2 2 , k ) m ( G u ~ v n 2 2 1 , k ) = m [ G ( n ; n 2 2 , n 2 2 ) , k ] m [ G ( n ; n 2 2 1 , n 2 2 + 1 ) , k ] = m ( P n 2 2 P n 2 2 , k 1 ) m ( P n 2 2 1 P n 2 2 + 1 , k 1 )

By Lemma 4, P n 2 2 P n 2 2 P n 2 2 1 P n 2 2 + 1 . Thus for some k, we have m ( G u ~ v n 2 2 , k ) > m ( G u ~ v n 2 2 1 , k ) . This means that G u ~ v n 2 2 G u ~ v n 2 2 1 . Sum up all, for n is an odd, if n 2 2 is also odd, then G u ~ v 2 G u ~ v 4 G u ~ v n 2 2 1 Math_110#;

If n 2 2 is an even, then G u ~ v 2 G u ~ v 4 G u ~ v n 2 2 G u ~ v n 2 2 1 G u ~ v n 2 2 3 G u ~ v 3 G u ~ v 1 .

By Theorems 6 and 7, we immediately get our main result as follow.

Theorem 8. Let G u ~ v i ( i = 1 , 2 , , n 3 ) be a circum graphs with chord.

1) If n is an even, then

M E ( G u ~ v 1 ) < M E ( G u ~ v 3 ) < < M E ( G u ~ v n 2 2 ) < M E ( G u ~ v n 2 2 1 ) < M E ( G u ~ v n 2 2 3 ) < < M E ( G u ~ v 2 ) < M E ( G u ~ v 2 ) (7)

2) If n and n 2 2 are both odd, then

M E ( G u ~ v 1 ) < M E ( G u ~ v 3 ) < < M E ( G u ~ v n 2 2 2 ) < M E ( G u ~ v n 2 2 ) < M E ( G u ~ v n 2 2 1 ) < < M E ( G u ~ v 4 ) < M E ( G u ~ v 2 ) (8)

3) If n is an odd and n 2 2 is an even, then

M E ( G u ~ v 1 ) < M E ( G u ~ v 3 ) < < M E ( G u ~ v n 2 2 3 ) < M E ( G u ~ v n 2 2 1 ) < M E ( G u ~ v n 2 2 ) < < M E ( G u ~ v 4 ) < M E ( G u ~ v 2 ) (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 P 2 . Then the general case, determining the quasi-order relation on the matching energy for circum graph with one generalized chord P k for 2 k n 3 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. (2017) The Quasi-Order of Matching Energy of Circum Graph with Chord. Applied Mathematics, 8, 1180-1185. doi: 10.4236/am.2017.88088.
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