Minimum Dominating Distance Energy of a Graph

Recently we introduced the concept of minimum dominating energy[21]. Motivatedby this paper,we introduced the concept of minimum dominating distance energyEDd(G) of a graph G and computed minimum dominating distance energies of a Stargraph,Complete graph,Crown graph and Cocktail graphs. Upper and lower boundsfor EDd(G) are also established.DOI : http://dx.doi.org/10.22342/jims.20.1.133.19-29


Introduction
The concept of energy of a graph was introduced by I. Gutman [9] in the year 1978. Let G be a graph with n vertices {v 1 , v 2 , ..., v n } and m edges. Let A = (a ij ) be the adjacency matrix of the graph. The eigenvalues λ 1 , λ 2 , · · · , λ n of A, assumed in non increasing order, are 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 to be the sum of the absolute values of the eigenvalues of G. i.e., For details on the mathematical aspects of the theory of graph energy see the reviews [10], paper [11] and the references cited there in. The basic properties including various upper and lower bounds for energy of a graph have been established in [16], and it has found remarkable chemical applications in the molecular orbital theory of conjugated molecules [5,6,7,12].
Further, studies on maximum degree energy, minimum dominating energy, Laplacian minimum dominating energy, minimum covering distance energies can be found in [18,19,20,21] and the references cited there in.
The distance matrix of G is the square matrix of order n whose (i, j) -entry is the distance (= length of the shortest path) between the vertices v i and v j . Let ρ 1 , ρ 2 , ..., ρ n be the eigenvalues of the distance matrix of G. The distance energy DE is defined by Detailed studies on distance energy can be found in [3,4,8,13,14,22].

The Minimum Dominating Distance Energy
Let G be a simple graph of order n with vertex set V = {v 1 , v 2 , ..., v n } and edge set E. A subset D of V is called a dominating set of G if every vertex of V -D is adjacent to some vertex in D. Any dominating set with minimum cardinality is called a minimum dominating set. Let D be a minimum dominating set of a graph G. The minimum dominating distance matrix of G is the n × n matrix defined by The characteristic polynomial ofA Dd (G) is denoted by f n (G, ρ)= det(ρI − A Dd (G)). The minimum dominating eigenvalues of the graph G are the eigenvalues of A Dd (G). Since A Dd (G) is real and symmetric, its eigenvalues are real numbers and we label them in non-increasing order ρ 1 ρ 2 · · · ρ n . The minimum dominating energy of G is defined as Note that the trace of A Dd (G) = Domination Number = k. Example 1. The possible minimum dominating sets for the following graph G in Figure 1 8946. Therefore, minimum dominating distance energy depends on the dominating set.

Minimum Dominating Distance Energy of Some Standard Graphs
Definition 3.1. The cocktail party graph, is denoted by K n×2 , is a graph having the Theorem 3.2. The minimum dominating distance energy of cocktail party graph K n×2 is 4n.
Theorem 3.3. For any integer n ≥ 3, the minimum dominating distance energy of star graph K 1,n−1 is equal to 4n − 7.
Proof. Consider the star graph Definition 3.4. The crown graph S 0 n for an integer n ≥ 2 is the graph with vertex set {u 1 , u 2 , ..., u n , v 1 , v 2 , ..., v n } and edge set {u i v j : 1 ≤ i, j ≤ n, i = j}. Hence S 0 n coincides with the complete bipartite graph K n,n with horizontal edges removed.
Theorem 3.5. For any integer n ≥ 2, the minimum dominating distance energy of the crown graph S 0 n is equal to 7(n − 1) + n 2 − 2n + 5.
Proof. For the crown graph S 0 Characteristic equation is Theorem 3.6. For any integer n ≥ 2, the minimum dominating distance energy of complete graph K n is (n − 2) + √ n 2 − 2n + 5.
Proof. For complete graphs the minimum dominating distance matrix is same as minimum dominating matrix [19], therefore the minimum dominating distance energy is equal to minimum dominating energy.

Properties of Minimum Dominating Eigenvalues
Theorem 4.1. Let G be a simple graph with vertex set V = {v 1 , v 2 , ..., v n }, edge set E and D = {u 1 , u 2 , ..., u k } be a minimum dominating set. If ρ 1 , ρ 2 , ..., ρ n are the eigenvalues of minimum dominating distance matrix A Dd (G) then Proof. i) We know that the sum of the eigenvalues of A Dd (G) is the trace of A Dd (G). Therefore, (ii) Similarly, the sum of squares of the eigenvalues of Corollary 4.2. Let G be a (n,m) simple graph with diameter 2 and D = {u 1 , u 2 , ..., u k } be a minimum dominating set. If ρ 1 , ρ 2 , ..., ρ n are the eigenvalues of minimum dominating distance matrix A Dd (G) then Proof. We know that in A Dd (G) there are 2m elements with 1 and n(n − 1) − 2m elements with 2 and hence corollary follows from the above theorem.

Bounds for Minimum Dominating Energy
Similar to McClelland's [17] bounds for energy of a graph, bounds for E Dd (G) are given in the following theorem.
Theorem 5.1. Let G be a simple (n,m) graph. If D is the minimum dominating set and P = |detA Dd (G)| then where k is a domination number. Proof.

Cauchy Schwarz inequality is
Since arithmetic mean is not smaller than geometric mean we have i.e., E Dd (G) ≥ (k + 2m + 2M ) + n(n − 1)P Proof. Let X be any nonzero vector. Then by [1], We have Therefore, Proof. Let G be a connected graph of diameter 2 and d i denotes the degree of vertex v i . Clearly i-th row of A dd consists of d i one's and n − d i − 1 two's. By using Raleigh's principle, for J = [1, 1, 1, · · · , 1] we have Similar to Koolen and Moulton's [15] upper bound for energy of a graph, upper bound for E Dd (G) is given in the following theorem.