# On the Spectral Characterizations of Graphs

• Volume: 37, Issue: 3, page 729-744
• ISSN: 2083-5892

top

## Abstract

top
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 ${𝒰}_{p}^{2q}$ be the set of graphs obtained from Cp by attaching two pendant edges to each of q (q ⩽ p) vertices on Cp, whereas [...] p2q ${𝒱}_{p}^{2q}$ the subset of [...] p2q ${𝒰}_{p}^{2q}$ 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 ${𝒰}_{p}^{2q}$ , 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 ${𝒱}_{p}^{p-1}=\left\{H\right\}$ , then G ≅ H.

## How to cite

top

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 -

## NotesEmbed?

top

You must be logged in to post comments.

To embed these notes on your page include the following JavaScript code on your page where you want the notes to appear.

Only the controls for the widget will be shown in your chosen language. Notes will be shown in their authored language.

Tells the widget how many notes to show per page. You can cycle through additional notes using the next and previous controls.

Note: Best practice suggests putting the JavaScript code just before the closing </body> tag.