χ ( G ) ) , η ( G ) = v χ ( G ) [ η ( G { v } ) + η ( G G { v } + v ) ] = η ( G χ ( G ) ) + η ( χ ( G ) ) . (11)

So the conclusion is proved.

Theorem 2 For a given triccyle graph G γ , then

c 3 ( G ) p ( G ) n ( G ) c 5 ( G ) . (12)

Proof. Let the kernel of tricycle graph be χ ( G ) .

1) If the graph G is Type I, and the vertex v on χ ( G ) such that v matched in G { v } , then

p ( G ) = p ( G { v } ) + p ( G G { v } ) , n ( G ) = n ( G { v } ) + n ( G G { v } ) . (13)

And G { v } is a tree, G G { v } is the union of trees, unicyclic graphs and bicyclic graphs . By Lemma 6, it's true for G { v } and G G { v } , so it's true for tricycle graph G.

2) If graph G is Type II, by Theorem 1,

p ( G ) = p ( G χ ( G ) ) + p ( χ ( G ) ) , n ( G ) = n ( G χ ( G ) ) + n ( χ ( G ) ) . (14)

Because of G χ ( G ) is a forest. So according to the Lemma 3, p ( G χ ( G ) ) = n ( G χ ( G ) ) , so p ( G ) n ( G ) = p ( χ ( G ) ) n ( χ ( G ) ) , hence the conclusion is confirmed by Lemma 7 and Table 1.

4. Conclusion

A new calculation method of the inertia indexes of one special kind of tricyclic graphs with large vertices is given, and the inertia indexes of this tricyclic graphs with fewer vertices can be calculated by Matlab.

Funding

This work is supported by National Natural Science Foundation of China (1561056, 11661066), National Natural Science Foundation of Qinghai Provence (2016-ZJ-914), and Scientific Research Fund of Qinghai University for Nationalities (2015G02).

Cite this paper
Ma, H. and Xie, C. (2019) The Inertia Indexes of One Special Kind of Tricyclic Graphs. Applied Mathematics, 10, 11-18. doi: 10.4236/am.2019.101002.
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