On two energy-like invariants of line graphs and related graph operations
- Xiaodan Chen^{1, 2}Email author,
- Yaoping Hou^{2} and
- Jingjian Li^{1}
https://doi.org/10.1186/s13660-016-0996-1
© Chen et al. 2016
Received: 20 September 2015
Accepted: 1 February 2016
Published: 11 February 2016
Abstract
Keywords
MSC
1 Introduction
Recall that giving an arbitrary orientation to each edge of G would yield an oriented graph G⃗. Let \(B(\vec{G})\) be the (vertex-edge) incidence matrix of G⃗. Then \(B(\vec{G})B(\vec{G})^{T}=L(G)\), and hence \(\mathit{LEL}(G)=E(B(\vec{G}))\). This provides a new interpretation for LEL: oriented incidence energy [8], and furnishes a new insight into its possible physical or chemical meaning.
Theorem 1.1
(see [10])
Let G be a graph of order n. Then \(\mathit{LEL}(G)\leq\mathit{IE}(G)\), with the equality holding if and only if G is bipartite.
For more results on IE, we refer the reader to [9, 11–13].
The line graph, subdivision graph, para-line graph and total graph are the well-known operations on graphs, which can produce many new types of graphs. In [14], Ramane et al. studied the spectra and energies of (iterated) line graphs of regular graphs, and based on the derived results they found a systematic construction of pairs of equienergetic graphs. Gao et al. [15] established formulas and lower bounds of the Kirchhoff index of line, subdivision and total graphs of a regular graph. Yan et al. [16] considered the asymptotic behavior of the number of spanning trees and the Kirchhoff index of iterated line graphs and iterated para-line graphs of a regular graph. Recently, in [17], Wang and Luo presented upper and lower bounds for the Laplaican-energy-like invariant of line graph, subdivision graph and total graph of a regular graph. In addition, upper and lower bounds for the incidence energy of (iterated) line graphs of a regular graph were also obtained by Gutman et al. [12].
In this paper, we give some new upper and lower bounds on the Laplacian-energy-like invariant and the incidence energy of line graph, subdivision graph, para-line graph and total graph of a regular graph, some of which improve the corresponding bounds in [17] and [12].
2 Preliminaries
In this section, we recall the concepts of line graphs and related graph operations, and list some lemmas that will be used in the subsequent sections.
Lemma 2.1
Let G be a regular graph of degree r with n vertices. If the eigenvalues of G are \(\lambda_{1},\lambda_{2} , \ldots,\lambda_{n}\), then the eigenvalues of \(\mathcal{L}(G)\) are \(r-2+\lambda_{1},r-2+\lambda_{2},\ldots,r-2+\lambda_{n}\), and −2 (with multiplicity \(n(r-2)/2\)).
Lemma 2.2
The para-line graph \(\mathcal{C}(G)\) of a graph G, which was first introduced in [20], is defined to be the line graph of the subdivision graph of G. In [21], the para-line graph is also called the clique-inserted graph.
Lemma 2.3
(see [21])
The total graph \(\mathcal{T}(G)\) of a graph G is the graph whose vertices are the vertices and edges of G, with two vertices of \(\mathcal{T}(G)\) adjacent if and only if the corresponding elements of G are adjacent or incident.
Lemma 2.4
(see [18])
The following is known as the Ozeki inequality.
Lemma 2.5
(see [22])
Lemma 2.6
Lemma 2.7
(see [23])
Let G be a graph of order n with at least one edge. Then \(\mu_{1}=\mu_{2}=\cdots=\mu_{n-1}\) if and only if \(G\cong K_{n}\).
Finally, we should remark that if G is a regular graph of degree \(r=1\), then G is nothing but the disjoint union of independent edges. Hence, in the following, for avoiding the triviality we always assume that \(r\geq2\).
3 The Laplacian-energy-like invariant
In this section, we consider LEL of line graph, subdivision graph, para-line graph, and total graph of a regular graph. We begin with the case of line graphs.
Theorem 3.1
(see [4])
By Theorem 3.1 and (5), we have the following corollary directly.
Corollary 3.2
Remark 1
We now consider the case of subdivision graphs.
Theorem 3.3
- (i)
\(\mathit{LEL}(\mathcal{S}(G))\leq(n-1)\sqrt{r+2+\frac{2}{n-1}\mathit {LEL}(G)} +\sqrt{r+2}+\frac{\sqrt{2}n(r-2)}{2}\), with the equality if and only if \(G\cong K_{n}\);
- (ii)
\(\mathit{LEL}(\mathcal{S}(G))>(n-1)\sqrt{r+\frac{3}{2}+\frac {2}{n-1}\mathit{LEL}(G)} +\sqrt{r+2}+\frac{\sqrt{2}n(r-2)}{2}\).
Proof
Theorem 3.3, together with (5), would yield the next immediate corollary.
Corollary 3.4
- (i)
\(\mathit{LEL}(\mathcal{S}(G))\leq(n-1)\sqrt{r+2+\frac{2 (\sqrt {r+1}+\sqrt{(n-2)(nr-r-1)} )}{n-1}} +\sqrt{r+2}+\frac{\sqrt{2}n(r-2)}{2}\), with the equality if and only if \(G\cong K_{n}\);
- (ii)
\(\mathit{LEL}(\mathcal{S}(G))>(n-1)\sqrt{r+\frac{3}{2}+\frac {2nr}{(n-1)\sqrt{r+1}}} +\sqrt{r+2}+\frac{\sqrt{2}n(r-2)}{2}\).
Remark 2
Theorem 3.5
- (i)
\(\mathit{LEL}(\mathcal{C}(G))\leq(n-1)\sqrt{r+2+\frac{2}{n-1}\mathit {LEL}(G)} +(\frac{n(r-2)}{2}+1)\sqrt{r+2}+\frac{n(r-2)}{2}\sqrt{r}\), with the equality if and only if \(G\cong K_{n}\);
- (ii)
\(\mathit{LEL}(\mathcal{C}(G))>(n-1)\sqrt{r+\frac{3}{2}+\frac {2}{n-1}\mathit{LEL}(G)} +(\frac{n(r-2)}{2}+1)\sqrt{r+2}+\frac{n(r-2)}{2}\sqrt{r}\).
Likewise, the next corollary follows evidently from Theorem 3.5 and (5).
Corollary 3.6
- (i)
\(\mathit{LEL}(\mathcal{C}(G))\leq(n-1)\sqrt{r+2+\frac{2 (\sqrt {r+1}+\sqrt{(n-2)(nr-r-1)} )}{n-1}} +(\frac{n(r-2)}{2}+1)\sqrt{r+2}+\frac{n(r-2)}{2}\sqrt{r}\), with the equality if and only if \(G\cong K_{n}\);
- (ii)
\(\mathit{LEL}(\mathcal{C}(G))>\sqrt{r+\frac{3}{2}+\frac {2nr}{(n-1)\sqrt{r+1}}} +(\frac{n(r-2)}{2}+1)\sqrt{r+2}+\frac{n(r-2)}{2}\sqrt{r}\).
We finally consider the case of total graphs.
Theorem 3.7
- (i)
\(\mathit{LEL}(\mathcal{T}(G))\leq n\sqrt{r+2}+\sqrt{2(n-1)nr}+\frac {n(r-2)}{2}\sqrt{2r+2}\), with the equality if and only if \(G\cong K_{n}\);
- (ii)
\(\mathit{LEL}(\mathcal{T}(G))>(n-1)\sqrt{4r+2}+\sqrt{r+2}+\frac {n(r-2)}{2}\sqrt{2r+2}\).
Proof
This completes the proof. □
Remark 3
4 The incidence energy
In this section, we investigate IE of line graph, subdivision graph, para-line graph and total graph of a regular graph.
Theorem 4.1
- (i)
(see [11]) \(\mathit{IE}(\mathcal{L}(G))\leq(n-1)\sqrt {\frac {3n-4}{n-1}r-4} +\sqrt{4r-4}+\frac{n(r-2)}{2}\sqrt{2r-4}\), with the equality if and only if \(G\cong K_{n}\);
- (ii)
\(\mathit{IE}(\mathcal{L}(G))>(n-1)\sqrt{\frac{5n-7}{2n-2}r-4} +\sqrt{4r-4}+\frac{n(r-2)}{2}\sqrt{2r-4}\).
Proof
Remark 4
Since the subdivision graph \(\mathcal{S}(G)\) of any graph G is bipartite, by Theorem 1.1 we have \(\mathit{IE}(\mathcal{S}(G))=\mathit {LEL}(\mathcal{S}(G))\). Consequently, the next theorem is obvious.
Theorem 4.2
- (i)
\(\mathit{IE}(\mathcal{S}(G))\leq(n-1)\sqrt{r+2+\frac{2 (\sqrt {r+1}+\sqrt {(n-2)(nr-r-1)} )}{n-1}} +\sqrt{r+2}+\frac{\sqrt{2}n(r-2)}{2}\), with the equality if and only if \(G\cong K_{n}\);
- (ii)
\(\mathit{IE}(\mathcal{S}(G))>(n-1)\sqrt{r+\frac{3}{2}+\frac {2nr}{(n-1)\sqrt{r+1}}} +\sqrt{r+2}+\frac{\sqrt{2}n(r-2)}{2}\).
Theorem 4.3
- (i)
\(\mathit{IE}(\mathcal{C}(G))\leq(n-1)\sqrt{3r-2+2\sqrt{2r^{2}-\frac{nr}{n-1}}} +\sqrt{2r}+\sqrt{r-2}+\frac{n(r-2)}{2} (\sqrt{r}+\sqrt{r-2} )\), with the equality if and only if \(G\cong K_{n}\);
- (ii)
\(\mathit{IE}(\mathcal{C}(G))>(n-1)\sqrt{3r-2-\frac{1}{4r}+2\sqrt {2r^{2}-\frac {nr}{n-1}-\frac{1}{4}}} +\sqrt{2r}+\sqrt{r-2}+\frac{n(r-2)}{2} (\sqrt{r}+ \sqrt{r-2} )\).
Proof
The proof is completed. □
Theorem 4.4
- (i)
\(\mathit{IE}(\mathcal{T}(G))\leq n\sqrt{3r-2}+\sqrt {2(n-1)(n-2)r}+\sqrt {4r}+\frac{n(r-2)}{2}\sqrt{2r-2}\), with the equality if and only if \(G\cong K_{n}\);
- (ii)
\(\mathit{IE}(\mathcal{T}(G))>(n-1)\sqrt{5r-2+2\sqrt{2}(r-1)}+\sqrt {4r}+\sqrt {3r-2}+\frac{n(r-2)}{2}\sqrt{2r-2}\).
Proof
This completes the proof. □
Declarations
Acknowledgements
The authors would like to thank the anonymous referees for their helpful comments and suggestions toward improving the original version of this paper. This work was supported in part by National Natural Science Foundation of China (Nos. 11501133, 11571101, 11461004), China Postdoctoral Science Foundation (No. 2015M572252), Guangxi Natural Science Foundation (No. 2014GXNSFBA118008) and the Scientific Research Fund of Guangxi Provincial Education Department (No. YB2014007).
Open Access This article is distributed under the terms of the Creative Commons Attribution 4.0 International License (http://creativecommons.org/licenses/by/4.0/), which permits unrestricted use, distribution, and reproduction in any medium, provided you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons license, and indicate if changes were made.
Authors’ Affiliations
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