Inverse Sum Indeg Energy of Graphs

Suppose <inline-formula> <tex-math notation="LaTeX">$G$ </tex-math></inline-formula> is an <inline-formula> <tex-math notation="LaTeX">$n$ </tex-math></inline-formula>-vertex simple graph with vertex set <inline-formula> <tex-math notation="LaTeX">$\{v_{1}, {\dots },v_{n}\}$ </tex-math></inline-formula> and <inline-formula> <tex-math notation="LaTeX">$d_{i}$ </tex-math></inline-formula>, <inline-formula> <tex-math notation="LaTeX">$i=1, {\dots },n$ </tex-math></inline-formula>, is the degree of vertex <inline-formula> <tex-math notation="LaTeX">$v_{i}$ </tex-math></inline-formula> in <inline-formula> <tex-math notation="LaTeX">$G$ </tex-math></inline-formula>. The ISI matrix <inline-formula> <tex-math notation="LaTeX">$S(G)= [s_{ij}]_{n\times n}$ </tex-math></inline-formula> of <inline-formula> <tex-math notation="LaTeX">$G$ </tex-math></inline-formula> is defined by <inline-formula> <tex-math notation="LaTeX">$s_{ij}= \frac {d_{i} d_{j}}{d_{i}+d_{j}}$ </tex-math></inline-formula> if the vertices <inline-formula> <tex-math notation="LaTeX">$v_{i}$ </tex-math></inline-formula> and <inline-formula> <tex-math notation="LaTeX">$v_{j}$ </tex-math></inline-formula> are adjacent and <inline-formula> <tex-math notation="LaTeX">$s_{ij}=0$ </tex-math></inline-formula> otherwise. The <inline-formula> <tex-math notation="LaTeX">$S$ </tex-math></inline-formula>-eigenvalues of <inline-formula> <tex-math notation="LaTeX">$G$ </tex-math></inline-formula> are the eigenvalues of its ISI matrix <inline-formula> <tex-math notation="LaTeX">$S(G)$ </tex-math></inline-formula>. Recently, the notion of inverse sum indeg (henceforth, ISI) energy of graphs is introduced and is defined by <inline-formula> <tex-math notation="LaTeX">$\sum \limits _{i=1}^{n}|\tau _{i}|$ </tex-math></inline-formula>, where <inline-formula> <tex-math notation="LaTeX">$\tau _{i}$ </tex-math></inline-formula> are the <inline-formula> <tex-math notation="LaTeX">$S$ </tex-math></inline-formula>-eigenvalues. We give ISI energy formula of some graph classes. We also obtain some bounds for ISI energy of graphs. In the end, we give some noncospectral equienergetic graphs with respect to inverse sum indeg energy.


Introduction
A graph G is a pair G = (V (G), E(G)), where V (G) denotes the vertex set {v 1 , . . ., v n } and E(G) denotes the edge set of G.The degree d i of a vertex v i is the number of edges incident on it.If vertices v i and v j are adjacent, we denote it by v i ∼ v j .If vertices v i and v j are not adjacent, we denote it by v i ∼ v j .An n-vertex path P n , (n ≥ 1), is a graph with vertex set {v 1 , . . .v n } and edge set {v j v j+1 |j = 1, 2 . . ., n − 1}.An n-vertex cycle C n (n ≥ 3) is a graph with vertex set {v 1 , . . ., v n } and edge set {v j v j+1 |j = 1, 2, . . ., n − 1} ∪ {v n v 1 }.The star graph S n of order n is isomorphic to K 1,n−1 .By G, we denote the complement of G.
The adjacency matrix A(G) = [a ij ] n×n of an n-vertex graph G is defined as The A-characteristic polynomial of G is the polynomial where I n is the diagonal matrix of order n with diagonal entries equal to 1.The A-eigenvalues of G are the A-eigenvalues of A(G).The spectrum of G, denoted by spec A (G), is the set of A-eigenvalues of G together with their multiplicities.The inverse sum indeg, (henceforth ISI) index, was studied in [24].The ISI index is defined as Zangi et al. [26] defined the ISI matrix S(G) = [s ij ] n×n of an n-vertex graph G as: The S-characteristic polynomial of G is given by: where I n is the diagonal matrix of order n with diagonal entries equal to 1.The S-eigenvalues of G are the S-eigenvalues of S(G).The S-spectrum spec S (G) of G, is the set of S-eigenvalues of G together with their multiplicities.Since ISI matrix of graph is symmetric and real, therefore its eigenvalues are real.If G is an n-vertex graph with distinct S-eigenvalues τ 1 , τ 2 , . . ., τ k and if their respective multiplicities are p 1 , p 2 , . . ., p k , we write the S-spectrum of G as spec S (G) = {τ In 1978, the energy of a simple graph is defined by Gutman [11] as Many results on the graph energy can be found in literature.The concept of Randic energy is given by Bozkurt et al. [2,3].In 2014, Gutman et al. [12] gave some of the properties of Randić matrix and Randić energy.Sedlar et al. [23] study the properties ISI index and finds extremal values of ISI index for some classes of graphs.Pattabiraman [17] gave some extremal bounds on ISI index.In 2018, Das et al. [8] summarized different types of energies of graphs introduced by many authors.Das et al. [8] find some of the lower and upper bounds for these energies of graphs.For recent results on different types of energies of graphs, one can study [6,9,10,16,18,19,20,21,27].Zangi et al. [26] introduce the concept of ISI energy of graphs.In this paper we obtain ISI energy formula of some well-known graphs.Upper and lower bounds are established.Finally, we give integral representation for ISI energy of graphs.

Inverse sum indeg energy
Let λ 1 , . . ., λ n be A-eigenvalues of an n-vertex graph G. Then Gutman [11] defined the energy of Let τ 1 , . . ., τ n be the S-eigenvalues of G. Then Zangi et al. [26] define ISI energy of G as (2.2) For convenience , we define some notations.We denote determinant of S(G) by det(S(G)).Let The trace of the matrix S(G) = [s ij ] n×n is defined by n i=1 s ii and is denoted by tr(S(G)).Zangi et al. [26] prove the following lemma.
Lemma 2.1 (Zongi et al. [26]).Let G be an n-vertex graph and let τ 1 , . . ., τ n be its S-eigenvalues.Then (1) Theorem 2.2.Let G be an n-vertex simple and connected graph and let m be the number of edges in G. Then where the equality holds if and only if As G ∼ = K n , it holds that m < n(n−1)

2
. Consequently Suppose G 1 and G 2 are two graphs with disjoint vertex sets.Then the graph union that contains it.A square diagonal matrix whose diagonal elements are square matrices and the non-diagonal elements are 0 is called a block diagonal matrix.
Next theorem gives the relation between ISI energy of a graph and its components. Hence Following result follows directly from ISI matrix of We now show that the ISI energy of a non-trivial graph, if it is an integer, must be an even positive integer.
Theorem 2.5.If G ∼ = K n and the ISI energy of a graph G is an integer then it must be an even positive integer.
Proof.Let τ 1 , . . ., τ n be S-eigenvalues of G and with no loss of generality, assume that τ 1 , . . ., τ s are positive and τ s+1 , . . ., τ n are non-negative.From Lemma 2.1, we have Therefore ISI energy of G is an even integer.
The distance between v and u of G is the length of the shortest path between them.The maximum distance between a vertex v to all other vertices of G is called the eccentricity of v.The diameter of G is the maximum eccentricity of any vertex in G.A matrix M is irreducible if the digraph associated with M is strongly connected.A matrix is non-negative if its all entries are non-negative.
In the following two results, we determine some properties of the S-eigenvalues.The idea of proof is taken from proof of Lemma 1.1 [7] Lemma 2.6.Let G be an n-vertex simple and connected graph, n ≥ 2, with non-incresing S- Proof.Since the graph G is connected therefore S(G) is an irreducible non-negative square matrix of order n.By Perron-Frobenius theorem, we have τ 1 > τ 2 .Since G has diameter at least 3, P 4 is the subgraph of G. Therefore we have τ 2 (G) ≥ τ 2 (P 4 ) = 4  3 > 0, where τ 2 (G) is the second largest S-eigenvalue of G and τ 2 (P 4 ) is the second largest S-eigenvalue of P 4 .Hence τ 1 > τ 2 > 0.
Lemma 2.7 (Brouwer and Haemers [4]).Let G be a connected graph with greatest eigenvalue λ 1 .Then −λ 1 is an eigenvalue of G if and only if G is bipartite.
Theorem 2.8.Suppose G is an n-vertex graph, n ≥ 2, with S-eigenvalues τ 1 , . . ., τ n and let its A-spectrum and S-spectrum are symmetric about the origin.Then Proof.First assume that The converse statement is easy to prove.

ISI energy of some graphs
In this section, we prove ISI energy formulae for some classes of graphs.
The A-spectrum of K n and K m,n is given by Bhat and Pirzada [1] gave the following energy formulae for cycle C n of order n: (3.4) Proof.Since degree of every vertex in C n is 2, therefore for any v i , v j ∈ V (C n ) with v i ∼ v j , we have m+n .Proof.Let B be an m × n matrix and C be an n × m matrix, where all entries of B and C are equal to mn m+n .Let O be a zero matrix of order m × m and O ′ be a zero matrix of order n × n.Then m + n . Therefore The proof is complete.
Following corollary is an easy consequence of Theorem 3.2 .
Proof.Since each vertex of K n has degree n − 1, for v i , v j ∈ V (K n ) with i = j, we have Consequently Remark 3.4.By Theorem 2.3 and Theorem 3.3, it is easily seen that A graph whose every vertex has equal degree is called a regular graph.A graph whose every vertex has degree k is called a k-regular graph.Zangi et al. [26] prove the following result.Theorem 3.5 (Zangi et al. [26]).Suppose G is an n-vertex k-regular graph.Then

Bounds and integral representation for ISI energy
In this section, we give some bounds for the ISI energy of graphs.
Let B is a matrix of order n × n such that where F is the function with the property F (y, z) = F (z, y).Das et al. [8] prove the following theorem for eigenvalues of degree based energies of graphs.Theorem 4.1 (Das et al. [8]).For the eigenvalues The following result is obtained by using Theorem 4.1,.
Using Theorem 2.2 and Theorem 4.2, we get the following result for an n-vertex connected graph G.
In next theorem, we find bounds for ISI energy in terms of trace of matrix S 2 (G) and determinant of S(G).Proof.As we know that arithmetic mean is always less than quadratic mean, therefore Arithmetic-quadratic mean inequality gives, The proof is complete.Now we have the following theorem.The proof is same as the proof of Theorem 3 [8] and is thus excluded.Theorem 4.5.Let G be a simple n-vertex graph with n ≥ 2 vertices.Then In Theorem 4.8, we obtain bounds for ISI energy in terms of number of edges, minimum and maximum degrees of a simple graph.Theorem 4.6.Suppose G is an n-vertex simple graph with m edges, minimum degree δ and maximum degree ∆.Then Proof.For each vertex v i of G, δ ≤ d i ≤ ∆, i = 1, 2, . . .n.Using this fact, we get Now using Theorem 4.5, we obtain the desired result.
Coulson [5] prove the following integral representation of energy of graphs.
Theorem 4.7 (Coulson [5]).Let G be an n-vertex simple graph, then where Next theorem is an analogue of Theorem 4.7.
Theorem 4.8.Let G be a simple graph of order n.Then The following result is similar to the graph energy.
Theorem 4.9.Let G be an n-vertex graph with S-characteristic polynomial The following result is based on our numerical testing.The application of Coulson-type integral expression for proving the conjecture was (so far) not successful.Conjecture 4.10.Among all n-vertex tress, the tree with minimal ISI energy is S n and the tree with maximal ISI energy is P n , where n ≥ 2.

S-Equienergetic graphs
Two graphs with same S-spectrum are said to be S-cospectral, otherwise S-noncospectral.Two S-equienergetic graphs of same order are the graphs which have same ISI energy.Two isomorphic graphs are always S-cospectral and thus are S-equienergetic.We construct few classes of Snoncospectral S-equienergetic graphs.
The line graph, denoted by L(G), of G, is the graph with V (L(G)) = E(G) and two vertices of L(G) are connected by an edge if edges incident on it are adjacent in G.
and the remaining S-eigenvalues are zero (if exist) if and only if G ∼ = p j=1 K r,s , where p(r + s) = n and one of the r or s is greater than 1.
3)and the remaining S-eigenvalues are zero (if exist).Then each component of G has atmost three distinct S-eigenvalues.Let H be a component of G. From (2.3) and Lemma 2.7, we see that H is bipartite.If H is not a complete bipartite graph, then the diameter of H is at least 3. Therefore by Lemma 2.6 and (2.3), we get a contradiction.Hence H is a complete bipartite graph.As H is arbitrary component of G, therefore G ∼ = p j=1 K r,s , where p(r + s) = n.