^{1}

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Let
G
be a graph and A=(a_{ij})n×n
be the adjacency matrix of G, the eigenvalue
s
of A are said to be the eigenvalues of the graph G, and to form the spectrum of this graph. The number
s
of positive, negative and zero eigenvalues in the spectrum of the graph G are called positive and negative inertia indexes and nullity of the graph G, are denoted by p(G), n(G), η(G),
respectively, and are collectively called inertia indexes of the graph G. The inertia indexes ha
ve
many important applications in chemistry and mathematics. The purpose of the research of this paper is
to
calculat
e
the inertia indexes of one special kind of tricyclic graphs. A new calculation method of the inertia indexes of this tricyclic graphs with large vertices is given, and the inertia indexes of this tricyclic graphs with fewer vertices can be calculated by Matlab.

Throughout the paper, graphs are simple, without loops and multiple edges. Let G = ( V , E ) be a simple graph of order n with vertex set V = { v 1 , v 2 , ⋯ , v n } and edge set E = E ( G ) . The adjacency matrix A = ( a i j ) n × n of G is defined as follows: a i j = 1 if v i is adjacent to v j , and a i j = 0 otherwise. The eigenvalues λ 1 , λ 2 , ⋯ , λ n of A are said to be the eigenvalues of the graph G, and to form the spectrum of this graph. The numbers of positive, negative and zero eigenvalues in the spectrum of the graph G are called positive and negative inertia indexes and nullity of the graph G, and are denoted by p ( G ) , n ( G ) , η ( G ) , respectively. Obviously p ( G ) + n ( G ) + η ( G ) = n . The positive and negative inertia indexes and nullity of the graph G are collectively called inertia indexes of the graph G.

Let G be a graph, then the nullity η ( G ) of G has many important applications in chemistry [

Let G be a connected graph of order n, the graphs respectively with size n − 1 , n, n + 1 and n + 2 are called the tree, the unicycle graph, the bicycle graph and tricyclic graph. U ⊆ V ( G ) , the vertex-induced subgraph G [ U ] is the subgraph of G whose vertex set is U and whose edge set consists of all edges of G which have both ends in U. G \ U denotes the graph G [ V ( G ) \ U ] . Let G and H be two vertex disjoint graphs; G ∪ H denotes the union graph of G and H. As shown in

Lemma 1 [

p ( G ) = ∑ i = 1 t p ( G i ) , n ( G ) = ∑ i = 1 t n ( G i ) , η ( G ) = ∑ i = 1 t η ( G i ) . (1)

Lemma 2 [

p ( G ) = p ( H ) + 2 , n ( G ) = n ( H ) + 2 , η ( G ) = η ( H ) . (2)

A matching of G is a collection of independent edges of G. A maximal matching is a matching with the maximum possible number of edges. A matching M of G is called perfect if every vertex of G is incident with (exactly) one edge in M. Obviously, a perfect matching is a maximum matching. The size of a maximal matching of G, i.e., the maximum number of independent edges in G, is called the matching number of G, denoted by μ ( G ) .

Lemma 3 [

For a tree T on at least two vertices, a vertex v ∈ T is called mismatched in T if there exists a maximum matching M of T that does not cover v; otherwise, v is called matched in T. If a tree consists of only one vertex, then this vertex is considered as mismatched. Let G 1 be a graph containing a vertex u, and let G 2 be a graph of order n that is disjoint from G 1 . For 1 ≤ k ≤ n , the k-joining graph of G 1 and G 2 with respect to u is obtained from G 1 ∪ G 2 by joining u and any k vertices of G 2 ; we denote it by G 1 ( u ) ⊙ k G 2 . Note that the graph G 1 ( u ) ⊙ k G 2 is not uniquely determined when n > k .

Lemma 4 [

p ( T ( u ) ⊙ k G ) = p ( T ) + p ( G ) , n ( T ( u ) ⊙ k G ) = n ( T ) + n ( G ) , η ( T ( u ) ⊙ k G ) = η ( T ) + η ( G ) . (3)

Lemma 5 [

p ( T ( u ) ⊙ k G ) = p ( T − u ) + p ( G + u ) = p ( T ) + p ( G + u ) , n ( T ( u ) ⊙ k G ) = n ( T − u ) + n ( G + u ) = n ( T ) + n ( G + u ) , η ( T ( u ) ⊙ k G ) = η ( T − u ) + η ( G + u ) = η ( T ) + η ( G + u ) . (4)

Let G be a graph. The number of all odd cycles in G is denoted by c 1 ( G ) . The number of all cycles of length 4k + 3 in G is denoted by c 3 ( G ) ; the number of all cycles of length 4k + 5 in G is denoted by c 5 ( G ) . Obviously, c 3 ( G ) + c 5 ( G ) = c 1 ( G ) .

Lemma 6 [

− c 3 ( G ) ≤ p ( G ) − n ( G ) ≤ c 5 ( G ) . (5)

Lemma 7 Let m ≥ 3 , n ≥ 3 , p ≥ 2 , q ≥ 3 and m = 4 k + m ′ ( 3 ≤ m ′ ≤ 6 ) , n = 4 s + n ′ ( 3 ≤ n ′ ≤ 6 ) , p = 4 t + p ′ ( 2 ≤ p ′ ≤ 5 ) , q = 4 y + q ′ ( 3 ≤ q ′ ≤ 6 ) . Then

p ( ζ ( m , n , p , q ) ) = 2 ( k + s + t + y ) + p ( ζ ( m ′ , n ′ , p ′ , q ′ ) ) , n ( ζ ( m , n , p , q ) ) = 2 ( k + s + t + y ) + n ( ζ ( m ′ , n ′ , p ′ , q ′ ) ) , η ( ζ ( m , n , p , q ) ) = 2 ( k + s + t + y ) + η ( ζ ( m ′ , n ′ , p ′ , q ′ ) ) . (6)

Proof. It can be obtained by Lemma 2. □

For graph ζ ( m , n , p , q ) with fewer vertices, ( 3 ≤ m , n , q ≤ 6 , 2 ≤ p ≤ 5 , m ≤ n ). We use Matlab to calculate their the positive and negative inertia indexes and nullity (

For a given tricycle graph G ∈ γ , the kernel of graph G is χ ( G ) , for each vertex v ∈ V ( χ ( G ) ) , let G { v } be the induced connected subgraph of G with the maximum possible number of vertices, which contains the vertex v and contains no other vertices of χ ( G ) . Then for all vertices v ∈ V ( χ ( G ) ) , G { v } is a tree and G is obtained by identifying the vertex v of G { v } with the vertex v on χ ( G ) . The tricycle graph G is called of Type I, if there exists a vertex v on χ ( G ) such that v is matched in G { v } ; otherwise, G is called of Type II.

Theorem 1 For a given tricycle graph G ∈ γ , the kernel of graph G is χ ( G ) .

1) If G is type I and the v ∈ V ( χ ( G ) ) is matched in G { v } , then

p ( G ) = p ( G { v } ) + p ( G − G { v } ) , n ( G ) = n ( G { v } ) + n ( G − G { v } ) , η ( G ) = η ( G { v } ) + η ( G − G { v } ) . (7)

And G { v } is a tree, G − G { v } is the union of trees, unicyclic graphs and bicyclic graphs.

2) If G is of type II, then

G | p | n | η | G | p | n | η | G | p | n | η | G | p | n | η | |||
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|

(3,3,2,3) | 3 | 5 | 0 | (3,5,4,3) | 6 | 6 | 0 | (4,5,2,3) | 5 | 6 | 1 | (5,5,4,3) | 7 | 7 | 0 | |||

(3,3,2,4) | 4 | 4 | 1 | (3,5,4,4) | 6 | 6 | 1 | (4,5,2,4) | 5 | 6 | 2 | (5,5,4,4) | 7 | 7 | 1 | |||

(3,3,2,5) | 5 | 5 | 0 | (3,5,4,5) | 7 | 7 | 0 | (4,5,2,5) | 7 | 7 | 0 | (5,5,4,5) | 9 | 7 | 0 | |||

(3,3,2,6) | 5 | 6 | 0 | (3,5,4,6) | 7 | 8 | 0 | (4,5,2,6) | 7 | 7 | 1 | (5,5,4,6) | 9 | 8 | 0 | |||

(3,3,3,3) | 4 | 5 | 0 | (3,5,5,3) | 6 | 7 | 0 | (4,5,3,3) | 6 | 6 | 1 | (5,5,5,3) | 8 | 7 | 0 | |||

(3,3,3,4) | 4 | 5 | 1 | (3,5,5,4) | 6 | 7 | 1 | (4,5,3,4) | 6 | 6 | 2 | (5,5,5,4) | 8 | 7 | 1 | |||

(3,3,3,5) | 5 | 6 | 0 | (3,5,5,5) | 8 | 7 | 0 | (4,5,3,5) | 7 | 7 | 1 | (5,5,5,5) | 9 | 8 | 0 | |||

(3,3,3,6) | 6 | 6 | 0 | (3,5,5,6) | 8 | 8 | 0 | (4,5,3,6) | 7 | 7 | 2 | (5,5,5,6) | 9 | 9 | 0 | |||

(3,3,4,3) | 4 | 6 | 0 | (3,6,2,3) | 5 | 6 | 0 | (4,5,4,3) | 6 | 8 | 0 | (5,6,2,3) | 6 | 7 | 0 | |||

(3,3,4,4) | 5 | 5 | 1 | (3,6,2,4) | 5 | 6 | 1 | (4,5,4,4) | 6 | 6 | 3 | (5,6,2,4) | 6 | 6 | 2 | |||

(3,3,4,5) | 6 | 6 | 0 | (3,6,2,5) | 7 | 6 | 0 | (4,5,4,5) | 8 | 7 | 1 | (5,6,2,5) | 8 | 7 | 0 | |||

(3,3,4,6) | 6 | 7 | 0 | (3,6,2,6) | 7 | 7 | 0 | (4,5,4,6) | 8 | 8 | 1 | (5,6,2,6) | 8 | 8 | 0 | |||

(3,3,5,3) | 5 | 6 | 0 | (3,6,3,3) | 6 | 6 | 0 | (4,5,5,3) | 7 | 7 | 1 | (5,6,3,3) | 7 | 7 | 0 | |||

(3,3,5,4) | 5 | 6 | 1 | (3,6,3,4) | 6 | 6 | 1 | (4,5,5,4) | 7 | 7 | 2 | (5,6,3,4) | 7 | 7 | 1 | |||

(3,3,5,5) | 6 | 7 | 0 | (3,6,3,5) | 8 | 6 | 0 | (4,5,5,5) | 8 | 8 | 1 | (5,6,3,5) | 8 | 8 | 0 | |||

(3,3,5,6) | 7 | 7 | 0 | (3,6,3,6) | 8 | 7 | 0 | (4,5,5,6) | 8 | 9 | 1 | (5,6,3,6) | 8 | 8 | 1 | |||

(3,4,2,3) | 3 | 5 | 1 | (3,6,4,3) | 6 | 7 | 0 | (4,6,2,3) | 5 | 6 | 1 | (5,6,4,3) | 7 | 8 | 0 | |||

(3,4,2,4) | 4 | 5 | 1 | (3,6,4,4) | 6 | 7 | 1 | (4,6,2,4) | 5 | 6 | 2 | (5,6,4,4) | 7 | 8 | 1 | |||

(3,4,2,5) | 5 | 6 | 0 | (3,6,4,5) | 8 | 7 | 0 | (4,6,2,5) | 7 | 7 | 0 | (5,6,4,5) | 9 | 8 | 0 | |||

(3,4,2,6) | 5 | 6 | 1 | (3,6,4,6) | 8 | 8 | 0 | (4,6,2,6) | 7 | 7 | 1 | (5,6,4,6) | 9 | 9 | 0 | |||

(3,4,3,3) | 5 | 5 | 0 | (3,6,5,3) | 7 | 7 | 0 | (4,6,3,3) | 6 | 6 | 1 | (5,6,5,3) | 8 | 8 | 0 | |||

(3,4,3,4) | 5 | 4 | 2 | (3,6,5,4) | 7 | 7 | 1 | (4,6,3,4) | 6 | 6 | 2 | (5,6,5,4) | 8 | 8 | 1 | |||

(3,4,3,5) | 6 | 6 | 0 | (3,6,5,5) | 8 | 8 | 0 | (4,6,3,5) | 7 | 7 | 1 | (5,6,5,5) | 9 | 9 | 0 | |||

(3,4,3,6) | 6 | 6 | 1 | (3,6,5,6) | 8 | 9 | 0 | (4,6,3,6) | 7 | 7 | 2 | (5,6,5,6) | 9 | 10 | 0 | |||

(3,4,4,3) | 4 | 6 | 1 | (4,4,2,3) | 4 | 5 | 1 | (4,6,4,3) | 6 | 7 | 1 | (6,6,2,3) | 6 | 8 | 0 | |||

(3,4,4,4) | 5 | 6 | 1 | (4,4,2,4) | 4 | 4 | 3 | (4,6,4,4) | 6 | 7 | 2 | (6,6,2,4) | 6 | 8 | 1 | |||

(3,4,4,5) | 6 | 6 | 1 | (4,4,2,5) | 5 | 6 | 1 | (4,6,4,5) | 8 | 7 | 1 | (6,6,2,5) | 8 | 8 | 0 | |||

(3,4,4,6) | 6 | 7 | 1 | (4,4,2,6) | 5 | 7 | 1 | (4,6,4,6) | 8 | 8 | 1 | (6,6,2,6) | 8 | 9 | 0 | |||

(3,4,5,3) | 5 | 6 | 1 | (4,4,3,3) | 4 | 7 | 0 | (4,6,5,3) | 7 | 7 | 1 | (6,6,3,3) | 7 | 7 | 1 | |||

(3,4,5,4) | 5 | 6 | 2 | (4,4,3,4) | 4 | 5 | 3 | (4,6,5,4) | 7 | 7 | 2 | (6,6,3,4) | 7 | 8 | 1 | |||

(3,4,5,5) | 7 | 7 | 0 | (4,4,3,5) | 6 | 6 | 1 | (4,6,5,5) | 8 | 8 | 1 | (6,6,3,5) | 8 | 9 | 0 | |||

(3,4,5,6) | 6 | 6 | 3 | (4,4,3,6) | 6 | 7 | 1 | (4,6,5,6) | 8 | 9 | 1 | (6,6,3,6) | 8 | 10 | 0 | |||

(3,5,2,3) | 5 | 5 | 0 | (4,4,4,3) | 5 | 5 | 2 | (5,5,2,3) | 6 | 8 | 0 | (6,6,4,3) | 7 | 8 | 1 | |||

(3,5,2,4) | 5 | 5 | 1 | (4,4,4,4) | 6 | 5 | 3 | (5,5,2,4) | 6 | 8 | 1 | (6,6,4,4) | 7 | 8 | 2 | |||

(3,5,2,5) | 6 | 5 | 1 | (4,4,4,5) | 6 | 6 | 2 | (5,5,2,5) | 8 | 8 | 0 | (6,6,4,5) | 9 | 9 | 0 | |||

(3,5,2,6) | 6 | 6 | 1 | (4,4,4,6) | 6 | 8 | 1 | (5,5,2,6) | 8 | 9 | 0 | (6,6,4,6) | 9 | 10 | 0 | |||

(3,5,3,3) | 5 | 6 | 0 | (4,4,5,3) | 5 | 8 | 0 | (5,5,3,3) | 7 | 7 | 1 | (6,6,5,3) | 8 | 8 | 1 | |||

(3,5,3,4) | 5 | 5 | 2 | (4,4,5,4) | 5 | 6 | 3 | (5,5,3,4) | 7 | 8 | 1 | (6,6,5,4) | 8 | 9 | 1 | |||

(3,5,3,5) | 7 | 6 | 0 | (4,4,5,5) | 7 | 8 | 0 | (5,5,3,5) | 8 | 9 | 0 | (6,6,5,5) | 9 | 10 | 0 | |||

(3,5,3,6) | 7 | 7 | 0 | (4,4,5,6) | 7 | 8 | 1 | (5,5,3,6) | 8 | 10 | 0 | (6,6,5,6) | 9 | 9 | 2 |

p ( G ) = p ( G − χ ( G ) ) + p ( χ ( G ) ) , n ( G ) = n ( G − χ ( G ) ) + n ( χ ( G ) ) , η ( G ) = η ( G − χ ( G ) ) + η ( χ ( G ) ) . (8)

Proof. 1) If G is of type I and v ∈ V ( χ ( G ) ) is matched in G { v } , then there will exist a positive integer k ∈ { 2,3,5 } such that . By Lemma 4,

p ( G ) = p ( G { v } ) + p ( G − G { v } ) , n ( G ) = n ( G { v } ) + n ( G − G { v } ) , η ( G ) = η ( G { v } ) + η ( G − G { v } ) . (9)

And G { v } is a tree, G − G { v } is the union of trees, unicyclic graphs and bicyclic graphs.

2) If G is of Type II, suppose G { v } contains vertices other than v, since v is mismatched in G { v } , by Lemma 5,

p ( G ) = p ( G { v } ) + p ( G − G { v } + v ) , n ( G ) = n ( G { v } ) + n ( G − G { v } + v ) , η ( G ) = η ( G { v } ) + η ( G − G { v } + v ) . (10)

Applying Lemma 5 and Lemma 1 repeatedly, we have

p ( G ) = ∑ v ∈ χ ( G ) [ p ( G { v } ) + p ( G − G { v } + v ) ] = p ( G − χ ( G ) ) + p ( χ ( G ) ) , n ( G ) = ∑ v ∈ χ ( G ) [ n ( G { v } ) + n ( G − G { v } + v ) ] = n ( G − χ ( G ) ) + n ( 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

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.

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).

The authors declare no conflicts of interest regarding the publication of this paper.

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