Bounds for various graph energies

. In this paper, we obtain some upper and lower bounds for the spectral radius of some special matrices such as maximum degree, minimum degree, Randic, sum-connectivity, degree sum, degree square sum, first Zagreb and second Zagreb matrices of a simple connected graph G by the help of matrix theory. We also get some upper bounds for the corresponding energies of G .


Introduction
Erich Hückel used the chemical applications of graph in describing the molecular orbitals and energies of the π-bonding frameworks.The structure of the simplest hydrocarbon was represented as a graph such that the eigenvalues of the graph is energy levels of electrons.Moreover, the carbon atoms denote the vertices of the graph and chemical bonds between the carbon atoms denote the edges of the graph.
In 1978, inspired by Hückel's study, Ivan Gutman defined the energy of a graph G as where λ 1 , λ 2 , • • • , λ n are the eigenvalues of the adjacency matrix of G, [8].
Many authors defined new type energies of a graph by means of different graph matrices and obtained some properties of these energies, see e.g.[1-7, 9, 14-17].Also, some bounds for various energies of the graphs were investigated in [10][11][12].Now, we give the definitions of various matrices and energies of the graphs which will be used in the following sections: Let G be a simple connected graph.The matrix M(G) = [M i j ] defined as otherwise is called the maximum degree matrix of G, [1].The maximum degree energy E M of G is the sum of the absolute values of the eigenvalues of M(G).
The matrix m(G) = [m i j ] defined as otherwise is called the minimum degree matrix of G, [2].The minimum degree energy E m of G is the sum of the absolute values of the eigenvalues of m(G).
The matrix R(G) = [r i j ] defined as is called the Randic matrix of G, [7].Similarly the Randic energy E R of G is the sum of the absolute values of the eigenvalues of R(G).
The matrix S C(G) = [sc i j ] defined by is called the sum-connectivity matrix of G, [17].The sum-connectivity energy E S C of G is the sum of the absolute values of the eigenvalues of S C(G).
The matrix DS (G) = [ds i j ] defined by otherwise is called the degree sum matrix of G, [9].Then the degree sum energy E DS of G is the sum of the absolute values of the eigenvalues of DS (G).
The matrix DS S (G) = [dss i j ] defined as The matrix Z (1) (G) = [z (1)  i j ] defined by otherwise is called the first Zagreb matrix of G.The first Zagreb energy ZE 1 of G is the sum of the absolute values of the eigenvalues of Z (1) (G).
The matrix Z (2) (G) = [z (2)  i j ] defined by otherwise is called the second Zagreb matrix of G.The second Zagreb energy ZE 2 of G is the sum of the absolute values of the eigenvalues of Z (2) (G).
Lemma 1.1 [13] Let A be an n × n square matrix, then the spectral radius of A satisfies In the following sections, we assume that G is a simple connected graph with n vertices and m edges and the degrees of the vertices of

The bounds for spectral radius of some matrices of a graph
In this section, we will obtain upper and lower bounds for the spectral radius of maximum degree, minimum degree, Randic, sum-connectivity, degree sum, degree square sum, first Zagreb and second Zagreb matrices of a graph by the help of matrix theory and basic information of graph theory.
Theorem 2.1 The bounds for the spectral radius of maximum degree matrix M of G is n∼ j Proof.By our assumption, d n = δ is the minimum degree in G, then R min (M(G)) is obtained from the n-th row of maximum degree matrix of G.For all vertices j which are adjacent to n, we have max{d n , d j } = d j .Then, by the definition of M(G), we obtain Also, we assume that d 1 = ∆ is the maximum degree in G implying that R max (M(G)) is obtained from the first row of maximum degree matrix of G.The number of the adjacent vertices of 1 is ∆ and for all k which are adjacent to 1, we have max{d k , d 1 } = d 1 = ∆.Then, by the definition of M(G), we obtain Hence by using Lemma 1.1, we get

ITM
Therefore, by using Lemma 1.1, we obtain Theorem 2.3 The bounds for the spectral radius of the Randic matrix R of G are n∼k Proof.Since d n = δ is the minimum degree in G, R min (R(G)) is obtained from the n − th row of Randic matrix of G.Then, by the definition of R(G), we obtain ) is obtained from the first row of randic matrix of G.Then, by the definition of R(G), we get As a result, by using Lemma 1.1, we have Theorem 2.4 The bounds for the spectral radius of sum-connectivity matrix S C of G are Proof.Since d n = δ is the minimum degree in G, R min (S C(G)) is obtained from the n − th row of sum-connectivity matrix of G.Then, by the definition of S C(G), we obtain ITM In conclusion, by using Lemma 1.1, we have Theorem 2.5 The bounds for the spectral radius of degree sum matrix DS of G is Proof.Since d n = δ is the minimum degree in G, R min (DS (G)) is obtained from the n − th row of degree sum matrix of G.Then, by the definition of DS (G), we have Since G is a simple connected graph with n vertices and m edges, we have By using (1), we get Also, since d 1 = ∆ is the maximum degree in G, R max (DS (G)) is obtained from the first row of degree sum matrix of G.Then, by the definition of DS (G), we get Again, by using (1), we obtain Hence, by using Lemma 1.1, we have Theorem 2.6 The bounds for the spectral radius of degree square sum matrix DS S of G is ITM Also, since d 1 = ∆ is the maximum degree in G, R max (DS S (G)) is obtained from the first row of degree square sum matrix of G.Then, by the definition of DS S (G), we get Hence, by using Lemma 1.1, we obtain Theorem 2.7 The bounds for the spectral radius of first zagreb matrix Z (1) of G is Proof.Since d n = δ is the minimum degree in G, R min (Z (1) (G)) is obtained from the n − th row of first zagreb matrix of G.Then, by the definition of Z (1) (G), we have Since the number of the adjacent vertices of n is δ, we have Moreover, since d 1 = ∆ is the maximum degree in G, R max (Z (1) (G)) is obtained from the first row of first zagreb matrix of G.Then, by the definition of Z (1) (G), we get Since the number of the adjacent vertices of 1 is ∆, we have Therefore, by using Lemma 1.1, we have ITM Web of Conferences 49, 01003 (2022) Fourth ICAMNM 2022 https://doi.org/10.1051/itmconf/20224901003 Theorem 2.8 The bounds for the spectral radius of second zagreb matrix Z (2) of G is Proof.Since d n = δ is the minimum degree in G, R min (Z (2) (G)) is obtained from the n − th row of second zagreb matrix of G.Then, by the definition of (2) (G), we have Moreover, since d 1 = ∆ is the maximum degree in G, R max (Z (2) (G)) is obtained from the first row of second zagreb matrix of G.Then, by the definition of (2) (G), we get As a result, by using Lemma 1.1, we have 3 Some upper bounds for some energies of graphs In this section, we will obtain upper bounds for maximum degree, minimum degree, Randic, sum-connectivity, degree sum, degree square sum, first Zagreb and second Zagreb energies of a graph.
Theorem 3.1 An upper bound for the maximum degree energy E M of G is Proof.Since G is a graph with n vertices, maximum degree matrix of G has n eigenvalues.By Theorem 2.1, we know that ρ(M(G)) can be at most ∆ 2 .Then, an upper bound for all eigenvalues is ∆ 2 .Hence, we obtain Theorem 3.2 An upper bound for the minimum degree energy E m of G is Proof.Since G is a graph with n vertices, minimum degree matrix of G has n eigenvalues.By Theorem 2.2, we know that ρ(m(G)) can be at most Proof.Since G is a graph with n vertices, sum-connectivity matrix of G has n eigenvalues.By Theorem 2.4, we know that ρ(S C(G)) can be at most Web of Conferences 49, 01003 (2022) Fourth ICAMNM 2022 https://doi.org/10.1051/itmconf/20224901003Theorem 2.2 The bounds for the spectral radius of minimum degree matrix m of G are δ 2 ≤ ρ(m(G)) ≤ 1∼ j d j .Proof.Since d n = δ is the minimum degree in G, R min (m(G)) is obtained from the n − th row of minimum degree matrix of G.The number of the adjacent vertices of n is δ, and for all k which are adjacent to n, we have min{d k , d n } = d n = δ.Then, by the definition of m(G), we obtain R min (m(G)) = δ 2 .Moreover, since d 1 = ∆ is the maximum degree in G, R max (M(G)) is obtained from the first row of minimum degree matrix of G.For all vertices j which are adjacent to 1, we have min{d 1 , d j } = d j .Then, by the definition of m(G), we obtain

Theorem 3 . 5 Theorem 3 . 6 Theorem 3 . 7
An upper bound for the degree sum energy E DS of G isE DS (G) ≤ n [∆(n − 2) + 2m] .Proof.Since G is a graph with n vertices, degree sum matrix of G has n eigenvalues.By Theorem 2.5, we know that ρ(DS (G)) can be at most ∆(n − 2) + 2m.Then, an upper bound for all eigenvalues is ∆(n − 2) + 2m.Hence, we obtainE DS (G) ≤ n [∆(n − 2) + 2m] .An upper bound for the degree square sum energy E DS S of G isE DS S (G) ≤ ∆ 2 n(n − 1)Since G is a graph with n vertices, degree square sum matrix of G has n eigenvalues.By Theorem 2.6, we know that ρ(DS S (G)) can be at most ∆ 2 (n − 1) Hence, we obtainE DS S (G) ≤ ∆ 2 n(n − 1) An upper bound for the first Zagreb energy ZE 1 of G is ZE 1 (G) ≤ n∆ 2 + n 1∼ j d j .ITM Web of Conferences 49, 01003 (2022) Fourth ICAMNM 2022 https://doi.org/10.1051/itmconf/20224901003 [3]erwise is the degree square sum matrix of G,[3].The degree square sum energy E DS S of G is the sum of the absolute values of the eigenvalues of DS S (G).
Web of Conferences 49, 01003 (2022) Fourth ICAMNM 2022 https://doi.org/10.1051/itmconf/20224901003Also, since d 1 = ∆ is the maximum degree in G, R max (S C(G)) is obtained from the first row of sum-connectivity matrix of G.Then, by the definition of S C(G), we get Proof.Since d n = δ is the minimum degree in G, R min (DS S (G)) is obtained from the n − th row of degree square sum matrix of G.Then, by the definition of DS S (G), we have Proof.Since G is a graph with n vertices, Randic matrix of G has n eigenvalues.By Theorem 2.3, we know that ρ(R(G)) can be at most Theorem 3.3 An upper bound for the Randic energyE R of G is E R (G) ≤ n