Some signed graphs whose eigenvalues are main

Abstract: Let

G

be a graph. For a subset

X

of

V (G)

, the switching

s i g m a

of

G

is the signed graph

G^{s i g m a}

obtained from

G

by reversing the signs of all edges between

X

and

V (G) s e t m i n u s X

. Let

A (G^{s i g m a})

be the adjacency matrix of

G^{s i g m a}

. An eigenvalue of

A (G^{s i g m a})

is called a main eigenvalue if it has an eigenvector the sum of whose entries is not equal to zero. Let

S_{n, k}

be the graph obtained from the complete graph

K_{n - r}

by attaching

r

pendent edges at some vertex of

K_{n - r}

. In this paper we prove that there exists a switching

s i g m a

such that all eigenvalues of

G^{s i g m a}

are main when

G

is a complete multipartite graph, or

G

is a harmonic tree, or

G

is

S_{n, k}

. These results partly confirm a conjecture of Akbari et al.

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