Integral Laplacian graphs with a unique repeated Laplacian eigenvalue, I
Integral Laplacian graphs with a unique repeated Laplacian eigenvalue, I
Abstract The set <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"><m:msub><m:mrow><m:mi>S</m:mi></m:mrow><m:mrow><m:mi>i</m:mi><m:mo>,</m:mo><m:mi>n</m:mi></m:mrow></m:msub><m:mo>=</m:mo><m:mrow><m:mo>{</m:mo><m:mrow><m:mn>0</m:mn><m:mo>,</m:mo><m:mn>1</m:mn><m:mo>,</m:mo><m:mn>2</m:mn><m:mo>,</m:mo><m:mrow><m:mo>…</m:mo></m:mrow><m:mo>,</m:mo><m:mi>n</m:mi><m:mo>−</m:mo><m:mn>1</m:mn><m:mo>,</m:mo><m:mi>n</m:mi></m:mrow><m:mo>}</m:mo></m:mrow><m:mo>\</m:mo><m:mrow><m:mo>{</m:mo><m:mrow><m:mi>i</m:mi></m:mrow><m:mo>}</m:mo></m:mrow></m:math> {S}_{i,n}=\left\{0,1,2,\ldots ,n-1,n\right\}\setminus \left\{i\right\} , <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"><m:mn>1</m:mn><m:mo>⩽</m:mo><m:mi>i</m:mi><m:mo>⩽</m:mo><m:mi>n</m:mi></m:math> 1\leqslant i\leqslant n , is called Laplacian realizable if there exists an undirected simple graph whose Laplacian spectrum is <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"><m:msub><m:mrow><m:mi>S</m:mi></m:mrow><m:mrow><m:mi>i</m:mi><m:mo>,</m:mo><m:mi>n</m:mi></m:mrow></m:msub></m:math> {S}_{i,n} . The existence of such graphs was established by Fallat et al. ( On graphs whose Laplacian …