Article ID | Journal | Published Year | Pages | File Type |
---|---|---|---|---|
4599563 | Linear Algebra and its Applications | 2014 | 15 Pages |
Abstract
Let X be a subset of {±α±β:α,βâBandαâ β} where B is an orthonormal set in an inner product space over R, such that xâXââxâX. Then the signed graph which is defined as described below is called a derived signed graph: its vertex set is X; two vertices x, y are joined by a positive (negative) edge when ãx,yã is positive (negative); when ãx,yã=0, x, y are not joined. Let D denote the family of all derived signed graphs-the order of a member of D may be infinite. (The family of all generalized line graphs-line graphs belong to this family-is a subfamily of D.) Let M be the class of all minimal nonderivable signed graphs. [M includes the 31 (finite) minimal nongeneralized line graphs computed by various methods in the literature.] In this article, we characterize D, determine M and classify the family of all signed graphs S for which, the following holds: for each finite subset X of V(S), the least eigenvalue of S[X] is at least â2. The third result substantially generalizes the well known result (Cameron et al. (1976) [1]) on classifying the family of all finite (signed) graphs with least eigenvalues ⩾â2.
Related Topics
Physical Sciences and Engineering
Mathematics
Algebra and Number Theory
Authors
G.R. Vijayakumar,