Abstract
Let be a simple graph. The incidence energy ( for short) of is defined as the sum of the singular values of the incidence matrix. In this paper, a new upper bound for of graphs in terms of the maximum degree is given. Meanwhile, bounds for of the line graph of a semiregular graph and the paraline graph of a regular graph are obtained.
1. Introduction
Let be a finite, simple, and undirected graph with vertices. The matrix (resp., ) is called the Laplacian matrix (resp., signless Laplacian matrix [1–4]) of , where is the adjacency matrix and is the diagonal matrix of the vertex degrees. (For details on Laplacian matrix, see [5, 6].) Since , and are all real symmetric matrices, their eigenvalues are real numbers. So, we can assume that (resp., , ) are the adjacency (resp., Laplacian, signless Laplacian) eigenvalues of . It follows from the Geršgorin disc theorem that and are semidefinite. Therefore, all Laplacian (resp., signless Laplacian) eigenvalues of are nonnegative. If the graph is connected, then for and [6]. If is a connected nonbipartite graph, then for [1].
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 [8–15], 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 and edge set , the -entry of is 0 if is not incident with and if is incident with . Jooyandeh et al. [18] introduced the incidence energy 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 may be found in [18–20].
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., [21–23]). Recently, several papers on line graph have been published [20, 24–27]. 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-energy-like invariant ( for short) of the line graph of a regular graph are obtained in [26]. For details on , 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 . In particular, an upper bound for of the line graph of a regular graph was established in [20].
In this paper, we continue to study the bounds for of graphs. In Section 2, we give a new upper bound for of graphs in terms of the maximum degree. Bounds for of the line graph of a semiregular graph and the paraline graph of a regular graph are obtained in Section 3.
2. A New Upper Bound for
In this section, we will give a new upper bound for of a nonempty graph. The following fundamental properties of the were established in [18].
Lemma 1 (see [18]). Let be a graph with vertices and edges. Then(i), and equality holds if and only if ;(ii)if are all components of , then .
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 . Then, ; the equality holds if and only if 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 . Then the equality holds if and only if .
Proof. Note that . By the Cauchy-Schwarz inequality
The equality holds if and only if .
We consider the function
Then
It is easily seen that the function is decreasing for . By Lemma 2,
Inequality (3) follows now from the monotonicity of and (4).
Equality in (3) holds if and only if is a cycle. It is well-known [4] that the signless Laplacian eigenvalues of are
If , 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 was given as follows.
Lemma 4 (see [20]). Let be a connected graph with 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 (3) is , but the bound in (10) is . Note that It follows from the monotonicity of that That is,
3. Bounds for of Line Graphs of Semiregular Graphs
In this section, we will investigate the 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 of a graph is the graph whose vertex set is in one-to-one correspondence with the set of edges of where two vertices of 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 on 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.
A bipartite graph with a bipartition is called an -semiregular graph if all vertices in have degree and all vertices in have degree . Denote by the signless Laplacian polynomial of . For an -semiregular graph , we can establish the following relationship between and .
Lemma 7. Let be an -semiregular graph with vertices. Then where 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 -regular graph. Thus
Combine (15) with (16), we have
It follows from Lemma 6, (15), and (17) that and have the same nonzero eigenvalues. Note that the difference between the dimensions of 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 are where are the signless Laplacian eigenvalues of .
Theorem 8. Let be an -semiregular graph with vertices. Then the equality holds if and only if , or is even and with .
Proof. Let be the number of edges of . Then, . Note that . It follows from (2) and (18) that
Note also that , . By the Cauchy-Schwarz inequality, we have
Inequality (19) follows now from (20) and (21).
Clearly, equality in (19) holds if and only if . Suppose that . Then, the number of distinct signless Laplacian eigenvalues of is at most . From the result, “the number of distinct signless Laplacian eigenvalues of a connected graphs with diameter is at least [1],” we know that the diameter of is at most . Note that is bipartite graph. If the diameter of is , then must be . If the diameter of is , then is a complete bipartite graph with exactly distinct signless Laplacian eigenvalues. It is well known that the signless Laplacian eigenvalues of are , , , and . Thus, , or is even and with .
Conversely, if , then . 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 . In this case, and . It is easy to check that the right-hand side of (19) is also equal to . If , then , where is even. Note also that the signless Laplacian spectrum of lattice graph [31] is
Similarly, it follows from (2) and (24) that the left-hand side of (19) is equal to . Note that is an -semiregular graph. Substituting into (19), we get the right-hand side of (19) is still equal to .
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 .

(a)

(b)

(c)
Remark 9. The subdivision graph of an -regular graph is -semiregular. Hence, the paraline graph of an -regular graph is the line graph of an -semiregular graph.
The following result is well known [5, 6].
Lemma 10. The spectra of and coincide if and only if the graph is bipartite.
Let be the Laplacian polynomial of .
Corollary 11. Let be an -regular graph with vertices. Then
Proof. Note that is a -semiregular graph with vertices and edges. By Lemma 7, we obtain It is shown in [33] that It follows from (27) and Lemma 10 that Combining (26) with (28), we have
It follows from Corollary 11 that the signless Laplacian spectrum of iswhere and , are the Laplacian eigenvalues of .
Theorem 12. Let be a connected -regular graph with vertices. Then the equality holds if and only if .
Proof. If , then . Note that . It is easily seen that
In this case, and . Hence
Suppose now that . For convenience, let 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 and . Then by (2) and (30), we have
Note that , by the Perron-Frobenius theorem. Consider the function
The derivative function of is
It is clear that is decreasing for . Therefore
That is, if and only if is regular graph and . It follows that is a regular graph with two distinct adjacency eigenvalues and with multiplicities and , respectively. Then by Proposition 1.3.3 [31], must be a complete graph. Note that the sum of the adjacency eigenvalues of is equal to zero; that is, . It follows that is a complete graph with two vertices; that is, . This is impossible since .
Summing up, we complete the proof.
Theorem 13. Let be an -regular graph with vertices, . Then
Proof. From the proof of Theorem 12, we know that the function is decreasing for . Therefore
Acknowledgments
The authors are grateful to the anonymous referees for many valuable comments and suggestions, which led to great improvements of the original paper. This research was partially supported by the National Natural Science Foundation of China (no. 11201201) and the Fundamental Research Funds for the Central Universities (no. lzujbky-2012-16).