Trees with maximum nullity

Abstract

The nullity of a graph is the multiplicity of the eigenvalue zero in its spectrum. Among nvertex trees, the star has greatest nullity (equal to n − 2). We generalize this by showing that among n-vertex trees whose vertex degrees do not exceed a certain value D, the greatest nullity is n − 2 (n − 1)/D . Methods for constructing such maximum-nullity trees are described.

A note on the nullity of unicyclic signed graphs

“…More results on the nullity (or rank) of graphs can be found in the book [28] and the paper [31]. For the nullity of some special classes of graphs, one can refer to the papers [10,12,[14][15][16]20,[23][24][25]27,32,34].…”

Positive and negative inertia index of a graph

“…From Table 1 we can obtain the following lemma. n (∞(p, l, q) 6) or (5,5), 3), (3,5) or (4,5), 3), (3,6), (5,5) or (5, 6), 1, 4) or (4,5), 5) or (6,6), 6). Proof.…”

“…There have been diverse studies on the nullity of a graph [1][2][3][5][6][7][8][10][11][12][13][14]; it is related to the stability of molecular represented by the graph. However, there is very little literature on positive and negative inertia index of a graph.…”

“…For 1 l 5, p(∞(p, l, q)) + 1, (p, q) = (3, 3), l 2 + 2, (p, q) = (3, 4) or (4, 4), l 2 + 3, (p, q) = (3, 5), l 2 + 3, (p, q) = (3,6),(4,5) or (4, 6), l 2 + 5, (p, q) = (5, 5),(5,6) or(6,6).…”

On bipartite graphs which attain minimum rank among bipartite graphs with given diameter

“…Recently Chang, Huang, and Yeh characterized the graphs with rank 4 in [4] and also the graphs with rank 5 in [5]. Other works on the rank or nullity of graphs can be found in [2,12,14,15,16,17,18,19,20].…”