Abstract
Abstract With this article in mind, we have found some results using eigenvalues of graph with sign. It is intriguing to note that these results help us to find the determinant of Normalized Laplacian matrix of signed graph and their coe cients of characteristic polynomial using the number of vertices. Also we found bounds for the lowest value of eigenvalue.
Highlights
The readers should refer to [6] for expression and notations of graph theory and only simple and finite graphs are considered
A Signed graph Γ = (G(V, E), ∇) is a graph with positive and negative signs in every edge, where G is the underlined graph without signs and ∇ is the function from the collection of edges E to the set having positive and negative signs
One of the main applications of signed graphs is to represent the relationship among people where we assign a positive sign if the relationship between two individuals is pleasant, otherwise we assign a negative sign. [10] & [5]
Summary
The readers should refer to [6] for expression and notations of graph theory and only simple and finite graphs are considered. A graph that has been marked Γν is a signed graph with positive or negative signs assigned to its vertices. A signed graph Γ2 is obtain from a signed graph Γ1 by reversing the sign of edges of Γ1 whose end vertices are having opposite sign, and their underlined graphs G1 and G2 are isomorphic. In a Laplacian matrix L(Γ), if vertices vi and vj are adjacent the entry aij is 1 with the opposite sign of corresponding adjacent edge vivj, otherwise aij is zero and the diagonal entries aii being the degree of the vertex. (Γ, −) is a signed graph in which each edge is assigned by minus sign and L(Γ, −) is the Laplacian matrix of (Γ, −). Eigenvalues of Laplacian matrix of a signed graph are λ1 ≥ λ2 ≥ λ3 .... ≥ λn
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