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Several matrices can be associated to a graph, such as the adjacency matrix or the Laplacian matrix. The spectrum of these matrices gives some informations about the structure of the graph and the question “Which graphs are determined by their spectrum?” is still a difficult problem in spectral graph theory. Let [...] p2q be the set of graphs obtained from Cp by attaching two pendant edges to each of q (q ⩽ p) vertices on Cp, whereas [...] p2q the subset of [...] p2q with odd p and its q vertices of degree 4 being nonadjacent to each other. In this paper, we show that each graph in [...] p2q , p even and its q vertices of degree 4 being consecutive, is determined by its Laplacian spectrum. As well we show that if G is a graph without isolated vertices and adjacency cospectral with the graph in [...] pp−1=H , then G ≅ H.
Jing Huang, and Shuchao Li. "On the Spectral Characterizations of Graphs." Discussiones Mathematicae Graph Theory 37.3 (2017): 729-744. <http://eudml.org/doc/288409>.
@article{JingHuang2017, abstract = {Several matrices can be associated to a graph, such as the adjacency matrix or the Laplacian matrix. The spectrum of these matrices gives some informations about the structure of the graph and the question “Which graphs are determined by their spectrum?” is still a difficult problem in spectral graph theory. Let [...] p2q $\{\mathcal \{U\}\}_p^\{2q\}$ be the set of graphs obtained from Cp by attaching two pendant edges to each of q (q ⩽ p) vertices on Cp, whereas [...] p2q $\{\mathcal \{V\}\}_p^\{2q\}$ the subset of [...] p2q $\{\mathcal \{U\}\}_p^\{2q\}$ with odd p and its q vertices of degree 4 being nonadjacent to each other. In this paper, we show that each graph in [...] p2q $\{\mathcal \{U\}\}_p^\{2q\}$ , p even and its q vertices of degree 4 being consecutive, is determined by its Laplacian spectrum. As well we show that if G is a graph without isolated vertices and adjacency cospectral with the graph in [...] pp−1=H $\{\mathcal \{V\}\}_p^\{p - 1\} = \lbrace H\rbrace $ , then G ≅ H.}, author = {Jing Huang, Shuchao Li}, journal = {Discussiones Mathematicae Graph Theory}, keywords = {Laplacian spectrum; adjacency spectrum; cospectral graphs; spectral characterization}, language = {eng}, number = {3}, pages = {729-744}, title = {On the Spectral Characterizations of Graphs}, url = {http://eudml.org/doc/288409}, volume = {37}, year = {2017}, }
TY - JOUR AU - Jing Huang AU - Shuchao Li TI - On the Spectral Characterizations of Graphs JO - Discussiones Mathematicae Graph Theory PY - 2017 VL - 37 IS - 3 SP - 729 EP - 744 AB - Several matrices can be associated to a graph, such as the adjacency matrix or the Laplacian matrix. The spectrum of these matrices gives some informations about the structure of the graph and the question “Which graphs are determined by their spectrum?” is still a difficult problem in spectral graph theory. Let [...] p2q ${\mathcal {U}}_p^{2q}$ be the set of graphs obtained from Cp by attaching two pendant edges to each of q (q ⩽ p) vertices on Cp, whereas [...] p2q ${\mathcal {V}}_p^{2q}$ the subset of [...] p2q ${\mathcal {U}}_p^{2q}$ with odd p and its q vertices of degree 4 being nonadjacent to each other. In this paper, we show that each graph in [...] p2q ${\mathcal {U}}_p^{2q}$ , p even and its q vertices of degree 4 being consecutive, is determined by its Laplacian spectrum. As well we show that if G is a graph without isolated vertices and adjacency cospectral with the graph in [...] pp−1=H ${\mathcal {V}}_p^{p - 1} = \lbrace H\rbrace $ , then G ≅ H. LA - eng KW - Laplacian spectrum; adjacency spectrum; cospectral graphs; spectral characterization UR - http://eudml.org/doc/288409 ER -
