - K)\phi ({G_1}\nabla {G_2};\mu ) = ((\mu - R)(\mu - K) - {n_1}{n_2})\phi ({G_1};\mu )\phi ({G_2};\mu ). That is, (μR)(μK)ϕ(G1G2;μ)=(μ2(R+K)μ+RKn1n2)ϕ(G1;μ)ϕ(G2;μ).(\mu - R)(\mu - K)\phi ({G_1}\nabla {G_2};\mu ) = ({\mu ^2} - (R + K)\mu + RK - {n_1}{n_2})\phi ({G_1};\mu )\phi ({G_2};\mu ).

Let P1(μ)=(μR)(μK)ϕ(G1G2;μ){P_1}(\mu ) = (\mu - R)(\mu - K)\phi ({G_1}\nabla {G_2};\mu ) and P2(μ)=(μ2(R+K)μ+RKn1n2)ϕ(G1;μ)ϕ(G2;μ).{P_2}(\mu ) = ({\mu ^2} - (R + K)\mu + RK - {n_1}{n_2})\phi ({G_1};\mu )\phi ({G_2};\mu ). 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)R + K + {E_q}({G_1}\nabla {G_2}) 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.{{R + K + \sqrt {{{(R + K)}^2} - 4(RK - {n_1}{n_2})} } \over 2},\;\;{{R + K - \sqrt {{{(R + K)}^2} - 4(RK - {n_1}{n_2})} } \over 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|.{E_q}({G_1}) + {E_q}({G_2}) + \left|{{R + K + \sqrt {{{(R + K)}^2} - 4(RK - {n_1}{n_2})} } \over 2}\right| + \left|{{R + K - \sqrt {{{(R + K)}^2} - 4(RK - {n_1}{n_2})} } \over 2}\right|. 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|.\matrix{ {R + K + {E_q}({G_1}\nabla {G_2})} \hfill & { = {E_q}({G_1}) + {E_q}({G_2})} \hfill \cr {} \hfill & { + \left|{{R + K + \sqrt {{{(R + K)}^2} - 4(RK - {n_1}{n_2})} } \over 2}\right|} \hfill \cr {} \hfill & { + \left|{{R + K - \sqrt {{{(R + K)}^2} - 4(RK - {n_1}{n_2})} } \over 2}\right|.} \hfill \cr }

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,\matrix{ {R + K + {E_q}({G_1}\nabla {G_2})} \hfill & { = {E_q}({G_1}) + {E_q}({G_2})} \hfill \cr {} \hfill & { + {{R + K + \sqrt {{{(R + K)}^2} - 4(RK - {n_1}{n_2})} } \over 2}} \hfill \cr {} \hfill & { + {{R + K - \sqrt {{{(R + K)}^2} - 4(RK - {n_1}{n_2})} } \over 2},} \hfill \cr } and therefore R+K+Eq(G1G2)=Eq(G1)+Eq(G2)+R+KR + K + {E_q}({G_1}\nabla {G_2}) = {E_q}({G_1}) + {E_q}({G_2}) + R + K implying that Eq(G1G2)=Eq(G1)+Eq(G2).{E_q}({G_1}\nabla {G_2}) = {E_q}({G_1}) + {E_q}({G_2}).

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.\matrix{ {R + K + {E_q}({G_1}\nabla {G_2})} \hfill & { = {E_q}({G_1}) + {E_q}({G_2})} \hfill \cr {} \hfill & { + {{R + K + \sqrt {{{(R + K)}^2} - 4(RK - {n_1}{n_2})} } \over 2}} \hfill \cr {} \hfill & { + {{R + K - \sqrt {{{(R + K)}^2} - 4(RK - {n_1}{n_2})} } \over 2}.} \hfill \cr } Hence we get R+K+Eq(G1G2)=Eq(G1)+Eq(G2)+(R+K)24(RKn1n2)R + K + {E_q}({G_1}\nabla {G_2}) = {E_q}({G_1}) + {E_q}({G_2}) + \sqrt {{{(R + K)}^2} - 4(RK - {n_1}{n_2})} and finally Eq(G1G2)=Eq(G1)+Eq(G2)(R+K)+(R+K)24(RKn1n2).{E_q}({G_1}\nabla {G_2}) = {E_q}({G_1}) + {E_q}({G_2}) - (R + K) + \sqrt {{{(R + K)}^2} - 4(RK - {n_1}{n_2})} .

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.

Figure 1.1

The crown graphS06S_0^6
The crown graphS06S_0^6

Figure 4.1

Join of two graphs
Join of two graphs

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), 799–814BapatR. B.LalA. K.PatiS.2006A q-analogue of the distance matrix of a treeLinear Algebra and its Applications4162–3799814Search in Google Scholar

R. B. Bapat, S. Pati, (2011), Energy of a graph is never an odd integer, Bull. Kerala Math. Assoc., 1, 129–132BapatR. B.PatiS.2011Energy of a graph is never an odd integerBull. Kerala Math. Assoc.1129132Search in Google Scholar

G. Caporossi, E. Chasset, B. Furtula, (2009), Some conjectures and properties on distance energy, Les Cahiers du GERAD, G-2009-64, 1–7CaporossiG.ChassetE.FurtulaB.2009Some conjectures and properties on distance energyLes Cahiers du GERADG-2009-64,17Search in Google Scholar

D. Cvetković, I. Gutman (eds.), (2009), Applications of Graph Spectra, Beograd: Matematicki institut SANUCvetkovićD.GutmanI.(eds.)2009Applications of Graph SpectraBeogradMatematicki institut SANUSearch in Google Scholar

D. Cvetković, I. Gutman (eds.), (2011), Selected Topics on Applications of Graph Spectra, Beograd: Matematicki institut SANUCvetkovićD.GutmanI.(eds.)2011Selected Topics on Applications of Graph SpectraBeogradMatematicki institut SANUSearch in Google Scholar

A. Graovac, I. Gutman, N. Trinajstić, (1977), Topological Approach to the Chemistry of Conjugated Molecules, Springer, BerlinGraovacA.GutmanI.TrinajstićN.1977Topological Approach to the Chemistry of Conjugated MoleculesSpringerBerlinSearch in Google Scholar

I. Gutman, (1978), The Energy of a Graph, Ber. Math-Statist. Sekt. Forschungsz. Graz, 103, 1–22GutmanI.1978The Energy of a GraphBer. Math-Statist. Sekt. Forschungsz. Graz103122Search in Google Scholar

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, 196–211GutmanI.2001The Energy of a Graph: Old and New Resultsed. byBettenA.KohnertA.LaueR.WassermannA.Algebraic Combinatorics and ApplicationsSpringerBerlin196211Search in Google Scholar

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, 154–174GutmanI.LiX.ZhangJ.2009Graph Energyed. byDehmerM.Emmert-StreibF.Analysis of Complex Networks. From Biology to LinguisticsWiley - VCHWeinheim154174Search in Google Scholar

I. Gutman, O. E. Polansky, (1986), Mathematical Concepts in Organic Chemistry, Springer-Verlag, BerlinGutmanI.PolanskyO. E.1986Mathematical Concepts in Organic ChemistrySpringer-VerlagBerlinSearch in Google Scholar

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), 131–150HosamaniS. M.KulkarniB. B.BoliR. G.GadagV. M.2017QSPR Analysis of Certain Graph Theoretical Matrices and Their Corresponding EnergyAppl. Maths and Nonlin. Sciences21131150Search in Google Scholar

J. H. Koolen, V. Moulton, (2001), Maximal Energy Graphs, Adv. Appl. Math., 26, 47–52KoolenJ. H.MoultonV.2001Maximal Energy GraphsAdv. Appl. Math.264752Search in Google Scholar

G. Indulal, (2009), Sharp Bounds on the Distance Spectral Radius and the Distance Energy of Graphs, Lin. Algebra Appl., 430, 106–113IndulalG.2009Sharp Bounds on the Distance Spectral Radius and the Distance Energy of GraphsLin. Algebra Appl.430106113Search in Google Scholar

G. Indulal, I. Gutman, A. Vijayakumar, (2008), On Distance Energy of Graphs, MATCH Commun. Math. Comput. Chem., 60, 461–472IndulalG.GutmanI.VijayakumarA.2008On Distance Energy of GraphsMATCH Commun. Math. Comput. Chem.60461472Search in Google Scholar

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), 707–718KannaM. R. R.DharmendraB. N.SridharaG.2013Minimum dominating energy of a graphInternational Journal of Pure and Applied Mathematics854707718Search in Google Scholar

H. Liu, M. Lu, F. Tian, (2007), Some Upper Bounds for the Energy of Graphs, Journal of Mathematical Chemistry, 41 (1), 45–57LiuH.LuM.TianF.2007Some Upper Bounds for the Energy of GraphsJournal of Mathematical Chemistry4114557Search in Google Scholar

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), 173–180LokeshaV.DeepikaT.RanjiniP. S.CangulI. N.2017Operations of Nanostructures via SDD, ABC4 and GA5 indicesAppl. Maths and Nonlin. Sciences21173180Search in Google Scholar

B. J. McClelland, (1971), Properties of the latent roots of a matrix: The estimation of π-electron energies, J. Chem. Phys., 54, 640–643McClellandB. J.1971Properties of the latent roots of a matrix: The estimation of π-electron energiesJ. Chem. Phys.54640643Search in Google Scholar

A. R. Nizami, A. Perveen, W. Nazeer, M. Baqir, (2018), WALK POLYNOMIAL: A New Graph Invariant, Appl. Maths and Nonlin. Sciences, 3 (1), 321–330NizamiA. R.PerveenA.NazeerW.BaqirM.2018WALK POLYNOMIAL: A New Graph InvariantAppl. Maths and Nonlin. Sciences31321330Search in Google Scholar

H. S. Ramane, I. Gutman, D. S. Revankar, (2008), Distance equienergetic graphs, MATCH Commun. Math. Comput. Chem., 60, 473–484RamaneH. S.GutmanI.RevankarD. S.2008Distance equienergetic graphsMATCH Commun. Math. Comput. Chem.60473484Search in Google Scholar

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, 59–68RamaneH. S.RevankarD. S.GutmanI.RaoS. B.AcharyaB. D.WalikarH. B.2008Bounds for the distance energy of a graphKragujevac J. Math.315968Search in Google Scholar

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