Positive and negative inertia index of a graph

Abstract

In this paper, the positive and negative inertia index of trees, unicyclic graphs and bicyclic graphs are discussed, the methods of calculating them are obtained, and an inequality about the difference between positive and negative inertia index is proved. Moreover, a conjecture is proposed.

On graphs with exactly two positive eigenvalues

“…In the next section, we will characterize the graphs of G s (n) (2 ≤ s ≤ n − 3) based on those graphs of G s−1 (n − 1) by using the three types of graph transformations defined above. 4 The characterization of graphs in G s (n) for 2 ≤ s ≤ n−3…”

“…There have been diverse studies on the positive inertia index, negative inertia index and nullity of a graph, for instance to see [3][4][5][6] and [9,[11][12][13] and references therein. Let B and D be two real symmetric matrices of order n. Then D is called congruent to B if there is an real invertible matrix C such that D = C T BC.…”

“…[4]). Let G be a graph containing a pendant vertex, and let H be the induced subgraph of G obtained by deleting the pendant vertex together with the vertex adjacent to it.…”

On the rank of weighted graphs

“…Lemma 2.2: [29] If C w n is of Type A, then r(C w n ) = n − 2; otherwise, r(C w n ) = n. Lemma 2.3: [31] Let G w be a weighted graph on n ≥ 2 vertices with a pendent vertex v, and let u be the unique vertex of G w adjacent to v. Then r (G w …”

Bounds for the positive and negative inertia index of a graph

“…Recently, Yu et al [11] investigated the minimum positive inertia index among all bicyclic graphs of fixed order with pendant vertices, and characterized the bicyclic graphs with positive index 1 or 2. Ma et al [6] discussed the positive or the negative inertia index for a graph with at most three cycles, and proved that |p(G) − n(G)| ≤ c 1 (G) for any graph G, where c 1 (G) denotes the number of odd cycles contained in G. They conjectured that −c 3 (G) ≤ p(G) − n(G) ≤ c 5 (G), (1.1) where c 3 (G) and c 5 (G) denote the number of cycles having length 3 modulo 4 and length 1 modulo 4 respectively. In [10] we proved that the conjecture (1.1) holds for line graphs and power trees.…”