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

M. R. Rajesh Kanna; R. Pradeep Kumar; Soner Nandappa; Ismail Naci Cangul

doi:10.2478/amns.2020.2.00017

Source Title:: Applied Mathematics and Nonlinear Sciences
Volume/Issue:: Volume 5: Issue 2

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

M. R. Rajesh Kanna¹, R. Pradeep Kumar², Soner Nandappa², and Ismail Naci Cangul³

¹ Department of Mathematics, Sri D Devaraj Urs Government First Grade College, Hunsur, India
² Department of Studies in Mathematics, University of Mysore, Mysuru, India
³ Bursa Uludag University, Faculty of Arts and Science, Department of Mathematics, Gorukle, Turkey

M. R. Rajesh Kanna

Department of Mathematics, Sri D Devaraj Urs Government First Grade College, Hunsur, 571105, India
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, R. Pradeep Kumar

Department of Studies in Mathematics, University of Mysore, Mysuru, 570 006, India
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, Soner Nandappa

Department of Studies in Mathematics, University of Mysore, Mysuru, 570 006, 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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DOI:: https://doi.org/10.2478/amns.2020.2.00017
Published online:: 10 Aug 2020

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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:: q-distance matrix; q-distance eigenvalues; q-distance energy; join of graphs; union of graphs; 05C07; 05C76; 05C90

1 Introduction and preliminaries

Let G be a graph with n vertices and m edges and let A = (a_ij) be the adjacency matrix of G. The eigenvalues λ₁,λ₂,...,λ_n 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) = \sum_{i = 1}^{n} | λ_{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 v_i and v_j which is defined as the length of the shortest path between these two vertices. Let μ₁, μ₂, ..., μ_n be the eigenvalues of the distance matrix of G. The distance energy DE is defined by

DE = DE (G) : = \sum_{i = 1}^{n} | μ_{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

A_{q} (G) = (q_{ij}) .

For an indeterminate q, the entries q_ij of this new matrix are defined by

q_{ij} = {\begin{matrix} 1 + q + q^{2} + \dots + q^{k - 1}, & if k = d_{ij}, \\ 0, & if i = j, \end{matrix}

where k = d_ij is the distance between the vertices v_i and v_j. Each entry of A_q(G) is a polynomial in q. Observe that A_q(G) is an entry-wise non-negative matrix for all q ≥ −1.

The characteristic polynomial of A_q(G) is defined by

f_{n} (G, μ) = \det (μ I - A_{q} (G)) .

The q-distance eigenvalues of the graph G are similarly the eigenvalues of A_q(G). Since A_q(G) is real and symmetric, its eigenvalues are also real numbers and we label them in non-increasing order μ₁ ≥ μ₂ ≥ ··· ≥ μ_n. The q-distance energy of G is denoted by E_q(G) and is defined by

E_{q} (G) = \sum_{i = 1}^{n} | μ_{i} | .

Note that the trace of A_q(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 $S_{6}^{0}$ as in Fig. 1.1.

A_{q} (S_{6}^{0}) = (\begin{matrix} 0 & 1 + q & 1 + q & 1 + q + q^{2} & 1 & 1 \\ 1 + q & 0 & 1 + q & 1 & 1 + q + q^{2} & 1 \\ 1 + q & 1 + q & 0 & 1 & 1 & 1 + q + q^{2} \\ 1 + q + q^{2} & 1 & 1 & 0 & 1 + q & 1 + q \\ 1 & 1 + q + q^{2} & 1 & 1 + q & 0 & 1 + q \\ 1 & 1 & 1 + q + q^{2} & 1 + q & 1 + q & 0 \end{matrix}),

the characteristic polynomial of

S_{0}^{6}

(μ + q^{2} - q + 1) (μ - q^{2} - 3 q - 5) (μ + q^{2} + 2 q + {1)}^{2} {(μ - q^{2} + 1)}^{2} .

Then the q-distance spectrum of

S_{6}^{0}

would be

(\begin{matrix} - q^{2} + q - 1 & q^{2} + 3 q + 5 & - q^{2} - 2 q - 1 & q^{2} - 1 \\ 1 & 1 & 2 & 2 \end{matrix})

and therefore the q-distance energy of

S_{6}^{0}

is found as

\begin{matrix} E_{q} (S_{6}^{0}) & = | - (q^{2} - q + 1) | + | q^{2} + 3 q + 5 | + 2 \cdot | - (q^{2} + 2 q + 1) | + 2 \cdot | q^{2} - 1 | \\ = q^{2} - q + 1 + q^{2} + 3 q + 5 + 2 q^{2} + 4 q + 2 + 2 q^{2} - 2 \\ = 6 q^{2} + 6 q + 6 . \end{matrix}

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,..., μ_n}. Then it is well-known that

\det A = \prod_{i = 1}^{n} μ_{i} .

In addition, for any polynomial P(x), the value P(μ) is an eigenvalue of P(A) and hence

\det P (A) = \prod_{i = 1}^{n} P (μ_{i}) .

Lemma 3

Let $B = (\begin{matrix} B_{0} & B_{1} \\ B_{1} & B_{0} \end{matrix})$ be a symmetric 2 × 2 block matrix. Then the spectrum of B is the union of the spectra of B₀ + B₁and B₀ − B₁.

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,..., μ_n. Then

\sum_{i = 1}^{n} μ_{i}^{2} = 2 m + (n^{2} - n - 2 m) (1 + q)^{2} .

Proof

In a q-distance adjacency matrix A_q(G), there are 2m elements equal to 1 and n² − n − 2m elements equal to (1 + q). Therefore

\begin{matrix} \sum_{i = 1}^{n} μ_{i}^{2} & = \sum_{i = 1}^{n} \sum_{j = 1}^{n} q_{ij} q_{ji} \\ = \sum_{i = 1}^{n} \sum_{j = 1}^{n} {(q_{ij})}^{2} \\ = (2 m {) (1)}^{2} + (n^{2} - n - 2 m) (1 + q)^{2} \\ = 2 m + (n^{2} - n - 2 m) (1 + q)^{2} . \end{matrix}

Theorem 5

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

μ_{1} \geq \frac{n (n - 1) (1 + q) - 2 mq}{n} .

Proof

Let G be a connected graph of diameter 2 with its vertices labeled as v₁,v₂,...,v_n and let d_i denote the degree of v_i. As G is of diameter 2, it is easy to observe that the i^th row of A_q consists of d_i times 1s and n − d_i − 1 times 2s. Let X = [1,1,1,...,1] be a vector containing only 1s. Then by the Rayleigh principle, we have

\begin{matrix} μ_{1} & \geq \frac{{XA}_{q} X^{T}}{{XX}^{T}} \\ = \frac{\sum_{i = 1}^{n} (d_{i} (1) + (n - d_{i} - 1) (1 + q))}{n} \\ = \frac{(2 m + (n - 1) n (1 + q) - 2 m (1 + q))}{n} \\ = \frac{n (n - 1) (1 + q) - 2 mq}{n} . \end{matrix}

Theorem 6

Let G be an r-regular graph of diameter 2 with r, μ₂,..., μ_n as its eigenvalues. Then the q-distance eigenvalues of G are −rq + (n − 1)(1 + q),−qμ₂ − (1 + q), −qμ₃ − (1 + q),...,−qμ_n − (1 + q).

Proof

Let G be an r-regular graph with diameter 2 and adjacency matrix A.

\bar{A}

is the adjacency matrix of

\bar{G}

. Then the q-distance adjacency matrix of G will be

A_{q} = A + (1 + q) \bar{A} .

(1)

If r, μ₂,..., μ_n are the eigenvalues of A with r ≥ μ₂ ≥ ··· ≥ μ_n, then n − 1 − r, −μ₂ − 1, −μ₃ − 1,...,−μ_n − 1 are the eigenvalues of

\bar{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,..., μ_n}. 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,−qμ_i(1 + q) − (1 + 2q + q²),−rq(1 − q) + q²(1 − n) − 1,−qμ_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) \bar{A}

. Then the q-distance matrix of H is of the form

(\begin{matrix} A + (1 + q) \bar{A} & J + qA + (q + q^{2}) \bar{A} \\ J + qA + (q + q^{2}) \bar{A} & A + (1 + q) \bar{A} \end{matrix}) .

By Lemma 3, the spectrum of H is the union of the spectra of

(1 + q) A + (1 + 2 q + q^{2}) \bar{A} + J

and

(1 - q) A + (1 - q^{2}) \bar{A} - J

. If r, μ₂,..., μ_n are the eigenvalues of A with r ≥ μ₂ ≥ ··· ≥ μ_n then n − r − 1,−μ₂ − 1,−μ₃ − 1,...,−μ_n − 1 are the eigenvalues of

\bar{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 E_q(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 = |detA_q(G)|, then

\sqrt{2 m + (n^{2} - n - 2 m) (1 + q)^{2} + n (n - 1) P^{\frac{2}{n}}} \leq E_{q} (G) \leq \sqrt{n (2 m + (n^{2} - n - 2 m) (1 + q)^{2})} .

Proof

Recall that the Cauchy-Schwarz inequality states that

{(\sum_{i = 1}^{n} a_{i} b_{i})}^{2} \leq (\sum_{i = 1}^{n} a_{i}^{2}) (\sum_{i = 1}^{n} b_{i}^{2}) .

If we substitute a_i = 1 and b_i = | μ_i |, then we obtain

{(\sum_{i = 1}^{n} | μ_{i} |)}^{2} \leq (\sum_{i = 1}^{n} 1) (\sum_{i = 1}^{n} μ_{i}^{2}) .

Hence by Theorem 4, we have

E_{q}^{2} (G) \leq n (2 m + (n^{2} - n - 2 m) (1 + q)^{2})

and therefore we obtain

E_{q} (G) \leq \sqrt{n (2 m + (n^{2} - n - 2 m) (1 + q)^{2})} .

Since the arithmetic mean is not smaller than the geometric mean, we have

\begin{matrix} \frac{1}{n (n - 1)} \sum_{i \neq j} | μ_{i} μ_{j} | & \geq {[\prod_{i \neq j} | μ_{i} μ_{j} |]}^{\frac{1}{n (n - 1)}} \\ = [\prod_{i = 1}^{n} | μ_{i} |^{2 (n - 1)}]^{\frac{1}{n (n - 1)}} \\ = [\prod_{i = 1}^{n} | μ_{i} |]^{\frac{2}{n}} \\ = [\prod_{i = 1}^{n} μ_{i}]^{\frac{2}{n}} \\ = | \det A_{q} (G {) |}^{\frac{2}{n}} \\ = P^{\frac{2}{n}} . \end{matrix}

Therefore

\sum_{i \neq j} | μ_{i} μ_{j} | \geq n (n - 1) P^{\frac{2}{n}} .

Now consider

\begin{matrix} E_{q}^{2} (G) & = (\sum_{i = 1}^{n} | μ_{i} |)^{2} \\ = \sum_{i = 1}^{n} | μ_{i} |^{2} + \sum_{i \neq j} | μ_{i} | | μ_{j} | . \end{matrix}

Therefore, by Theorem 4, we obtain

E_{q}^{2} (G) \geq 2 m + (n^{2} - n - 2 m) (1 + q)^{2} + n (n - 1) P^{\frac{2}{n}}

and hence

E_{q} (G) \geq \sqrt{2 m + (n^{2} - n - 2 m) (1 + q)^{2} + n (n - 1) P^{\frac{2}{n}}} .

Now, we find another bound for E_q(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

\frac{n (n - 1) (1 + q) - 2 mq}{n} \geq 1,

then

E_{q} (G) \leq \frac{n (n - 1) (1 + q) - 2 mq}{n} + \sqrt{(n - 1) [(2 m + (n^{2} - n - 2 m) (1 + q)^{2} - {(\frac{n (n - 1) (1 + q) - 2 mq}{n})}^{2}]} .

Proof

By substituting a_i = 1 and b_i =| μ_i | in Cauchy-Schwarz inequality, we have

{(\sum_{i = 2}^{n} | μ_{i} |)}^{2} \leq \sum_{i = 2}^{n} 1 \sum_{i = 2}^{n} μ_{i}^{2} {[E_{q} (G) - μ_{1}]}^{2} \leq (n - 1) (2 m + (n^{2} - n - 2 m) (1 + q)^{2}) .

Hence

E_{q} (G) \leq μ_{1} + \sqrt{(n - 1) (2 m + (n^{2} - n - 2 m) (1 + q)^{2} - μ_{1}^{2})} .

Let

f (x) = x + \sqrt{(n - 1) (2 m + (n^{2} - n - 2 m) (1 + q)^{2} - x^{2})} .

Then for a decreasing function f (x), the fact f^′(x) ≤ 0 implies that

1 - \frac{x (n - 1)}{\sqrt{(n - 1) (2 m + (n^{2} - n - 2 m) (1 + q)^{2} - x^{2})}} \leq 0 .

From this we obtain

x \geq \sqrt{\frac{2 m + (n^{2} - n - 2 m) (1 + q)^{2}}{n}} .

Therefore the function f (x) is decreasing in the interval

(\sqrt{\frac{2 m + (n^{2} - n - 2 m) (1 + q)^{2}}{n}}, \sqrt{2 m + (n^{2} - n - 2 m) (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 (n - 1) (1 + q) - 2 m q) / n \leq μ_{1} \leq \sqrt{2 m + (n^{2} - n - 2 m) (1 + q)^{2}}

. By Lemma 5.3, we can write f (μ₁) ≤ f((n(n − 1)(1 + q) − 2mq)/n). Hence

E_{q} (G) \leq f (μ_{1}) \leq f ((n (n - 1) (1 + q) - 2 mq) / 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 E_q(G) of G is a rational number, then

E_{q} (G) \equiv | 0 | (\mod 2) .

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 G₁∇G₂ 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 G₁∇G₂is

\frac{(μ - (1 + q) (n_{1} - 1) + r_{1} q) (μ - (1 + q) (n_{2} - 1) + r_{2} q) - n_{1} n_{2}}{(μ - (1 + q) (n_{1} - 1) + r_{1} q) (μ - (1 + q) (n_{2} - 1) + r_{2} q)} ϕ (G_{1} : μ) ϕ (G_{2} : μ) .

(2)

Proof

Let us assume that v₁, v₂, ..., v_n₁ be the vertices of the graph G₁ and u₁, u₂, ..., u_n₂ be the vertices of the graph G₂. Let q_ij denote the q-distance between the vertices v_i and v_j in G₁ and

q_{ij}^{'}

denote the q-distance between the vertices u_i and u_j 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

\begin{matrix} \sum_{j = 1}^{n_{1}} q_{ij} & = 1 (r_{1}) + (1 + q) (n_{1} - 1 - r_{1}) \\ = r_{1} + (1 + q) (n_{1} - 1) - r_{1} - r_{1} q \\ = (1 + q) (n_{1} - 1) - r_{1} q \end{matrix}

and similarly, for i = 1,2,3,...,n₂, we have

\sum_{j = 1}^{n_{2}} q_{ij}^{'} = (1 + q) (n_{2} - 1) - r_{2} q .

Let E_q(G₁∇G₂) be the q-distance adjacency matrix of the join graph G₁∇G₂. Then this matrix A_q(G₁∇G₂) has the form

{\begin{matrix} \begin{matrix} v_{1} & v_{2} & v_{3} & \dots & v_{n_{1}} & u_{1} & u_{2} & u_{3} & \dots & u_{n_{2}} \end{matrix} \\ \begin{matrix} v_{1} \\ v_{2} \\ v_{3} \\ ⋮ \\ v_{n_{1}} \\ u_{1} \\ u_{2} \\ u_{3} \\ ⋮ \\ u_{n_{1}} \end{matrix} & (\begin{matrix} 0 & q_{12} & q_{13} & \dots & q_{1}_{n_{1}} & 1 & 1 & 1 & \dots & 1 \\ q_{21} & 0 & q_{23} & \dots & q_{2 n_{1}} & 1 & 1 & 1 & \dots & 1 \\ q_{31} & q_{32} & 0 & \dots & q_{3 n_{1}} & 1 & 1 & 1 & \dots & 1 \\ ⋮ & ⋮ & ⋮ & ⋱ & ⋮ & ⋮ & ⋮ & ⋮ & ⋱ & ⋮ \\ q_{n_{1}}_{1} & q_{n_{1}}_{2} & q_{n_{1}}_{3} & \dots & 0 & 1 & 1 & 1 & \dots & 1 \\ 1 & 1 & 1 & \dots & 1 & 0 & q_{12}^{'} & q_{13}^{'} & \dots & q_{1 n_{2}}^{'} \\ 1 & 1 & 1 & \dots & 1 & q_{21}^{'} & 0 & q_{23}^{'} & \dots & q_{2 n_{2}}^{'} \\ 1 & 1 & 1 & \dots & 1 & q_{31}^{'} & q_{32}^{'} & 0 & \dots & q_{3 n_{2}}^{'} \\ ⋮ & ⋮ & ⋮ & ⋱ & ⋮ & ⋮ & ⋮ & ⋮ & ⋱ & ⋮ \\ 1 & 1 & 1 & \dots & 1 & q_{n_{2} 1}^{'} & q_{n_{2} 2}^{'} & q_{n_{2} 3}^{'} & \dots & 0 \end{matrix}) \end{matrix}}_{(n_{1} + n_{2}) \times (n_{1} + n_{2})}

Let φ(G₁∇G₂ : μ) denote the q-distance characteristic polynomial of G₁∇G₂; i.e.,

ϕ (G_{1} \nabla G_{2} : μ) = | μ - E_{q} (G_{1} \nabla G_{2}) | .

This polynomial is equal to the following determinant

{| \begin{matrix} μ & - q_{12} & - q_{13} & \dots & - q_{1 n_{1}} & - 1 & - 1 & - 1 & \dots & - 1 \\ - q_{21} & μ & - q_{23} & \dots & - q_{2 n_{1}} & - 1 & - 1 & - 1 & \dots & - 1 \\ - q_{31} & - q_{32} & μ & \dots & - q_{3 n_{1}} & - 1 & - 1 & - 1 & \dots & - 1 \\ ⋮ & ⋮ & ⋮ & ⋱ & ⋮ & ⋮ & ⋮ & ⋮ & ⋱ & ⋮ \\ - q_{n_{1} 1} & - q_{n_{1} 2} & - q_{n_{1} 3} & \dots & μ & - 1 & - 1 & - 1 & \dots & - 1 \\ - 1 & - 1 & - 1 & \dots & - 1 & μ & - q_{12}^{'} & - q_{13}^{'} & \dots & - q_{1 n_{2}}^{'} \\ - 1 & - 1 & - 1 & \dots & - 1 & - q_{21}^{'} & μ & - q_{23}^{'} & \dots & - q_{2 n_{2}}^{'} \\ - 1 & - 1 & - 1 & \dots & - 1 & - q_{31}^{'} & - q_{32}^{'} & μ & \dots & - q_{3 n_{2}}^{'} \\ ⋮ & ⋮ & ⋮ & ⋱ & ⋮ & ⋮ & ⋮ & ⋮ & ⋱ & ⋮ \\ - 1 & - 1 & - 1 & \dots & - 1 & - q_{n_{2} 1}^{'} & - q_{n_{2} 2}^{'} & - q_{n_{2} 3}^{'} & \dots & μ \end{matrix} |}_{(n_{1} + n_{2}) \times (n_{1} + n_{2})} .

Applying the row operations

R_{n_{1} + 2}^{'} = R_{n_{1} + 2} - R_{n_{1} + 1}

;

R_{n_{1} + 3}^{'} = R_{n_{1} + 3} - R_{n_{1} + 1}

;··· ;

R_{n_{1} + n_{2}}^{'} = R_{n_{1} + n_{2}} - R_{n_{1} + 1}

to the above determinant, we see that the determinant becomes

| \begin{matrix} μ & - q_{12} & - q_{13} & \dots & - q_{1 n_{1}} & - 1 & - 1 & - 1 & \dots & - 1 \\ - q_{21} & μ & - q_{23} & \dots & - q_{2 n_{1}} & - 1 & - 1 & - 1 & \dots & - 1 \\ - q_{31} & - q_{32} & μ & \dots & - q_{3 n_{1}} & - 1 & - 1 & - 1 & \dots & - 1 \\ ⋮ & ⋮ & ⋮ & ⋱ & ⋮ & ⋮ & ⋮ & ⋮ & ⋱ & ⋮ \\ - q_{n_{1} 1} & - q_{n_{1} 2} & - q_{n_{1} 3} & \dots & μ & - 1 & - 1 & - 1 & \dots & - 1 \\ - 1 & - 1 & - 1 & \dots & - 1 & μ & - q_{12}^{'} & - q_{13}^{'} & \dots & - q_{1 n_{2}}^{'} \\ 0 & 0 & 0 & \dots & 0 & - q_{21}^{'} - μ & μ + q_{12}^{'} & - q_{23}^{'} + q_{13}^{'} & \dots & - q_{2 n_{2}}^{'} + q_{1 n_{2}}^{'} \\ 0 & 0 & 0 & \dots & 0 & - q_{31}^{'} - μ & - q_{32}^{'} + q_{12}^{'} & μ + q_{13}^{'} & \dots & - q_{3 n_{2}}^{'} + q_{1 n_{2}}^{'} \\ ⋮ & ⋮ & ⋮ & ⋱ & ⋮ & ⋮ & ⋮ & ⋮ & ⋱ & ⋮ \\ 0 & 0 & 0 & \dots & 0 & - q_{n_{2} 1}^{'} - μ & - q_{n_{2} 2}^{'} + q_{12}^{'} & - q_{n_{2} 3}^{'} + q_{13}^{'} & \dots & μ + q_{1 n_{2}}^{'} \end{matrix} | .

Applying the column operation

C_{n_{1} + 1}^{'} = C_{n_{1} + 1} + C_{n_{1} + 2} + \dots + C_{n_{1} + n_{2}}

, using the fact that

q_{ij}^{'} = q_{ji}^{'}

and the equations above, the same determinant becomes

\begin{matrix} | \begin{matrix} μ & - q_{12} & \dots & - q_{1 n_{1}} & - n_{2} & - 1 & - 1 & \dots & - 1 \\ - q_{21} & μ & \dots & - q_{2 n_{1}} & - n_{2} & - 1 & - 1 & \dots & - 1 \\ - q_{31} & - q_{32} & \dots & - q_{3 n_{1}} & - n_{2} & - 1 & - 1 & \dots & - 1 \\ ⋮ & ⋮ & ⋱ & ⋮ & ⋮ & ⋮ & ⋮ & ⋱ & ⋮ \\ - q_{n_{1} 1} & - q_{n_{1} 2} & \dots & μ & - n_{2} & - 1 & - 1 & \dots & - 1 \\ - 1 & - 1 & \dots & - 1 & μ - (1 + q) (n_{2} - 1) - r_{2} q & - q_{12}^{'} & - q_{13}^{'} & \dots & - q_{1 n_{2}}^{'} \\ 0 & 0 & \dots & 0 & 0 & μ + q_{12}^{'} & - q_{23}^{'} + q_{13}^{'} & \dots & - q_{2 n_{2}}^{'} + q_{1 n_{2}}^{'} \\ 0 & 0 & \dots & 0 & 0 & - q_{32}^{'} + q_{12}^{'} & μ + q_{13}^{'} & \dots & - q_{3 n_{2}}^{'} + q_{1 n_{2}}^{'} \\ ⋮ & ⋮ & ⋱ & ⋮ & ⋮ & ⋮ & ⋮ & ⋱ & ⋮ \\ 0 & 0 & \dots & 0 & 0 & - q_{n_{2} 2}^{'} + q_{12}^{'} & - q_{n_{2} 3}^{'} + q_{13}^{'} & \dots & μ + q_{1 n_{2}}^{'} \end{matrix} | \\ = | \begin{matrix} μ & - q_{12} & - q_{13} & \dots & - q_{1 n_{1}} & - n_{2} \\ - q_{21} & μ & - q_{23} & \dots & - q_{2 n_{1}} & - n_{2} \\ - q_{31} & - q_{32} & μ & \dots & - q_{3 n_{1}} & - n_{2} \\ ⋮ & ⋮ & ⋮ & ⋮ & ⋮ & ⋮ \\ - q_{n_{1} 1} & - q_{n_{1} 2} & q_{n_{1} 3} & \dots & μ & - n_{2} \\ - 1 & - 1 & - 1 & \dots & - 1 & μ - (1 + q) (n_{2} - 1) + r_{2} q \end{matrix} | | B |, \end{matrix}

where

| B | = {| \begin{matrix} μ + q_{12}^{'} & - q_{23}^{'} + q_{13}^{'} & \dots & - q_{2 n_{2}}^{'} + q_{1 n_{2}}^{'} \\ - q_{32}^{'} + q_{12}^{'} & μ + q_{13}^{'} & \dots & - q_{3 n_{2}}^{'} + q_{1 n_{2}}^{'} \\ ⋮ & ⋮ & ⋮ & ⋮ \\ - q_{n_{2} 2}^{'} + q_{12}^{'} & - q_{n_{2} 3}^{'} + q_{12}^{'} & \dots & μ + q_{1 n_{2}}^{'} \end{matrix} |}_{(n_{2} - 1) \times (n_{2} - 1)} .

Applying the row operations

R_{2}^{'} = R_{2} - R_{1}

R_{3}^{'} = R_{3} - R_{1}

, ···,

R_{n_{1}}^{'} = R_{n_{1}} - R_{1}

, the above determinant transforms to

| \begin{matrix} μ & - q_{12} & - q_{13} & \dots & - q_{1 n_{1}} & - n_{2} \\ - q_{21} - μ & μ + q_{12} & - q_{23} + q_{13} & \dots & - q_{2 n_{1}} + q_{1 n_{1}} & 0 \\ - q_{31} - μ & - q_{32} + q_{12} & μ + q_{13} & \dots & - q_{3 n_{1}} + q_{1 n_{1}} & 0 \\ ⋮ & ⋮ & ⋮ & ⋮ & ⋮ & ⋮ \\ - q_{n_{11}} - μ & - q_{n_{12}} + q_{12} & - q_{n_{13}} + q_{13} & \dots & μ + q_{1 n_{1}} & 0 \\ - 1 & - 1 & - 1 & \dots & - 1 & μ - (1 + q) (n_{2} - 1) + r_{2} q \end{matrix} | | B | .

Applying the column operation

C_{1}^{'} = C_{1} + C_{2} + \dots + C_{n_{1}}

and using the above equations, the determinant becomes

| \begin{matrix} μ - (1 + q) (n_{1} - 1) + r_{1} q & - q_{12} & - q_{13} & \dots & - q_{1 n_{1}} & - n_{2} \\ 0 & μ + q_{12} & - q_{23} + q_{13} & \dots & - q_{2 n_{1}} + q_{1 n_{1}} & 0 \\ 0 & - q_{32} + q_{12} & μ + q_{13} & \dots & - q_{3 n_{1}} + q_{1 n_{1}} & 0 \\ ⋮ & ⋮ & ⋮ & ⋮ & ⋮ \\ 0 & - q_{n_{12}} + q_{12} & - q_{n_{13}} + q_{13} & \dots & μ + q_{1 n_{1}} & 0 \\ - n_{1} & - 1 & - 1 & \dots & - 1 & μ - (1 + q) (n_{2} - 1) + r_{2} q \end{matrix} | | B | .

Expanding it along the first column C₁, we obtain

ϕ (G_{1} \nabla G_{2} : μ) = {(μ - (1 + q) (n_{1} - 1) + r_{1} q) Δ_{1} - {(- 1)}^{n_{1}} n_{1} Δ_{2}} | B |,

(3)

where

\begin{matrix} Δ_{1} & = | \begin{matrix} μ + q_{12} & - q_{23} + q_{13} & \dots & - q_{2 n_{1}} + q_{1 n_{1}} & 0 \\ - q_{32} + q_{12} & μ + q_{13} & \dots & - q_{3 n_{1}} + q_{1 n_{1}} & 0 \\ ⋮ & ⋮ & ⋮ & ⋮ & ⋮ \\ - q_{n_{12}} + q_{12} & - q_{n_{13}} + q_{13} & \dots & μ + q_{1 n_{1}} & 0 \\ - 1 & - 1 & \dots & - 1 & μ - (1 + q) (n_{1} - 1) + r_{2} q \end{matrix} | \\ = (μ - (1 + q) (n_{1} - 1) + r_{2} q) | A | (- {1)}^{n_{1} + n_{2}} \\ = (μ - (1 + q) (n_{1} - 1) + r_{2} q) | A | \end{matrix}

and

\begin{matrix} Δ_{2} & = | \begin{matrix} - q_{12} & - q_{13} & \dots & - q_{1 n_{1}} & - n_{2} \\ μ + q_{12} & - q_{23} + q_{13} & \dots & - q_{2 n_{1}} + q_{1 n_{1}} & 0 \\ - q_{32} + q_{12} & μ + q_{13} & \dots & - q_{3 n_{1}} + q_{1 n_{1}} & 0 \\ ⋮ & ⋮ & ⋮ & ⋮ & ⋮ \\ - q_{n_{12}} + q_{12} & - q_{n_{13}} + q_{13} & \dots & μ + q_{1 n_{1}} & 0 \end{matrix} | \\ = {(- 1)}^{- n_{1} + 1} (- n_{2}) | A | \\ = n_{2} {(- 1)}^{n_{1}} | A | . \end{matrix}

Therefore we have

\begin{matrix} ϕ (G_{1} \nabla G_{2} : μ) & = ((μ - (1 + q) (n_{1} - 1) + r_{1} q) (μ - (1 + q) (n_{2} - 1) + r_{2} q) | A |) \\ - {(- 1)}^{n_{1}} n_{1} n_{2} {(- 1)}^{n_{1}} | A |) | B |, \end{matrix}

i.e.

ϕ (G_{1} \nabla G_{2} : μ) = | A | | B | [(μ - (1 + q) (n_{1} - 1) + r_{1} q) \cdot (μ - (1 + q) (n_{2} - 1) + r_{2} q) - n_{1} n_{2}]

where

| A | = | \begin{matrix} μ + q_{12} & - q_{23} + q_{13} & \dots & - q_{2 n_{1}} + q_{1 n_{1}} \\ - q_{32} + q_{12} & μ + q_{13} & \dots & - q_{3 n_{1}} + q_{1 n_{1}} \\ ⋮ & ⋮ & ⋮ & ⋮ \\ - q_{n_{12}} + q_{12} & - q_{n_{13}} + q_{13} & \dots & μ + q_{1 n_{1}} \end{matrix} | .

Clearly

| A | = \frac{1}{(μ - (1 + q) (n_{1} - 1) + r_{1} q)} \times | \begin{matrix} μ - (1 + q) (n_{1} - 1) + r_{1} q & - q_{12} & - q_{13} & \dots & - q_{1 n_{1}} \\ 0 & μ + q_{12} & - q_{23} + q_{13} & \dots & - q_{2 n_{1}} + q_{1 n_{1}} \\ 0 & - q_{32} + q_{12} & μ + q_{13} & \dots & - q_{3 n_{1}} + q_{1 n_{1}} \\ ⋮ & ⋮ & ⋮ & ⋮ & ⋮ \\ 0 & - q_{n_{12}} + q_{12} & - q_{n_{13}} + q_{13} & \dots & μ + q_{1 n_{1}} \end{matrix} | .

Applying the operation

C_{1}^{'} = C_{1} - (C_{2} + C_{3} + \dots + C_{n_{1}})

, the determinant becomes

| A | = \frac{1}{μ - (1 + q) (n_{1} - 1) + r_{1} q} \times | \begin{matrix} μ & - q_{12} & - q_{13} & \dots & - q_{1 n_{1}} \\ - μ - q_{21} & μ + q_{12} & - q_{23} + q_{13} & \dots & - q_{2 n_{1}} + q_{1 n_{1}} \\ - μ - q_{n_{31}} & - q_{32} + q_{12} & μ + q_{13} & \dots & - q_{3 n_{1}} + q_{1 n_{1}} \\ ⋮ & ⋮ & ⋮ & ⋮ & ⋮ \\ - μ - q_{n_{1} 1} & - q_{n_{1} 2} + q_{12} & - q_{n_{1} 3} + q_{13} & \dots & μ + q_{1 n_{1}} \end{matrix} | .

Applying the operations R^′ = R₂ + R₁;

R_{3}^{'} = R_{3} + R_{1}

;··· ;

R_{n}^{'} = R_{n_{1}} + R_{1}

, we have

\begin{matrix} | A | & = \frac{1}{μ - (1 + q) (n_{1} - 1) + r_{1} q} \times | \begin{matrix} μ & - q_{12} & - q_{13} & \dots & - q_{1 n_{1}} \\ - q_{21} & μ & - q_{23} & \dots & - q_{2 n_{1}} \\ - q_{31} & - q_{32} & μ & \dots & - q_{3 n_{1}} \\ ⋮ & ⋮ & ⋮ & ⋮ \\ - q_{n_{1} 1} & - q_{n_{1} 2} & - q_{n_{1} 3} & \dots & μ \end{matrix} | \\ = \frac{1}{μ - (1 + q) (n_{1} - 1) + r_{1} q} ϕ (G_{1}; μ) . \end{matrix}

Similarly,

| B | = \frac{1}{μ - (1 + q) (n_{2} - 1) + r_{2} q} ϕ (G_{2}; μ) .

By substituting these values in the above equation, we have the required result.

Theorem 12

Let G₁and G₂be r₁ – and r₂ – regular graphs with n₁and n₂vertices, respectively. If diam(G₁) ≤ 2 and diam(G₂) ≤ 2, then

E_{q} (G_{1} \nabla G_{2}) = {\begin{matrix} E_{q} (G_{1}) + E_{q} (G_{2}), & if RK \geq n_{1} n_{2}, \\ E_{q} (G_{1}) + E_{q} (G_{2}) - (R + K) + \sqrt{{(R + K)}^{2} - 4 (RK - n_{1} n_{2})}, & if RK < n_{1} n_{2}, \end{matrix}

where R = (1 + q)(n₁ − 1) − r₁q and K = (1 + q)(n₂ − 1) − r₂q.

Proof

From Theorem 2, we have

ϕ (G_{1} \nabla G_{2}; μ) = \frac{((μ - (1 + q) (n_{1} - 1) + r_{1} q) (μ - (1 + q) (n_{2} - 1) + r_{2} q) - n_{1} n_{2}))}{(μ - (1 + q) (n_{1} - 1) + r_{1} q) (μ - (1 + q) (n_{2} - 1) + r_{2} q)} ϕ (G_{1}; μ) ϕ (G_{2}; μ)

implying that

ϕ (G_{1} \nabla G_{2}; μ) = \frac{((μ - R) (μ - K) - n_{1} n_{2}) ϕ (G_{1}; μ) ϕ (G_{2}; μ)}{(μ - R) (μ - K)},

where R = (1 + q)(n₁ − 1) − r₁q and K = (1 + q)(n₂ − 1) − r₂q; i.e.,

(μ - R) (μ - K) ϕ (G_{1} \nabla G_{2}; μ) = ((μ - R) (μ - K) - n_{1} n_{2}) ϕ (G_{1}; μ) ϕ (G_{2}; μ) .

That is,

(μ - R) (μ - K) ϕ (G_{1} \nabla G_{2}; μ) = (μ^{2} - (R + K) μ + RK - n_{1} n_{2}) ϕ (G_{1}; μ) ϕ (G_{2}; μ) .

Let

P_{1} (μ) = (μ - R) (μ - K) ϕ (G_{1} \nabla G_{2}; μ)

and

P_{2} (μ) = (μ^{2} - (R + K) μ + R K - n_{1} n_{2}) ϕ (G_{1}; μ) ϕ (G_{2}; μ) .

The roots of the equation P₁(μ) = 0 are R,K and q–distance eigenvalues of G₁∇G₂. Therefore, the sum of the absolute values of the roots of P₁(μ) = 0 is

R + K + E_{q} (G_{1} \nabla G_{2})

(4)

and similarly the roots of the equation P₂(μ) = 0 are q–distance eigenvalues of G₁ and G₂ and hence

\frac{R + K + \sqrt{{(R + K)}^{2} - 4 (RK - n_{1} n_{2})}}{2}, \frac{R + K - \sqrt{{(R + K)}^{2} - 4 (RK - n_{1} n_{2})}}{2} .

Therefore, the sum of the absolute values of the roots of P₂(μ) = 0 is

E_{q} (G_{1}) + E_{q} (G_{2}) + | \frac{R + K + \sqrt{{(R + K)}^{2} - 4 (RK - n_{1} n_{2})}}{2} | + | \frac{R + K - \sqrt{{(R + K)}^{2} - 4 (RK - n_{1} n_{2})}}{2} | .

Since P₁(μ) = P₂(μ), from above equations, we get,

\begin{matrix} R + K + E_{q} (G_{1} \nabla G_{2}) & = E_{q} (G_{1}) + E_{q} (G_{2}) \\ + | \frac{R + K + \sqrt{{(R + K)}^{2} - 4 (RK - n_{1} n_{2})}}{2} | \\ + | \frac{R + K - \sqrt{{(R + K)}^{2} - 4 (RK - n_{1} n_{2})}}{2} | . \end{matrix}

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

\begin{matrix} R + K + E_{q} (G_{1} \nabla G_{2}) & = E_{q} (G_{1}) + E_{q} (G_{2}) \\ + \frac{R + K + \sqrt{{(R + K)}^{2} - 4 (RK - n_{1} n_{2})}}{2} \\ + \frac{R + K - \sqrt{{(R + K)}^{2} - 4 (RK - n_{1} n_{2})}}{2}, \end{matrix}

and therefore

R + K + E_{q} (G_{1} \nabla G_{2}) = E_{q} (G_{1}) + E_{q} (G_{2}) + R + K

implying that

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

\begin{matrix} R + K + E_{q} (G_{1} \nabla G_{2}) & = E_{q} (G_{1}) + E_{q} (G_{2}) \\ + \frac{R + K + \sqrt{{(R + K)}^{2} - 4 (RK - n_{1} n_{2})}}{2} \\ + \frac{R + K - \sqrt{{(R + K)}^{2} - 4 (RK - n_{1} n_{2})}}{2} . \end{matrix}

Hence we get

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

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})} .

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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Article information

Received:: 2019-04-04

Accepted:: 2019-04-17

Published Online:: 2020-08-10

Citation Information:: Applied Mathematics and Nonlinear Sciences, Volume 5, Issue 2, Pages 85–98, eISSN 2444-8656, DOI: https://doi.org/10.2478/amns.2020.2.00017.