@article{Ismail2023, 
author = {Rashad Ismail and Saira Hameed and Uzma Ahmad and Khadija Majeed and Muhammad Javaid},
title = {Unbalanced signed graphs with eigenvalue properties},
year = {2023},
journal = {AIMS Mathematics},
volume = {8},
number = {10},
pages = {24751-24763},
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 (SR)), the strong anti-reciprocal eigenvalue property of graph (the property −SR)), balanced and unbalanced signed graph},
url = {https://www.sciopen.com/article/10.3934/math.20231262},
doi = {10.3934/math.20231262},
abstract = {For a signature function    Ψ  :  E  (      H    )  ⟶  {  ±  1  } with underlying graph    H, a signed graph (S.G)              H      ^        =  (  H  ,  Ψ  ) is a graph in which edges are assigned the signs using the signature function    Ψ. An S.G              H      ^       is said to fulfill the symmetric eigenvalue property if for every eigenvalue              h      ^        (            H      ^        ) of              H      ^      ,    −            h      ^        (            H      ^        ) is also an eigenvalue of              H      ^      . A non singular S.G              H      ^       is said to fulfill the property    (      S    R    ) if for every eigenvalue              h      ^        (            H      ^        ) of              H      ^      , its reciprocal is also an eigenvalue of              H      ^       (with multiplicity as that of              h      ^        (            H      ^        )). A non singular S.G              H      ^       is said to fulfill the property    (  −      S    R    ) if for every eigenvalue              h      ^        (            H      ^        ) of              H      ^      , its negative reciprocal is also an eigenvalue of              H      ^       (with multiplicity as that of              h      ^        (            H      ^        )). In this article, non bipartite unbalanced S.Gs                                C                ^                    3              (      m      ,      1      )       and                                C                ^                    5              (      m      ,      2      )      , 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                                C                ^                    3              (      m      ,      1      )       also fulfill the properties    (  −      S    R    ) and    (      S    R    ), whereas the S.Gs                                C                ^                    5              (      m      ,      2      )       are close to fulfill the properties    (  −      S    R    ) and    (      S    R    ).}
}