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# On Solutions of Fractional order Telegraph Partial Differential Equation by Crank-Nicholson Finite Difference Method

###### Accepted: 17 Apr 2019
Journal Details
Format
Journal
First Published
01 Jan 2016
Publication timeframe
2 times per year
Languages
English
Copyright
© 2020 Sciendo

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.

#### 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 $qij={1+q+q2+⋯+qk−1,if k=dij,0,if i=j,${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 $fn(G,μ)=det(μI−Aq(G)).${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 graph$S60$S_6^0as in Fig. 1.1.

As$Aq(S60)=(01+q1+q1+q+q2111+q01+q11+q+q211+q1+q0111+q+q21+q+q21101+q1+q11+q+q211+q01+q111+q+q21+q1+q0),${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 of$S06$S_0^6is$(μ+q2−q+1)(μ−q2−3q−5)(μ+q2+2q+1)2(μ−q2+1)2.$(\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 of$S60$S_6^0would be$(−q2+q−1q2+3q+5−q2−2q−1q2−11122)$\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 of$S60$S_6^0is found as$Eq(S60)=|−(q2−q+1)|+|q2+3q+5|+2⋅|−(q2+2q+1)|+2⋅|q2−1|=q2−q+1+q2+3q+5+2q2+4q+2+2q2−2=6q2+6q+6.$\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 that$det A=∏i=1nμi.$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 hence$det P(A)=∏i=1nP(μi).$det\;P(A) = \prod\limits_{i = 1}^n P({\mu _i}).

Lemma 3

Let$B=(B0B1B1B0)$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. Then$∑i=1nμi2=2m+(n2−n−2m)(1+q)2.$\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 $∑i=1nμi2=∑i=1n∑j=1nqijqji=∑i=1n∑j=1n(qij)2=(2m)(1)2+(n2−n−2m)(1+q)2=2m+(n2−n−2m)(1+q)2.$\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. Then$μ1≥n(n−1)(1+q)−2mqn.${\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 $μ1≥XAqXTXXT=∑i=1n(di(1)+(n−di−1)(1+q))n=(2m+(n−1)n(1+q)−2m(1+q))n=n(n−1)(1+q)−2mqn.$\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 $(1−q)A+(1−q2)A¯−J$(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)|, then$2m+(n2−n−2m)(1+q)2+n(n−1)P2n≤Eq(G)≤n(2m+(n2−n−2m)(1+q)2).$\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 $Eq2(G)≤n(2m+(n2−n−2m)(1+q)2)$E_q^2(G) \le n(2m + ({n^2} - n - 2m)(1 + q{)^2}) and therefore we obtain $Eq(G)≤n(2m+(n2−n−2m)(1+q)2).${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 $1n(n−1)∑i≠j|μiμj|≥[∏i≠j|μiμj|]1n(n−1)=[∏i=1n|μi|2(n−1)]1n(n−1)=[∏i=1n|μi|]2n=[∏i=1nμi]2n=|detAq(G)|2n=P2n.$\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 $∑i≠j|μiμj|≥n(n−1)P2n.$\sum\limits_{i \ne j} |{\mu _i}{\mu _j}| \ge n(n - 1){P^{{2 \over n}}}. Now consider $Eq2(G)=(∑i=1n|μi|)2=∑i=1n|μi|2+∑i≠j|μi||μj|.$\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 $Eq2(G)≥2m+(n2−n−2m)(1+q)2+n(n−1)P2n$E_q^2(G) \ge 2m + ({n^2} - n - 2m)(1 + q{)^2} + n(n - 1){P^{{2 \over n}}} and hence $Eq(G)≥2m+(n2−n−2m)(1+q)2+n(n−1)P2n.${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 that$n(n−1)(1+q)−2mqn≥1,${{n(n - 1)(1 + q) - 2mq} \over n} \ge 1,then$Eq(G)≤n(n−1)(1+q)−2mqn+(n−1)[(2m+(n2−n−2m)(1+q)2−(n(n−1)(1+q)−2mqn)2].${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 $(∑i=2n|μi|)2≤∑i=2n1∑i=2nμi2[Eq(G)−μ1]2≤(n−1)(2m+(n2−n−2m)(1+q)2).${\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 $Eq(G)≤μ1+(n−1)(2m+(n2−n−2m)(1+q)2−μ12).${E_q}(G) \le {\mu _1} + \sqrt {(n - 1)(2m + ({n^2} - n - 2m)(1 + q{)^2} - \mu _1^2)} . Let $f(x)=x+(n−1)(2m+(n2−n−2m)(1+q)2−x2).$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 $1−x(n−1)(n−1)(2m+(n2−n−2m)(1+q)2−x2)≤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 $x≥2m+(n2−n−2m)(1+q)2n.$x \ge \sqrt {{{2m + ({n^2} - n - 2m)(1 + q{)^2}} \over n}} . Therefore the function f (x) is decreasing in the interval $(2m+(n2−n−2m)(1+q)2n,2m+(n2−n−2m)(1+q)2).$\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 $(n(n−1)(1+q)−2mq)/n≤μ1≤2m+(n2−n−2m)(1+q)2$(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 $Eq(G)≤f(μ1)≤f((n(n−1)(1+q)−2mq)/n)${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, then$Eq(G) ≡ |0| (mod 2).${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.

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$(μ−(1+q)(n1−1)+r1q)(μ−(1+q)(n2−1)+r2q)−n1n2(μ−(1+q)(n1−1)+r1q)(μ−(1+q)(n2−1)+r2q)ϕ(G1:μ)ϕ(G2:μ).${{(\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 $qij′$q_{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 $∑j=1n1qij=1(r1)+(1+q)(n1−1−r1)=r1+(1+q)(n1−1)−r1−r1q=(1+q)(n1−1)−r1q$\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 $∑j=1n2qij′=(1+q)(n2−1)−r2q.$\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 \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., $ϕ(G1∇G2:μ)=|μ−Eq(G1∇G2)|.$\phi ({G_1}\nabla {G_2}:\mu ) = |\mu - {E_q}({G_1}\nabla {G_2})|.

This polynomial is equal to the following determinant $|μ−q12−q13…−q1n1−1−1−1…−1−q21μ−q23…−q2n1−1−1−1…−1−q31−q32μ…−q3n1−1−1−1…−1⋮⋮⋮⋱⋮⋮⋮⋮⋱⋮−qn11−qn12−qn13…μ−1−1−1…−1−1−1−1…−1μ−q12′−q13′…−q1n2′−1−1−1…−1−q21′μ−q23′…−q2n2′−1−1−1…−1−q31′−q32′μ…−q3n2′⋮⋮⋮⋱⋮⋮⋮⋮⋱⋮−1−1−1…−1−qn21′−qn22′−qn23′…μ|(n1+n2)×(n1+n2).${\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 $Rn1+2′=Rn1+2−Rn1+1$R_{{n_1} + 2}^\prime = {R_{{n_1} + 2}} - {R_{{n_1} + 1}} ; $Rn1+3′=Rn1+3−Rn1+1$R_{{n_1} + 3}^\prime = {R_{{n_1} + 3}} - {R_{{n_1} + 1}} ;··· ; $Rn1+n2′=Rn1+n2−Rn1+1$R_{{n_1} + {n_2}}^\prime = {R_{{n_1} + {n_2}}} - {R_{{n_1} + 1}} to the above determinant, we see that the determinant becomes $|μ−q12−q13…−q1n1−1−1−1…−1−q21μ−q23…−q2n1−1−1−1…−1−q31−q32μ…−q3n1−1−1−1…−1⋮⋮⋮⋱⋮⋮⋮⋮⋱⋮−qn11−qn12−qn13…μ−1−1−1…−1−1−1−1…−1μ−q12′−q13′…−q1n2′000…0−q21′−μμ+q12′−q23′+q13′…−q2n2′+q1n2′000…0−q31′−μ−q32′+q12′μ+q13′…−q3n2′+q1n2′⋮⋮⋮⋱⋮⋮⋮⋮⋱⋮000…0−qn21′−μ−qn22′+q12′−qn23′+q13′…μ+q1n2′|.$\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 $Cn1+1′=Cn1+1+Cn1+2+⋯+Cn1+n2$C_{{n_1} + 1}^\prime = {C_{{n_1} + 1}} + {C_{{n_1} + 2}} + \cdots + {C_{{n_1} + {n_2}}} , using the fact that $qij′=qji′$q_{ij}^\prime = q_{ji}^\prime and the equations above, the same determinant becomes $|μ−q12…−q1n1−n2−1−1…−1−q21μ…−q2n1−n2−1−1…−1−q31−q32…−q3n1−n2−1−1…−1⋮⋮⋱⋮⋮⋮⋮⋱⋮−qn11−qn12…μ−n2−1−1…−1−1−1…−1μ−(1+q)(n2−1)−r2q−q12′−q13′…−q1n2′00…00μ+q12′−q23′+q13′…−q2n2′+q1n2′00…00−q32′+q12′μ+q13′…−q3n2′+q1n2′⋮⋮⋱⋮⋮⋮⋮⋱⋮00…00−qn22′+q12′−qn23′+q13′…μ+q1n2′|= |μ−q12−q13⋯−q1n1−n2−q21μ−q23⋯−q2n1−n2−q31−q32μ⋯−q3n1−n2⋮⋮⋮⋮⋮⋮−qn11−qn12qn13⋯μ−n2−1−1−1⋯−1μ−(1+q)(n2−1)+r2q||B|,$\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 $|B|=|μ+q12′−q23′+q13′⋯−q2n2′+q1n2′−q32′+q12′μ+q13′⋯−q3n2′+q1n2′⋮⋮⋮⋮−qn22′+q12′−qn23′+q12′⋯μ+q1n2′|(n2−1)×(n2−1).$|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 $R2′=R2−R1$R_2^\prime = {R_2} - {R_1} , $R3′=R3−R1$R_3^\prime = {R_3} - {R_1} , ···, $Rn1′=Rn1−R1$R_{{n_1}}^\prime = {R_{{n_1}}} - {R_1} , the above determinant transforms to $|μ−q12−q13⋯−q1n1−n2−q21−μμ+q12−q23+q13⋯−q2n1+q1n10−q31−μ−q32+q12μ+q13⋯−q3n1+q1n10⋮⋮⋮⋮⋮⋮−qn11−μ−qn12+q12−qn13+q13⋯μ+q1n10−1−1−1⋯−1μ−(1+q)(n2−1)+r2q||B|.$\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 $C1′=C1+C2+⋯+Cn1$C_1^\prime = {C_1} + {C_2} + \cdots + {C_{{n_1}}} and using the above equations, the determinant becomes $|μ−(1+q)(n1−1)+r1q−q12−q13⋯−q1n1−n20μ+q12−q23+q13⋯−q2n1+q1n100−q32+q12μ+q13⋯−q3n1+q1n10⋮⋮⋮⋮⋮0−qn12+q12−qn13+q13⋯μ+q1n10−n1−1−1⋯−1μ−(1+q)(n2−1)+r2q||B|.$\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 $ϕ(G1∇G2:μ)={(μ−(1+q)(n1−1)+r1q)Δ1−(−1)n1n1Δ2}|B|,$\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 $Δ1=|μ+q12−q23+q13⋯−q2n1+q1n10−q32+q12μ+q13⋯−q3n1+q1n10⋮⋮⋮⋮⋮−qn12+q12−qn13+q13⋯μ+q1n10−1−1⋯−1μ−(1+q)(n1−1)+r2q|=(μ−(1+q)(n1−1)+r2q)|A|(−1)n1+n2=(μ−(1+q)(n1−1)+r2q)|A|$\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 $Δ2=|−q12−q13⋯−q1n1−n2μ+q12−q23+q13⋯−q2n1+q1n10−q32+q12μ+q13⋯−q3n1+q1n10⋮⋮⋮⋮⋮−qn12+q12−qn13+q13⋯μ+q1n10|=(−1)−n1+1(−n2)|A|=n2(−1)n1|A|.$\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 $ϕ(G1∇G2:μ)=((μ−(1+q)(n1−1)+r1q)(μ−(1+q)(n2−1)+r2q)|A|)−(−1)n1n1n2(−1)n1|A|)|B|,$\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. $ϕ(G1∇G2:μ)=|A||B|[(μ−(1+q)(n1−1)+r1q)⋅(μ−(1+q)(n2−1)+r2q)−n1n2]$\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 $|A|=|μ+q12−q23+q13⋯−q2n1+q1n1−q32+q12μ+q13⋯−q3n1+q1n1⋮⋮⋮⋮−qn12+q12−qn13+q13⋯μ+q1n1|.$|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 $|A|=1(μ−(1+q)(n1−1)+r1q)×|μ−(1+q)(n1−1)+r1q−q12−q13⋯−q1n10μ+q12−q23+q13⋯−q2n1+q1n10−q32+q12μ+q13⋯−q3n1+q1n1⋮⋮⋮⋮⋮0−qn12+q12−qn13+q13⋯μ+q1n1|.$\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 $|A|=1μ−(1+q)(n1−1)+r1q×|μ−q12−q13⋯−q1n1−μ−q21μ+q12−q23+q13⋯−q2n1+q1n1−μ−qn31−q32+q12μ+q13⋯−q3n1+q1n1⋮⋮⋮⋮⋮−μ−qn11−qn12+q12−qn13+q13⋯μ+q1n1|.$\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; $R3′=R3+R1$R_3^\prime = {R_3} + {R_1} ;··· ; $Rn′=Rn1+R1$R_n^\prime = {R_{{n_1}}} + {R_1} , we have $|A|=1μ−(1+q)(n1−1)+r1q×|μ−q12−q13⋯−q1n1−q21μ−q23⋯−q2n1−q31−q32μ⋯−q3n1⋮⋮⋮⋮−qn11−qn12−qn13⋯μ|=1μ−(1+q)(n1−1)+r1qϕ(G1;μ).$\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, $|B|=1μ−(1+q)(n2−1)+r2qϕ(G2;μ).$|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, then$Eq(G1∇G2)={Eq(G1)+Eq(G2), if RK≥n1n2,Eq(G1)+Eq(G2)−(R+K)+(R+K)2−4(RK−n1n2), if RK{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 $ϕ(G1∇G2;μ)=((μ−(1+q)(n1−1)+r1q)(μ−(1+q)(n2−1)+r2q)−n1n2))(μ−(1+q)(n1−1)+r1q)(μ−(1+q)(n2−1)+r2q)ϕ(G1;μ)ϕ(G2;μ)$\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 $ϕ(G1∇G2;μ)=((μ−R)(μ−K)−n1n2)ϕ(G1;μ)ϕ(G2;μ)(μ−R)(μ−K),$\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., $(μ−R)(μ−K)ϕ(G1∇G2;μ)=((μ−R)(μ−K)−n1n2)ϕ(G1;μ)ϕ(G2;μ).$(\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, $(μ−R)(μ−K)ϕ(G1∇G2;μ)=(μ2−(R+K)μ+RK−n1n2)ϕ(G1;μ)ϕ(G2;μ).$(\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 $P1(μ)=(μ−R)(μ−K)ϕ(G1∇G2;μ)${P_1}(\mu ) = (\mu - R)(\mu - K)\phi ({G_1}\nabla {G_2};\mu ) and $P2(μ)=(μ2−(R+K)μ+RK−n1n2)ϕ(G1;μ)ϕ(G2;μ).${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 $R+K+Eq(G1∇G2)$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 $R+K+(R+K)2−4(RK−n1n2)2, R+K−(R+K)2−4(RK−n1n2)2.${{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 $Eq(G1)+Eq(G2)+|R+K+(R+K)2−4(RK−n1n2)2|+|R+K−(R+K)2−4(RK−n1n2)2|.${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, $R+K+Eq(G1∇G2)=Eq(G1)+Eq(G2)+|R+K+(R+K)2−4(RK−n1n2)2|+|R+K−(R+K)2−4(RK−n1n2)2|.$\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 $R+K+Eq(G1∇G2)=Eq(G1)+Eq(G2)+R+K+(R+K)2−4(RK−n1n2)2+R+K−(R+K)2−4(RK−n1n2)2,$\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 $R+K+Eq(G1∇G2)=Eq(G1)+Eq(G2)+R+K$R + K + {E_q}({G_1}\nabla {G_2}) = {E_q}({G_1}) + {E_q}({G_2}) + R + K implying that $Eq(G1∇G2)=Eq(G1)+Eq(G2).${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 $R+K+Eq(G1∇G2)=Eq(G1)+Eq(G2)+R+K+(R+K)2−4(RK−n1n2)2+R+K−(R+K)2−4(RK−n1n2)2.$\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 $R+K+Eq(G1∇G2)=Eq(G1)+Eq(G2)+(R+K)2−4(RK−n1n2)$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 $Eq(G1∇G2)=Eq(G1)+Eq(G2)−(R+K)+(R+K)2−4(RK−n1n2).${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.

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