Applied Mathematics
Vol.08 No.08(2017), Article ID:78730,6 pages
10.4236/am.2017.88088
The Quasi-Order of Matching Energy of Circum Graph with Chord
Ning Zhao, Yinkui Li
School of Mathematics and Statistics, Qinghai Nationalities University, Xining, China
Copyright © 2017 by authors and Scientific Research Publishing Inc.
This work is licensed under the Creative Commons Attribution International License (CC BY 4.0).
http://creativecommons.org/licenses/by/4.0/
Received: July 21, 2017; Accepted: August 22, 2017; Published: August 25, 2017
ABSTRACT
The matching energy of graph G is defined as , where be the roots of matching polynomial of graph G. In order to compare the energies of a pair of graphs, Gutman and Wager further put forward the concept of quasi-order relation. In this paper, we determine the quasi- order relation on the matching energy for circum graph with one chord.
Keywords:
Matching Polynomial, Matching Energy, Matching Root
1. Introduction
All graphs considered are finite, undirected, loopless and without multiple edges. The terminology and nomenclature of [1] will be used. Throughout this paper, G will denote a graph with vertex set and edge set . By denote the induced subgraph obtained from G by deleting vertex u together with its incident edges and by the edge-induced subgraph obtained from G by deleting edge e. As usual, use and to denote the path and cycle on n vertices, respectively.
Let be the number of k-matchings in graph G. The matching polynomial of a graph G is defined in [2] as
(1)
where and for all .
Gutman and Wager defined the quasi-order “ ” of two graphs G and H as follows:
If G and H have the matching polynomials in the form (1), then the quasi- order “ ” is defined by
(2)
In particular, if and for some k, then we write .
We call G and H matching-equivalent if both and hold and denoted by . Further, Gutman and Wagner introduced the concept of matching energy of a graph G in [3] and defined in different expressions as follows:
(3)
and
where be the roots of matching polynomial of graph G.
The matching energy of a graph G is an important index, which is widely used in the field of molecular orbital theory. There are many literatures about this parameter. See [4] - [14] .
By the above definitions, it is immediately to get
(4)
In fact, this property provide an important technique to determine the order relation of the matching energy for graphs. In this paper, we discuss the order of the matching energy for circum graph with chord.
Let be a cycle with order n, the circum graph with chord is obtained by adding one edge for some to , which is denoted by and simplified as .
2. Preliminaries
Lemma 1. [2] Let be an edge of graph G and . Then Math_39#.
By Lemma 1, it is easy to get
Lemma 2. Let e be an edge of graph G. Then .
Lemma 3. [15] Let u be a vertex of graph G. Then , where the summation goes over all vertices v adjacent to the vertex u.
Lemma 4. [16] Let for . Then .
By the definition of graph , we can immediately get
Lemma 5. for .
Proof. Since and , so we get .,
3. Main Results
Theorem 6. Let be a circum graph with chord and n is an even. Then .
Proof. First we consider the graph and . Since n is even, by Lemma 5, we only consider . By Lemma 3, we obtain that
(5)
Similarly,
(6)
Based on (5) and (6), we immediately get .
Case 1. s is even.
By Lemma 4, we can obtain that . Thus, for some k, there be . This means that
.
Case 2. is odd.
By Lemma 4, using a similar argument as in the previous proof we conclude that . Thus, for some k, there be . This imply that .
For graph and , we get that
Repeating the same argument as in the previous proof, combine the fact n is even, we have . Thus . Sum up all, we get . ,
Theorem 7. Let n is an odd number. Then
1) If is also odd, then ;
2) If is even, then .
Proof. First consider the graph and . Since n is odd, similar as Lemma 5, we only consider . By Lemma 1, we get .
Case 1. s is even.
By Lemma 4, we have . Thus, Math_87#.
If is odd, then .
If is even, then .
Case 2. s is odd.
By Lemma 4, we have . Thus, .
If is odd, then .
If is even, then .
Based on the above analysis, if is odd,
By lemma 4, . Thus for some k, we have . This means .
If is even, then
By Lemma 4, . Thus for some k, we have . This means that . Sum up all, for n is an odd, if is also odd, then Math_110#;
If is an even, then .
By Theorems 6 and 7, we immediately get our main result as follow.
Theorem 8. Let be a circum graphs with chord.
1) If n is an even, then
(7)
2) If n and are both odd, then
(8)
3) If n is an odd and is an even, then
(9)
4. Conclusions and Suggestions
In this paper, we determine the quasi-order relation on the matching energy for circum graph with one chord. If the chord here can be see . Then the general case, determining the quasi-order relation on the matching energy for circum graph with one generalized chord for is more meaningful.
Acknowledgements
Sincere thanks to the members of JAMP for their professional performance, and special thanks to managing editor for a rare attitude of high quality. This research supported by NSFC (11561056, 11661066) and QHAFP (2017-ZJ-701).
Cite this paper
Zhao, N. and Li, Y.K. (2017) The Quasi-Order of Matching Energy of Circum Graph with Chord. Applied Mathematics, 8, 1180-1185. https://doi.org/10.4236/am.2017.88088
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