Some new results on energy of graphs with self loops

The graph Gσ\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$G_\sigma $$\end{document} is obtained from graph G by attaching self loops on σ\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\sigma $$\end{document} vertices. The energy E(Gσ)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$ E(G_\sigma )$$\end{document} of the graph Gσ\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$G_\sigma $$\end{document} with order n and eigenvalues λ1,λ2,⋯,λn\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\lambda _1,\lambda _2,\dots ,\lambda _n$$\end{document} is defined as E(Gσ)=∑i=1n|λi-σn|\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$ E(G_\sigma )= \displaystyle \sum _{i=1}^n\bigg |\lambda _i-\dfrac{\sigma }{n}\bigg |$$\end{document}. It has been proved that if σ=0orn\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\sigma =0\; or\; n$$\end{document} then E(G)=E(Gσ)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$ E(G)=E(G_\sigma ) $$\end{document}. The obvious question arise: Are there any graph such that E(G)=E(Gσ)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$E(G)=E(G_\sigma )$$\end{document} and 0<σ<n\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$<\sigma <n$$\end{document}? We have found an affirmative answer of this question and contributed a graph family which satisfies this property.


Introduction
For standard terminology and notations related to graph theory, we follow Balakrishnan and Ranganathan [1] while any terms related to algebra we depend on Lang [2].
An undirected graph without multiple edges and self-loops is called a simple graph. The adjacency matrix A(G) of a simple graph G with vertex set {v 1 , v 2 , . . . , v n } is n-ordered symmetric matrix A(G) = [a i j ] such that, of graph G. The energy E(G) of graph G is developed by Gutman [3] in 1978 as E(G) = n i=1 |λ i |. This graph energy is an emerging subject for a researchers of applied mathematics and mathematical chemistry. A brief account of graph energy of simple graphs can be found in [4][5][6] as well as in the books [7,8]. The variants of graph energy can be found in [9][10][11].
Recently the concept of energy of graphs with self-loops is open-up by Gutman et al. [12]. Let G σ be the graph obtained from graph G by attaching self loops on σ chosen vertices. The adjacency matrix A(G σ ) of graph G σ is an n × n symmetric matrix such that A(G σ ) = A(G) + I σ , where I σ is a square matrix of order n with exactly σ ones on the main diagonal and all other entries are zero. The eigenvalues of Gutman et al. have [12] conjectured that for any graph G of order n, E(G) < E(G σ ).
Irena et al. [13] have disproved this conjuncture by showing examples of graphs such that E(G) > E(G σ ). It has been shown that [12] if σ = 0 or n then E(G) = E(G σ ).
In the present paper we have obtained a graph family such that E(G) = E(G σ ) and 0 < σ < n.

Main results
Theorem 1 Let G be the simple graph of order n with eigenvalues λ 1 , λ 2 , · · · , λ n and G l be the graph obtained from G by adding a loop on each vertex of G then Proof Let H n = G ∪G l . The graph H n contains 2n vertices and n loops. The adjacency matrix of H is given by: The characteristic polynomial of above matrix is given by: It follows that if λ 1 , λ 2 , · · · , λ n are eigenvalues of A then, The roots of above characteristic polynomial are: Here,

Therefore, E(H ) = E(H 3 ).
Theorem 2 Let G be the simple graph of order n with eigenvalues λ 1 , λ 2 , · · · , λ n and G l be the graph obtained from G by adding a loop on each vertex of G. Let p and q be non-negative integer and p + q = m then E(( pG ∪ qG l ) qn ) = m E(G), if |λ i | ≥ max p m , q m , for each i = 1, 2, · · · , n.
Proof Let H qn = pG ∪ qG l . The graph H qn contains mn vertices and qn loops. The adjacency matrix of H qn is given by: The characteristic polynomial of above matrix is given by: It follows that if λ 1 , λ 2 , · · · , λ n are eigenvalues of A then, The roots of above characteristic polynomial are: ,for each i = 1, 2, · · · , n Here, Therefore, Case − ii If p = q then we assume |λ i | ≥ p m , for all 1 ≤ i ≤ n then