Relations between ordinary energy and energy of a self-loop graph

Let G be a graph on n vertices with vertex set V(G) and let S⊆V(G) with |S|=α. Denote by GS, the graph obtained from G by adding a self-loop at each of the vertices in S. In this note, we first give an upper bound and a lower bound for the energy of GS (E(GS)) in terms of ordinary energy (E(G)), order (n) and number of self-loops (α). Recently, it is proved that for a bipartite graph GS, E(GS)≥E(G). Here we show that this inequality is strict for an unbalanced bipartite graph GS with 0<α<n. In other words, we show that there exits no unbalanced bipartite graph GS with 0<α<n and E(GS)=E(G).


Introduction
Let  be a simple graph on  vertices with vertex set  () = { 1 ,  2 , … ,   }.Let  ⊆  () with || = .The graph   is obtained from  by adding a self-loop at each of the vertices in .Adjacency matrix of   is denoted by (  ) and is defined as (  ) = () +   (𝐺), where () is the well-known adjacency matrix of  and   () is the diagonal matrix with its -th diagonal entry equal to 1 or 0 according as   ∈  or   ∉ , respectively.
We denote the -th largest eigenvalue of (  ) (resp.A(G)) by    (resp.  ).The sum of absolute values of   for  = 1, 2, … ,  is called the energy of a graph .Nowadays it is called as ordinary energy.For its basic mathematical properties, including various lower and upper bounds, see [1,[3][4][5]7] and especially the book [11].Applications of graph energy can be found in the articles [13,14].The sum is called the energy of graph   and is denoted by (  ).The definition of energy of a graph with self-loops was put forward very recently by Gutman et al. in [8].The authors in [8] showed that if  is a bipartite graph, then (  ) = (  ()∖ ).Also, they conjectured that (  ) > () for 1 ≤  ≤  − 1. However this conjecture was disproved in [10] by means of counterexamples.In [2], Akbari et al. showed that this conjecture was nevertheless not a complete miss by proving that energy of a bipartite graph   is always greater than or equal to its ordinary energy.In [12], Popat et al. obtained a family of graphs which satisfies the property (  ) = () and 0 <  < .Some bounds on energy of self-loop graph are presented in [2,8] Motivated by this, in this paper, we give an upper bound and a lower bound for the energy of   ((  )) in terms of ordinary energy (()), order () and number of self-loops ().Also, we show that for an unbalanced bipartite graph   with 0 <  < , (  ) > ().In other words, we show that there exits no unbalanced bipartite self-loops graph   with 0 <  <  and (  ) = ().
We denote the complete graph on  vertices by   and its complement graph by   or  1 .The following theorem gives an upper bound and a lower bound for (  ) in terms of (),  and . ] where  is a (0, 1)-matrix.
Right inequality: We have Applying Lemma 2.1 to equation (1), we get That is, ( Proving the right equality.Suppose the equality in (3) holds.Then the equality in (2) also holds and thus by Lemma 2.1 there exists an orthogonal matrix  such that the matrices  () and  ) Therefore,
Thus  = 0 or .Conversely, if  = 0, , then one can easily see that equality holds.This completes the proof.□ The following corollary follows immediately from the above theorem.
Corollary 2.6.Let G be a graph on n vertices.Then for any vertex  of , we have ] where  is a (0, 1)-matrix of order  1 ×  2 .
Let  ′ (  ) = Applying Lemma 2.1 to the equation ( 6 Therefore these matrices share the same singular eigenvalues.Thus from equation ( 7), we have Suppose that the equality in relation ( 8) holds.Then the equality in relation (7)
Since the matrices () and (  ) Let  and  be two  ×  matrices.Then the equality holds if and only if there exists an unitary matrix  such that both the matrices   and   are positive semi-definite.Moreover, if  and  are real matrices, then the matrix  can be taken as real orthogonal.Let  = [  ] × be a positive semi-definite matrix such that   = 0 for some 1 ≤  ≤ .Then   =   = 0 for all 1 ≤  ≤ .