The Minimum Edge Dominating Energy of a Graph A. Sharmila

(1)

The Minimum Edge Dominating Energy of a Graph

A. Sharmila

¹

and S. Lavanya

²

1

Research Scholar, Bharathiar University Coimbatore - 641 046, Tamil Nadu, INDIA.

1

Department of Mathematics,

Justice Basheer Ahmed Sayeed College for Women Chennai - 600018, Tamil Nadu, INDIA.

2

Department of Mathematics, Bharathi Women’s College Chennai - 600108, Tamil Nadu, INDIA.

(Received on: December 16, 2017) ABSTRACT

Let G be a graph with n vertices and m edges. In analog to the definition of Energy of a graph the edge energy is defined as the sum of the absolute the eigen values of the edge adjacency matrix. In this paper we introduce the minimum edge dominating energy of a graph 𝐸_𝐷′(G) analogous to the minimum dominating energy 𝐸_𝐷(G) and compute minimum edge dominating energies of a complete graph (K4), Ladder graph (L3) and Star graph (K1,n-1).

AMS Subject Classification : 05C50, 05C69

Keywords: Edge adjacency matrix, edge dominating set, minimum edge dominating eigen values, minimum edge dominating energy.

1. INTRODUCTION

The concept of energy of a graph was introduced by I. Gutman

¹

in the year 1978. The interest in graph energy has incresaed in the past decades and several variants of graph energies like Laplacian energy, incidence energy, Randic energy, Siedel energy etc. were defined.

Rajesh Kanna et al. (2013)

³

introduced Minimum dominating energy and computed minimum dominating energies of star graph, complete graph, crown graph and cocktail graphs. They established upper and lower bounds for the minimum dominating energy. Motivated by this, we define the minimum edge dominating energy.

Let G be a graph with n vertices and m edges and let A = (aij) be the adjacency matrix

of the graph. The eigen values λ

1

, λ

2

,· · ·,λ

n

of A, assumed in non increasing order, are the

eigen values of the graph G. As A is real symmetric, the eigen values of G are real with sum

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equal to zero. The energy E(G) of G is defined to be the sum of the absolute values of the eigen values of G. i.e.,

E (G) =∑

^𝑛_𝑖=1

⃒𝜆

_𝑖

⃒

In this paper we are concerned with finite, simple and undirected graphs.

The Edge adjacency matrix of a graph G, determined by the adjacency of edges, is defined as (𝑎′

_ij

) = { 1 𝑖𝑓 𝑒𝑑𝑔𝑒𝑠 𝑖 𝑎𝑛𝑑 𝑗 𝑎𝑟𝑒 𝑎𝑑𝑗𝑎𝑐𝑒𝑛𝑡

0 𝑜𝑡ℎ𝑒𝑟𝑤𝑖𝑠𝑒 .

The eigen values ρ

1

, ρ

2

, . . . , ρ

m

of the edge adjacency matrix are said to be edge eigen values of the graph G. The edge energy (Bond energy) is defined as EE(G) =∑

^𝑚_𝑖=1

|𝜌

_𝑖

| .

The concept of edge domination was introduced by Mitchell and Hedet-niemi

²

. 2. THE MINIMUM EDGE DOMINATING ENERGY OF A GRAPH

Let G be a simple graph of order n with vertex set V= {v

1

, v

2

, ..., v

n

} and edge set E = {e

1

, e

2

, ..., e

m

}. A subset 𝐷′ of E is called an edge dominating set of G if every edge of

E - 𝐷′ is adjacent to some edge in 𝐷′ .Any edge dominating set with minimum cardinality is called a minimum edge dominating set. Let 𝐷′ be a minimum edge dominating set of a graph G. The minimum edge dominating matrix of G is the m x m matrix defined by

𝐷′(𝐺) = (𝑑′

_ij

), where (𝑑′

_ij

)= {

1 𝑖𝑓 𝑒

_𝑖

𝑎𝑛𝑑 𝑒

_𝑗

𝑎𝑟𝑒 𝑖𝑛𝑐𝑖𝑑𝑒𝑛𝑡 1 𝑖𝑓 𝑖 = 𝑗 𝑎𝑛𝑑 𝑒

_𝑖

∈ 𝐷′

0 𝑜𝑡ℎ𝑒𝑟𝑤𝑖𝑠𝑒 The characteristic polynomial of 𝐷′(𝐺) is denoted by f

n

(G, ρ) = det (ρI - 𝐷′ (G) ).

The minimum edge dominating eigen values of a graph G are the eigen values of 𝐷′(G). Minimum edge dominating energy of G is defined as

𝐸

_𝐷′

(G) = ∑

^𝑚_𝑖=1

|𝜌

_𝑖

|

Example 1. We first compute the minimum edge dominating energy of the graph K

4

in figure 1.

Figure 1

Let the minimum edge dominating set be D

₁^ˈ

= { e

1

, e

2

}. Then

𝐷

₁^ˈ

(G) = [

1 1 0 1 1 1

1 1 1 0 1 1

0 1 0 1 1 1

1 0 1 0 1 1

1 1 1 1 0 0

1 1 1 1 0 0]

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The characteristic equation is ρ

⁶

- 2 ρ

⁵

– 11 ρ

⁴

+ 11 ρ

²

- 4 ρ = 0.

The minimum edge dominating eigen values are ρ

1

≈ - 1.8741, ρ

2

≈ -1.6180, ρ

3

≈ 0.0000, ρ

4

≈ 0.4865, ρ

5

≈ 0.6180, ρ

6

≈ 4.3876.

The minimum edge dominating energy, E

D

ˈ(G) ≈ 8.9842.

If we take another minimum edge dominating set D

₂^ˈ

= { e

1

, e

3

}. Then

𝐷

₂^ˈ

(G) = [

1 1 0 1 1 1

1 0 1 0 1 1

0 1 1 1 1 1

1 0 1 0 1 1

1 1 1 1 0 0

1 1 1 1 0 0]

The characteristic equation is ρ

⁶

- 2 ρ

⁵

– 11 ρ

⁴

+ 12 ρ

²

= 0.

The minimum edge dominating eigen values are ρ

1

≈ - 2, ρ

2

≈ -1.3723, ρ

3

≈ 0.0000, ρ

4

≈ 0.0000, ρ

5

≈ 1.0000, ρ

6

≈ 4.3723.

The minimum edge dominating energy, E

D

ˈ(G) ≈ 8.7446.

This example illustrates the fact that the minimum edge dominating energy of a graph G depends on the choice of the minimum edge dominating set. i.e. the minimum edge dominating energy is not a graph invariant.

Example 2. We compute the minimum edge dominating energy of the Ladder graph (L

3

) in figure 2.

Figure 2

Let the minimum edge dominating set be D

₁^ˈ

= { e

1

, e

2

, e

3

, e

4

}. Then

AE𝐷

₂^ˈ

(G) =

[

1 1 0 0 0 1 1 1 1 1 0 0 0 1 0 1 1 1 0 0 0 0 0 1 1 1 0 1 0 0 0 1 0 1 1 1 0 0 0 1 0 0 1 1 0 1 1 0 0]

The characteristic equation is ρ

⁷

- 4 ρ

⁶

– 4 ρ

⁵

+ 21ρ

⁴

+ 2ρ

³

– 26 ρ

²

+ ρ + 2 = 0.

The minimum edge dominating eigen values are ρ

1

≈ - 1.8261, ρ

2

≈ -1.2588, ρ

3

≈ - 0.2635,

ρ

4

≈ 0.3141, ρ

5

≈ 1.5511, ρ

6

≈ 1.8830, ρ

7

≈ 3.6002.

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The minimum edge dominating energy, MEE

D

ˈ(G) ≈ 10.6968.

3. MAIN RESULTS

In the following section, we introduce some properties of characteristic polynomial of minimum edge dominating matrix of a graph G.

Theorem 2.1. Let G be a graph of order n, let Dˈ be the minimum edge dominating set and let

f

n

(G, ρ) = c

0

ρ

^m

+ c

1

ρ

^m-1+

c

2

ρ

^m-2

+……+ c

n

be the characteristic polynomial of the minimum edge dominating matrix of the graph G.Then

1. c

0

= 1 2. c

1

= - ⃒D⃒.

Proof:

1. From the definition of f

n

(G, ρ).

2. Since the sum of diagonal elements of AEDˈ (G) is equal to ⃒D⃒, the sum of determinants of all 1 x 1 principal sub matrices of AEDˈ (G) is the trace of AEDˈ (G), which evidently is equal to ⃒D⃒. Thus (-1)

¹

c

1

= - ⃒D⃒.

Theorem 2.2. Let G be a graph of order n. Let ρ

1

, ρ

2

, ρ

1

,……, ρ

m

be the eigen values of AEDˈ (G). Then ∑

^𝑚_𝑖=1

𝜌

_𝑖

= ⃒D⃒.

Proof:

Since the sum of the eigen values of AEDˈ (G) is the trace of AEDˈ (G),

∑

^𝑚_𝑖=1

𝜌

_𝑖

= ∑

^𝑚_𝑖=1

𝑎

_𝑖𝑖

= ⃒D⃒.

3. Minimum Edge Dominating Energy of the Star graph (K

1, n-1

).

Theorem 3. 1. For n≥ 3, the minimum edge dominating energy of the Star graph (K

1, n-1

) is equal to (n – 3) + √𝑛

²

− 4𝑛 + 8 .

Proof:

Consider the Star graph K

1, n-1

with vertex set V = {v

0,

v

1

, v

2

,…….., vn-

1

} and edge set E = { e

1

, e

2

,…….., en-

1

}.

Let the minimum edge dominating set be Dˈ = {e

1

}. Then

Dˈ(K

1, n-1

) = [

1 1 1 1 0 1 1 1 0

⋯ 1 1

⋮ ⋱ 1 ⋮

1 1 1 ⋯ 0]

(𝑛−1)×(𝑛−1)

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Characteristic polynomial is

| |

𝜌 − 1 −1 −1

−1 𝜌 −1

−1 −1 𝜌

⋯

−1

⋮ ⋱ ⋮

−1 −1 −1 ⋯ 𝜌

| |

Characteristic equation is ( 1 + ρ )

^{n – 3}

( ρ

²

– (n – 2 ) ρ – 1) = 0 . The minimum edge dominating eigen values are

ρ = -1 [(n – 3) times], ρ =

^{(𝑛−2)±√𝑛}²^−4𝑛+8

2

[ one time each] . Minimum edge dominating energy is

E

D

ˈ(G) = |−1|( n – 3) + ⃒

^{(𝑛−2)+√𝑛}²^−4𝑛+8

2

⃒ + ⃒

^{(𝑛−2)−√𝑛}₂²^−4𝑛+8

⃒ = ( n – 3 ) + √𝑛

²

− 4𝑛 + 8

4. CONCLUSION

Thus in this paper, the new energy namely the minimum edge dominating energy is defined and has been found for some graphs.

REFERENCES

1. C. Adiga, A. Bayad, I. Gutman, S. A. Srinivas, The minimum covering energy of a graph.

Kragujevac J. Sci.34, 39-56 (2012).

2. I. Gutman, The energy of a graph, Ber. Math-Satist. Sekt. Forschungsz. Graz 103,pp.1-22, (1978).

3. M. R. Rajesh Kanna, B. N. Dharmendra, G. Sridhara, The Minimum Dominating Energy of A Graph, International Journal of Pure and Applied Mathematics 85 No. 4, 707-718 (2013).

4. S. Arumugam and S. Velammal, Edge domination in graphs, Taiwanese Journal of

Mathematics, vol. 2, no. 2, pp. 173–179, 1998.