2024/05/15 by Saieed Akbari, Akbari, Saieed, Irena M. Jovanović +3 · 1 citation
Computer Science · Mathematics · #05C50 #05C90 #05C92 #Advanced Graph Theory Research #Combinatorics (math.CO) #FOS: Mathematics #Graph theory and applications #Interconnection Networks and Systems
paper · pdf · doi:10.48550/arxiv.2405.09093
openalex publication_date 2024/05/15 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
Let GS be the graph obtained by attaching a self-loop at every vertex in S ⊆ V(G) of a simple graph G of order n. In this paper, we explore several new results related to the line graph L(GS) of GS. Particularly, we show that every eigenvalue of L(GS) must be at least -2, and relate the characteristic polynomial of the line graph L(G) of G with the characteristic polynomial of the line graph L(\widehatG) of a self-loop graph \widehatG, which is obtained by attaching a self-loop at each vertex of G. Then, we provide some new bounds for the eigenvalues and energy of GS. As one of the consequences, we obtain that the energy of a connected regular complete multipartite graph is not greater than the energy of the corresponding self-loop graph. Lastly, we establish a lower bound of the spectral radius in terms of the first Zagreb index M1(G) and the minimum degree δ(G), as well as proving two Nordhaus-Gaddum-type bounds for the spectral radius and the energy of GS, respectively.