Spectral deviations of graphs - Stanić Zoran (Special Matrices)
For the graphs GG and HH, the spectral deviation of HH from GG is defined as ϱG(H)=∑μ∈Hminλ∈G∣λ−μ∣,{ρ }_{G}\left(H)=\sum _{μ \in H}\mathop{\min }\limits_{λ \in G}| λ -μ | , where ∈\in designates that the given number is an eigenvalue of the adjacency matrix of the corresponding graph. In this study, we consider the problem of existence of a proper induced subgraph HH of a prescribed graph GG such that ϱG(H)=0{ρ }_{G}\left(H)=0, and the problem of determination of all such subgraphs. We investigate these problems in the framework of Smith graphs and their induced subgraphs, graphs with small second largest eigenvalue, graphs with small number of either positive or distinct eigenvalues, integral graphs, and chain graphs. Our results can be interesting in the context of graphs with a fixed number of distinct eigenvalues, eigenvalue distribution, or spectral distances of graphs.