On inverse sum indeg energy of graphs

: For a simple graph with vertex set v v v , , , n 1 2 { } … and degree sequence d i n 1, 2, , v i = … , the inverse sum indeg matrix ( ISI matrix ) A G a ij ISI ( ) ( ) = of G is a square matrix of order n , where a , ij d d d d vi vj vi vj = + if v i is adjacent to v j and 0, otherwise. The multiset of eigenvalues τ τ τ n 1 2 ≥ ≥⋯≥ of A G ISI ( ) is known as the ISI spectrum of G . The ISI energy of G is the sum τ in i 1 ∣ ∣ ∑ = of the absolute ISI eigenvalues of G . In this article, we give some properties of the ISI eigenvalues of graphs. Also, we obtain the bounds of the ISI eigenvalues and characterize the extremal graphs. Furthermore, we construct pairs of ISI equienergetic graphs for each n 9 ≥ .

and degree sequence d i n 1, 2, , vi = … , the inverse sum indeg matrix (ISI matrix) A G a ij ISI ( ) ( ) = of G is a square matrix of order n, where a , adjacent to v j and 0, otherwise.The multiset of eigenvalues τ τ τ n ≥ ≥⋯≥ of A G ISI ( ) is known as the ISI spectrum of G.The ISI energy of G is the sum and an edge set E G ( ).We consider only simple and undirected graphs, unless otherwise stated.The number of elements in V G ( ) is the order n, and the number of elements in E G ( ) is the size m of G.By u v ~, we mean vertex u is adjacent to vertex v, we also denote an edge by e.The neighbourhood N v is the set of vertices adjacent to v. The degree d vi (or simply d i ) of a vertex v i is the number of elements in the set N v i ( ).A graph G is called r-regular if the degree of every vertex is r.For two distinct vertices u and v in a connected graph G, the distance d u v , ( ) between them is the length of a shortest path connecting them.The largest distance between any two vertices in a connected graph is called the diameter of G.We denote the complete graph by K n , the complete bipartite graph by K a b , , and the star by K n 1, 1 − .We follow the standard graph theory notation, and more graph theoretic notations can be found in [1].
The adjacency matrix A G ( ) of G is a square matrix of order n n × , with i j , ( )th entry equals 1, if v i and v j are adjacent and 0 otherwise.Clearly, A G ( ) is a real symmetric matrix, and its multiset of eigenvalues is known as the spectrum of G. Let λ λ λ n ≥ ≥⋯≥ be the eigenvalue of A G ( ), where the eigenvalue λ 1 is called the spectral radius of G.More about the adjacency matrix A G ( ) can be seen in [1][2][3].The energy [4] of G is defined as follows: The energy is intensively studied in both mathematics and theoretical chemistry since it is the trace norm of real symmetric matrices in linear algebra and the total π-electron energy of a molecule, see [5,6].For more about the energy of G, including the recent development, see [7-9].
The inverse sum indeg index (ISI index) [10] is a topological index defined as follows: The ISI index is a well-studied topological index and has many applications in quantitative structureactivity or structure-property relationships (QSAR/QSPR) [11][12][13].
The inverse sum indeg matrix (ISI matrix) of a graph G, introduced by Zangi et al. [14], is a square matrix of order n defined as follows: The ISI matrix is a real symmetric, and its eigenvalues are also real.We order its eigenvalues from largest to smallest by

≥ ≥⋯≥
The multiset of all eigenvalues of the ISI matrix of G is known as the ISI spectrum of G, and the largest eigenvalue τ 1 is called the ISI-spectral radius of G.If an eigenvalue, say τ, of the ISI matrix occurs with algebraic multiplicity k 2 ≥ , then we denote it by τ k [ ] .The ISI energy of G is defined as follows: Zangi et al. [14] gave the basics properties of the ISI matrix including the bounds for the ISI energy of graphs.Hafeez and Farooq [15] obtained ISI spectrum and ISI energy from special graphs.They also gave some bounds on the ISI energy of graphs.Bharali et al. [16] gave some bounds on ISI energy and introduced ISI Estrada index of G. Havare [17] obtained the ISI index and ISI energy of the molecular graphs of Hyaluronic Acid-Paclitaxel conjugates.For some other types of energies and indices, see [18][19][20][21][22][23][24][25][26].
In Section 2, we characterize graphs with two distinct ISI eigenvalues and three distinct ISI eigenvalues among bipartite graphs and give some sharp bounds on the ISI spectral radius and the ISI energy of graphs, which are better than already known results.In Section 3, we give the ISI spectrum of the join of two graphs, and as a consequence, we construct ISI equienergetic graphs for every integer n 9.
≥ We end up article with a conclusion for future work.

Inverse sum indeg energy of graphs
It is trivial that nK 1 is the only graph with exactly one ISI eigenvalue and its ISI spectrum is Next, we have result about graphs whose all ISI eigenvalues are equal in absolute value.Proposition 2.1.Let G be a graph of order n.Then, τ τ τ n Conversely, assume that τ τ τ n = =⋯= = and G nK 1 ≅ .The other possibility is that k 0, = and if maximum degree is 1, then Now, if maximum degree is greater than or equal to two, then G contains a connected component G′ with order at least 3.By Perron-Frobenius theorem, τ G τ G which is not possible.Thus, G K .

≅ □
The following well-known result provides a relationship between the number of distinct eigenvalues in a graph and its diameter.It can be found in [2].Theorem 2.2.[2] Let G be a connected graph with diameter D.Then, G has at least D 1 + distinct adjacency eigenvalues.
From the proof of Theorem 2.2 (Proposition 1.3.3,[2]), it follows that Theorem 2.2 is true for any nonnegative symmetric matrix M m ij n n ( ) = × indexed by the order of a graph G, in which m 0 ij > if and only if v i is adjacent to v j .The following result is the consequence of Theorem 2.2.
Another immediate important consequence is given as follows.
Corollary 2.4.Let G be a connected graph of order n 2 ≥ .Then, G has exactly two distinct ISI eigenvalues if and only if G is the complete graph.
, and G has two distinct ISI eigenvalues.
Conversely, if G has exactly two distinct eigenvalues, from Corollary 2.3, its diameter is 1.Therefore, G is necessarily K n .□ The following observation states that G has a symmetric ISI spectrum towards the origin if G is bipartite.
Remark 2.5.Clearly, the ISI matrix of the bipartite graph G can be written as follows: This implies that the ISI eigenvalues of a bipartite graph are symmetric about the origin.Proposition 2.6.Let G be a bipartite graph.Then, G has three distinct ISI eigenvalues if and only if G is the complete bipartite graph.
be the complete bipartite graph with partite cardinality a and b a b n , ( ) + = .Then, the ISI spectrum (see [15] − and clearly G has three distinct ISI eigenvalues.Conversely, if we assume that G has three distinct ISI eigenvalues, then by Corollary 2.3, the diameter of G is at most two.Also, by Corollary 2.4, diameter of G cannot be one as in this case G cannot have three distinct ISI eigenvalues.So, diameter of G is exactly two.As G is a bipartite graph of diameter two, so any two non-adjacent vertices of G must have the same neighbour; otherwise, if a vertex u has neighbour w not adjacent to v, then w along with uv-path induces the path P 4 subgraph, which cannot happen as the diameter of G is two.Thus, any two non-adjacent vertices in G share the common neighbour, and it follows that G is the complete bipartite graph.□ On inverse sum indeg energy of graphs  3 The sum of the squares of the eigenvalues (Frobenius norm of real symmetric matrix) of the ISI matrix (Theorem 5, [14]) is where B .
The following result gives the bounds for the ISI spectral radius of graphs in terms of the Frobenius norm and ISI index.
Lemma 2.7.Let G be a graph of order n.Then the following holds ≅ Also, by the Cauchy-Schwartz inequality, we have . If equality holds, then all above inequalities are equalities, that is τ τ τ n It follows that G has two distinct ISI eigenvalues.By Corollary 2.4, G is the complete graph.
be an arbitrary vector of n and let J denote the vector with all entries equal to 1, that is J 1, 1, ,1 ( ) = … .Furthermore, we note that A G ISI ( ) is non-negative and an irreducible matrix.Thus, by Perron-Frobenius theorem, τ τ i for all i and τ 0. 1 > Therefore, by Rayleigh quotient for Hermitian matrices [27], we have . Also, it is well known that the largest eigenvalue λ 1 of A G ( ) is bounded above by the maximum degree Δ with equality if and only if G is regular.So, for regular graphs, we have τ  , , , , with equality holding if and only if τ τ τ .
Also, by equation (1), we have with equality holding if and only if τ τ n 1 = − and τ τ 0.
The second part of the next result is the analogue of Koolen-Moulton bound for the ISI energy of graphs.
Theorem 2.9.Let G be a graph of order n.Then, the following hold.
with equality if and only if G is regular and has only one positive ISI eigenvalue, like the complete regular multipartite graphs, the Peterson graph and its complement.
or a non-complete connected graph with three distinct ISI and the other two distinct eigenvalues with absolute value .
with equality holding if and only if G has only two non-zero ISI eigenvalues (one positive and one negative as τ 0 ).By Lemma 2.7, we obtain with equality holding if and only if G is a regular connected graph with only one positive ISI eigenvalue.
(ii) By the Cauchy-Schwartz inequality, we have with equality if and only if τ τ τ n ) .Thus, inequality (3) remains valid if on the right side of F x ( ), the variable is replaced with any lower bound of τ 1 .So from Lemma 2.7.we have The equality occurs if and only if all inequalities are equalities.By Lemma 2.7, G is a regular graph and by Next, lemma is an application of interlacing theorem, it relates the independence number (the cardinality of a largest pairwise non-adjacent vertex set) to the number of positive and non-positive ISI eigenvalues of G. Lemma 2.10.Let G be a graph with n vertices, and let p and q be the number of ISI eigenvalues that are greater than and less than equal to 0, respectively.Then, where μ is the independence number of G.
Proof.As G has independence number μ, so the ISI matrix of G has the principal submatrix M 0 μ μ ′ = × .By interlacing theorem [27], . This completes the proof.□ Theorem 2.11.Let G be a connected graph with independence number μ, p, and q number of ISI eigenvalues which are greater than and less than equal to 0, respectively.Then with equality holding if and only if G is the star graph K n 1, 1 ≥ ≥⋯≥ and τ τ τ q ′ ≥ ′ ≥⋯≥ ′ be the positive and non-positive ISI eigenvalues of G, respec- tively.Since τ 0 and by the definition of ISI energy, we have Now, by using the Cauchy-Schwartz inequality, we have with equality holding if and only if τ τ τ. p with equality holding if and only if τ τ τ .
and the required inequality (4) follows.
If equality holds in (4), then from above, we have τ τ τ p ⋯= ′ , and p q n μ. = = − But by the Perron-Frobenious theorem, τ 1 is a simple eigenvalue of G, so p 1 = , and it implies that q μ n 1, 1 = = − , and the ISI eigenvalue 0 has multiplicity n 2. − By Lemma 2.6, G is the complete bipartite graph, thereby it follows that G K , Therefore, we have This proves the equality case.□ + is the complete graph, otherwise its diameter is 2. Suppose we have a matrix M partitioned in some block form, and we form a new matrix Q whose entries are the average row sums of the blocks of the partitioned matrix, then such a matrix is known as the quotient matrix.If the average row sums of blocks are some constant, not necessarily same for all blocks, and this happens for every block we say that the quotient matrix is equitable.In general, the eigenvalues of Q matrix interlace those of M. While for equitable quotient matrix, each of the eigenvalues of Q is the eigenvalue of M [1,2].

ISI equienergetic graphs
The following theorem gives the ISI spectrum of the join of two regular non-complete graphs.
Theorem 3.1.Let G 1 and G 2 be r 1 -regular and r 2 -regular graphs of order n 1 and n 2 , respectively.Let λ r λ λ , , , n = + .We first index the vertices of G 1 and then the vertices G 2 .With this indexing, the ISI matrix is where J n n × and J n n × are the matrices whose each entry equals 1, and A G i ( ) is the adjacency matrix of G i , for i 1, 2.
= Since G 1 is r 1 regular, it follows that r 1 is an eigenvalue of A G 1 ( ) with the corresponding eigenvector J (whose all entries are equal to 1), and J is orthogonal to all other eigenvectors of G 1 .Let x be a non-zero column vector satisfying A G x λ x i 1 ( ) = and J x 0.   Theorem 3.3.For every n 8 > , there exists a pair of ISI equienergetic graphs of order n.
Proof.Consider two 4-regular equienergetic graphs (Example 4.1, [28]) as in Figure 1.Also, G G 16  [5,26,25]).The more important is relating the spectral parameters of the ISI matrix to the underlying graph structure and the relation of the ISI matrix to the adjacency matrix for irregular graphs remains challenging.
have the analogue of the McClelland bound for the ISI energy of a graph.The upper bound of (i) part of Theorem 2.8 is given in[15], but extremal graphs were not characterized.Theorem 2.8.Let G be a graph of order n.Then, By applying the Cauchy-Schwarz inequality to vector τ τ τ

By Proposition 2 . 6 ,
G must be the complete bipartite graph, since ISI spectrum is symmetric towards origin and 0 is the ISI eigenvalue of G with multiplicity n 2 − .Conversely, for G K a b ,

−Proof. Let p 1 ≥
be the number of positive ISI eigenvalues.Then,

□
we have two cases: first possibility is G has two distinct ISI eigenvalues and by Corollary 2.4, G K n ≅ .The second possibility is that G has three distinct ISI eigenvalues, τ G n 1 2ISI( )=and the other two distinct eigenvalues with absolute value .On inverse sum indeg energy of graphs  5 be the adjacency eigenvalues of G 1 and G 2 , respectively.Then, the ISI spectrum of G G

2 + consists of r n 2 1 2 +
By Theorem 3.1, the ISI spectrum of G G 1 times the adjacency eigenvalues of G 1 except r 1 , r n
Two graphs of the same order are said to be equienergetic (or adjacency equienergetic) if they have the same energy but have a different adjacency spectrum.Likewise, two graphs of order n are said to be ISI equi- energetic if they have the same ISI energy but distinct ISI spectrum.Let G 1 and G 2 be the connected graphs of order n 1 and n 2 , respectively.The join of G 1 and G 2 , denoted by G G + , is the graph obtained by joining each vertex of G 1 to every vertex of G 2 .If both G 1 and G 2 are complete graphs, then G G The extremal energy (ISI energy) problem is long standing, and it is very non-trivial to explicitly characterize the graphs with maximum and minimum energy (ISI energy) among general graphs.The problem of maximal (minimal) ISI energy of arbitrary graphs remains open.Besides, new concepts like the Estrada index, sum of k largest ISI eigenvalues (Ky Fan k-norm), Laplacian ISI matrices, distribution of ISI eigen- values, spectral radius, and application of ISI spectra in chemical theory are yet to be introduced/investigated (like in