Spectra of graphs with loops with respect to some unary graph operations and beyond
Abstract
Characteristic polynomials of certain self-loop graphs of a given graph with respect to some unary graph operations are derived by use of the relations connected with
the incidence matrix and its transpose of both simple graphs and graphs with loops. The rank of the incidence matrix of a self-loop graph of a connected graph is determined, and it is proved that the multiplicity of the eigenvalue $-2$ in the spectrum of the line graph $L(G_S)$ of the self-loop graph $G_S$ of a connected graph $G$ is equal to the multiplicity of the same eigenvalue in the spectrum of the line graph $L(G)$ of the graph $G$. The nullity of certain connected bipartite graphs with pendant vertices is determined, and some structural perturbations of these graphs, which preserve the nullity, or which increase (decrease) it by 2, are considered. A part of the presented results is applied for considering some other problems in the domain of graph theory. Precisely, an example of self-loop graphs whose energy is strictly greater than the energy of their underlying bipartite graphs is found, in which way a recently posed conjecture is partially confirmed; an alternative proof to the existing proofs in the literature, which concerns the nullity set of the set of all connected bipartite graphs, is proposed; and finally, based on computational results for trees up to $9$ vertices, it is conjectured in which way all trees of the given order and nullity can be constructed.
the incidence matrix and its transpose of both simple graphs and graphs with loops. The rank of the incidence matrix of a self-loop graph of a connected graph is determined, and it is proved that the multiplicity of the eigenvalue $-2$ in the spectrum of the line graph $L(G_S)$ of the self-loop graph $G_S$ of a connected graph $G$ is equal to the multiplicity of the same eigenvalue in the spectrum of the line graph $L(G)$ of the graph $G$. The nullity of certain connected bipartite graphs with pendant vertices is determined, and some structural perturbations of these graphs, which preserve the nullity, or which increase (decrease) it by 2, are considered. A part of the presented results is applied for considering some other problems in the domain of graph theory. Precisely, an example of self-loop graphs whose energy is strictly greater than the energy of their underlying bipartite graphs is found, in which way a recently posed conjecture is partially confirmed; an alternative proof to the existing proofs in the literature, which concerns the nullity set of the set of all connected bipartite graphs, is proposed; and finally, based on computational results for trees up to $9$ vertices, it is conjectured in which way all trees of the given order and nullity can be constructed.
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