부산대학교 도서관

The theoretical framework of the paper is spectral graph theory, moreparticularly, the spectral radius of graphs and its extensions.\par A signed graph $\Gamma = (G,\sigma)$ is a graph $G = (V,E)$ withvertex 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 thesigned 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 theeigenvector related to $\lambda_1(\Gamma)$ might have positive, zero andnegative components.\par A signed cycle $(C_n,\sigma)$ is called positive (resp. negative) ifit contains an even (resp. odd) number of negative edges. A signedgraph is balanced if all of its cycles are positive; otherwise it isunbalanced. A unicyclic graph is a connected graph containing exactlyone cycle.\par In this work the authors analyze how the index changes when modifications aremade to a signed graph. Also, the problem of identifying the extremalgraphs with respect to the index is solved for unbalanced unicyclicgraphs and signed unicyclic graphs.