Sesión Matemática DiscretaAbout of the determinant of graphs with a unique maximum matching.
Diego Gabriel Martinez
Departamento de Matemáticas, Universidad Nacional de San Luis -- Instituto de Matemáticas Aplicadas de San Luis -- CONICET, Argentina, Argentina - Esta dirección de correo electrónico está siendo protegida contra los robots de spam. Necesita tener JavaScript habilitado para poder verlo.
The structure of graphs with a unique perfect matching - UPM graphs-, was studied by Kotzig in 1959 (see [1]). His mayor result was that every connected UPM graph has a bridge that belongs to the perfect matching. This result was strengthen by Wang, Shang and Yuan in 2015 via the Gallai-Edmonds Structure Theorem (see [2]). In this work we prove that if \(G\) is a KE and a UPM graph, then \(det(G)=(-1)^{\mu(G)}\), where \(\mu(G)\) is the matching number of \(G\). The FP-KE decomposition applied to UPM graph give us the following result: if \(G\) is a UPM graph, then \[ \det(G)=(-1)^{\mu(\text{KE}(G))}\det(\text{FP}(G)). \] Hence, if \(G\) is a UPM graph, then \(\det(G)=1\mod 2\).
Trabajo en conjunto con: Daniel A. Jaume(Departamento de Matemáticas, Universidad Nacional de San Luis -- Instituto de Matemáticas Aplicadas de San Luis -- CONICET, Argentina), Gonzalo Molina (Departamento de Matemáticas, Universidad Nacional de San Luis -- Instituto de Matem\'{a}ticas Aplicadas de San Luis -- CONICET, Argentina) y Cristian Panelo (Departamento de Matemáticas, Universidad Nacional de San Luis, Argentina)..
Referencias
[1] A. Kotzig. On the theory of finite graphs with linear factor II. Mat.- Fyz. Casopis. Slovensk. Akad. Vied, 9(3)(1959), p. 136-159
[2] Xiumei Wang, Weiping Shang, Jinjiang Yuan. On Graphs with Unique perfect Matching. Graphs ans Combinatorics(2015)