Bounds for Incidence Energy of Some Graphs

LetG be a finite, simple, and undirected graphwith n vertices. Thematrix L(G) = D(G)−A(G) (resp., L+(G) = D(G)+A(G)) is called the Laplacianmatrix (resp., signless Laplacianmatrix [1–4]) of G, where A(G) is the adjacency matrix and D(G) is the diagonal matrix of the vertex degrees. (For details on Laplacian matrix, see [5, 6].) Since A(G), L(G) and L+(G) are all real symmetric matrices, their eigenvalues are real numbers. So, we can assume that λ 1 (G) ≥ λ 2 (G) ≥ ⋅ ⋅ ⋅ ≥

One of the most remarkable chemical applications of graph theory is based on the close correspondence between the graph eigenvalues and the molecular orbital energy levels of -electrons in conjugated hydrocarbons.For the Hüchkel molecular orbital approximation, the total -electron energy in conjugated hydrocarbons is given by the sum of absolute values of the eigenvalues corresponding to the molecular graph  in which the maximum degree is not more than four in general.The energy of  was defined by Gutman in [7] as Research on graph energy is nowadays very active, as seen from the recent papers [8][9][10][11][12][13][14][15], monograph [16], and the references quoted therein.
The singular values of a real matrix (not necessarily square)  are the square roots of the eigenvalues of the matrix   , where   denotes the transpose of .Recently, Nikiforov [17] extended the concept of graph energy to any matrix  by defining the energy () to be the sum of singular values of .Obviously, () = (()).
Let () be the (vertex-edge) incidence matrix of the graph .For a graph  with vertex set {V 1 , V 2 , . . ., V  } and edge set { 1 ,  2 , . . .,   }, the (, )-entry of () is 0 if V  is not incident with   and 1 if V  is incident with   .Jooyandeh et al. [18] introduced the incidence energy IE of , which is defined as the sum of the singular values of the incidence matrix of .Gutman et al. [19] showed that Some basic properties of IE may be found in [18][19][20].
A line graph is a classical unary operation of graphs with finite number and infinite number of vertices.Its basic properties can be found in any text book on graph theory (see, e.g., [21][22][23]).Recently, several papers on line graph have been published [20,[24][25][26][27].For example, Gao et al. [24] established a formula and lower bounds for the Kirchhoff index of the line graph of a regular graph.Bounds for Laplacian-energylike invariant (LEL for short) of the line graph of a regular graph  are obtained in [26].For details on LEL, see the comprehensive survey [28].
From (2), one can immediately get the incidence energy of a graph by computing the signless Laplacian eigenvalues of the graph.However, even for special graphs, it is still complicated to find the signless Laplacian eigenvalues of them.Hence, it makes sense to establish lower and upper bounds to estimate the invariant for some classes of graphs.Zhou [29] obtained the upper bounds for the incidence energy using the first Zagreb index.Gutman et al. [20] gave several lower and upper bounds for IE.In particular, an upper bound for IE of the line graph of a regular graph was established in [20].
In this paper, we continue to study the bounds for IE of graphs.In Section 2, we give a new upper bound for IE of graphs in terms of the maximum degree.Bounds for IE of the line graph of a semiregular graph and the paraline graph of a regular graph are obtained in Section 3.

A New Upper Bound for 𝐼𝐸
In this section, we will give a new upper bound for IE of a nonempty graph.The following fundamental properties of the IE were established in [18].
Lemma 1 (see [18]).Let  be a graph with  vertices and  edges.Then (i) IE() ≥ 0, and equality holds if and only if  = 0; From Lemma 1(ii), when we study the incidence energy of a graph , we may assume that  is connected.
The following lemma will be used later.
Lemma 2 (see [3]).Let  be a connected graph without vertices of degree 1 and the maximum degree Δ.Then,

)); the equality holds if and only if 𝐺 is a cycle.
Denote by   the cycle with  vertices.Theorem 3. Let  be a connected graph without vertices of degree 1 and the maximum degree Δ.Then the equality holds if and only if  ≅  3 .
Proof.Note that ∑  =2  +  = 2 −  + 1 .By the Cauchy-Schwarz inequality The equality holds if and only if  + 2 = ⋅ ⋅ ⋅ =  +  .We consider the function Then It is easily seen that the function () is decreasing for  > 2/.By Lemma 2, If  ≅  3 , then we may verify directly that the equality in (3) holds.
Conversely, if the equality in (3) holds, then It follows from Lemma 2 that  is a cycle.Note that  has at most two distinct signless Laplacian eigenvalues.By virtue of (8), we now conclude that  is a triangle.
Recall from [20] that an upper bound for IE was given as follows.
Lemma 4 (see [20]).Let  be a connected graph with  ≥ 3 vertices and  edges.Then Remark 5.It should be pointed out that, for a connected graph without vertices of degree 1, the bound in (3) is better than the bound in (10).Indeed, it is easily seen that the bound in It follows from the monotonicity of () that That is,

Bounds for 𝐼𝐸 of Line Graphs of Semiregular Graphs
In this section, we will investigate the IE of the line graph of an (, )-semiregular graph and the paraline graph of an regular graph.We first consider the case for line graph.The line graph L() of a graph  is the graph whose vertex set is in oneto-one correspondence with the set of edges of  where two vertices of L() are adjacent if and only if the corresponding edges in  have a vertex in common.For instance, the line graph of a star   on  vertices is a complete graph  −1 on  − 1 vertices.
The following result is well known [30].
Lemma 6 (see [30]).Let  be a matrix.Then, the matrices   and    have the same nonzero eigenvalues.

Lemma 7.
Let  be an (, )-semiregular graph with  vertices.Then where L() is the line graph of  and  is the number of edges of .
Proof.Let () be the (vertex-edge) incidence matrix of a graph .Then where   stands for the unit matrix of order .
Note that if  is an (, )-semiregular graph, then the line graph of  is ( +  − 2)-regular graph.Thus Combine ( 15) with ( 16), we have It follows from Lemma 6,(15), and (17) that  + () and  + (L()) − ( +  − 4)  have the same nonzero eigenvalues.Note that the difference between the dimensions of  + (L()) and  + () is  − .The proof is finished by the fact that the leading coefficient of the characteristic polynomial is equal to one.
By Lemma 7, the signless Laplacian eigenvalues of L() are where Note also that  +  = 0, ∑  =1  +  = 2.By the Cauchy-Schwarz inequality, we have Inequality (19) follows now from ( 20) and (21).Clearly, equality in (19) holds if and only if Then, the number of distinct signless Laplacian eigenvalues of  is at most 3. From the result, "the number of distinct signless Laplacian eigenvalues of a connected graphs with diameter  is at least  + 1 [1], " we know that the diameter of  is at most 2. Note that  is bipartite graph.If the diameter of  is 1, then  must be  2 .If the diameter of  is 2, then  is a complete bipartite graph  , with exactly 3 distinct signless Laplacian eigenvalues.It is well known that the signless Laplacian eigenvalues of  , are  + ,  −1 ,  −1 , and 0. Thus,  ≅  1,−1 , or  is even and  ≅  (/2),(/2) with  ≥ 4.
Conversely, if  ≅  1,−1 , then L() ≅  −1 .Note that the signless Laplacian spectrum of complete graph   [4] is It follows from ( 2) and ( 22) that That is, the left-hand side of ( 19) is equal to √ 2 − 4 + (n-2) √  − 3.In this case,  = −1 and  = 1.It is easy to check that the right-hand side of ( 19) is also equal to where  ≥ 4 is even.Note also that the signless Laplacian spectrum of lattice graph  2 () [31] is Similarly, it follows from ( 2) and ( 24) that the left-hand side of ( 19) is equal to ) is an (/2, /2)semiregular graph.Substituting  =  = /2 into (19), we get the right-hand side of ( 19) is still equal to (/2 − 1) Hence, we complete the proof of Theorem 8. Now, we consider the case for paraline graph.Let  be a simple graph.A paraline graph, denoted by (), is defined as a line graph of the subdivision graph () (the subdivision graph () of a graph  is the graph obtained from  by inserting a vertex to every edge of ) of  (e.g., see Figure 1).The concept of the paraline graph (or clique-inserted graph [32]) of a graph was first introduced in [25], where the author obtained the spectrum of the paraline graph of a regular graph  with infinite number of vertices in terms of the spectrum of .
Remark 9.The subdivision graph of an -regular graph is (, 2)-semiregular.Hence, the paraline graph of an -regular graph is the line graph of an (, 2)-semiregular graph.
In this case,  = 1 and  = 2. Hence Suppose now that  ≥ 2. For convenience, let  = /2 be the edges of .Note that  is an -regular graph.It follows from the definition of paraline graph that () is still an regular graph.Note also that   () =  −  −+1 (),  = 1, 2, . . .,  and  1 () = .Then by ( 2) and ( 30 [31],  must be a complete graph.Note that the sum of the adjacency eigenvalues of  is equal to zero; that is,  + ( − 1)(−) = 0.It follows that  is a complete graph with two vertices; that is,  ≅  2 .This is impossible since  ≥ 2.
Summing up, we complete the proof.