Unbalanced signed graphs with eigenvalue properties

Abstract

For a signature function $ \Psi:E({H}) \longrightarrow \{\pm 1\} $ with underlying graph $ H $, a signed graph (S.G) $ \hat{H} = (H, \Psi) $ is a graph in which edges are assigned the signs using the signature function $ \Psi $. An S.G $ \hat{H} $ is said to fulfill the symmetric eigenvalue property if for every eigenvalue $ \hat{h}(\hat{H}) $ of $ \hat{H} $, $ -\hat{h}(\hat{H}) $ is also an eigenvalue of $ \hat{H} $. A non singular S.G $ \hat{H} $ is said to fulfill the property $ (\mathcal{SR}) $ if for every eigenvalue $ \hat{h}(\hat{H}) $ of $ \hat{H} $, its reciprocal is also an eigenvalue of $ \hat{H} $ (with multiplicity as that of $ \hat{h}(\hat{H}) $). A non singular S.G $ \hat{H} $ is said to fulfill the property $ (-\mathcal{SR}) $ if for every eigenvalue $ \hat{h}(\hat{H}) $ of $ \hat{H} $, its negative reciprocal is also an eigenvalue of $ \hat{H} $ (with multiplicity as that of $ \hat{h}(\hat{H}) $). In this article, non bipartite unbalanced S.Gs $ \hat{\mathfrak{C}}^{(m, 1)}_{3} $ and $ \hat{\mathfrak{C}}^{(m, 2)}_{5} $, where $ m $ is even positive integer have been constructed and it has been shown that these graphs fulfill the symmetric eigenvalue property, the S.Gs $ \hat{\mathfrak{C}}^{(m, 1)}_{3} $ also fulfill the properties $ (-\mathcal{SR}) $ and $ (\mathcal{SR}) $, whereas the S.Gs $ \hat{\mathfrak{C}}^{(m, 2)}_{5} $ are close to fulfill the properties $ (-\mathcal{SR}) $ and $ (\mathcal{SR}) $.

Keywords

• signed graph (s.g)
• bipartite graph
• non bipartite graph
• adjacency matrix
• the symmetric eigenvalue property
• the strong reciprocal eigenvalue property of graph (the property $ (\mathcal{sr}) $)
• the strong anti-reciprocal eigenvalue property of graph (the property $ (-\mathcal{sr}) $)
• balanced and unbalanced signed graph