부산대학교 도서관

The theoretical framework of the paper is spectral graph theory, more particularly, the spectral radius of graphs and its extensions. \par A signed graph $\Gamma = (G,\sigma)$ is a graph $G = (V,E)$ with vertex set $V$ and edge set $E$ together with a function $\sigma$ defined on the edge set $E$, $$ \sigma \: E \to \{-1, +1\}, $$ called the signature. The adjacency matrix $A_\sigma = (a_{ij} )$ of $(G, \sigma)$ is defined as $$ a_{ij}= \cases \sigma(ij) & \text{if }ij\in E,\\ 0 & \text{otherwise}. \endcases $$ The characteristic polynomial $\varphi((G, \sigma), \lambda) = \det(\lambda I - A_\sigma )$ is called the characteristic polynomial of $(G, \sigma)$. The spectrum of $A_\sigma$ is called the spectrum of the signed graph $(G,\sigma)$. The eigenvalues $\lambda_1((G, \sigma)) \geqslant \lambda_2((G, \sigma)) \geqslant \cdots \geqslant \lambda_n((G, \sigma))$ of the signed graph $(G, \sigma)$ of order $n$ are all real numbers because $A_\sigma$ is a real symmetric matrix. The largest eigenvalue $\lambda_1$ is often called the index. \par Let $\Gamma$ be a signed graph of order $n$, and $X = (x_1,\dots,x_n)^\top$ be a unit eigenvector corresponding to $\lambda_1(\Gamma)$. From the Rayleigh quotient we get the following: $$ \lambda_1(\Gamma)=X^\top A_\sigma X=2\sum_{uv\in E}x_u x_v\sigma(uv). $$ In this case $\lambda_1(\Gamma)$ might not be simple; and the eigenvector related to $\lambda_1(\Gamma)$ might have positive, zero and negative components. \par A signed cycle $(C_n,\sigma)$ is called positive (resp. negative) if it contains an even (resp. odd) number of negative edges. A signed graph is balanced if all of its cycles are positive; otherwise it is unbalanced. A unicyclic graph is a connected graph containing exactly one cycle. \par In this work the authors analyze how the index changes when modifications are made to a signed graph. Also, the problem of identifying the extremal graphs with respect to the index is solved for unbalanced unicyclic graphs and signed unicyclic graphs.