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Escribir la matriz de incidencia de los digrafos G4 y G5. +G4 +\begin{bmatrix} + 0& 0& 0& 1& 1& 0& 0\\ + 0& 0& 0& 1& 0& 1& 0\\ + 0& 0& 0& 0& 1& 1& 0\\ + 0& 0& 1& 0& 1& 0& 0\\ + 0& 1& 0& 0& 0& 1& 0\\ + 0& 1& 1& 0& 0& 0& 0\\ + 1& 0& 0& 0& 0& 1& 0\\ + 1& 1& 0& 0& 0& 0& 0\\ + 0& 1& 0& 0& 0& 0& 1\\ + 1& 0& 0& 0& 0& 0& 1\\ +\end{bmatrix} + +G5 +\begin{bmatrix} + 0& 0& 0& 0& 1& 0& 0& 1\\ + 0& 0& 1& 0& 0& 0& 0& 1\\ + 0& 0& 0& 1& 1& 0& 0& 0\\ + 0& 0& 0& 0& 1& 1& 0& 0\\ + 0& 0& 0& 1& 0& 1& 0& 0\\ + 0& 1& 0& 1& 0& 0& 0& 0\\ + 0& 0& 1& 1& 0& 0& 0& 0\\ + 0& 1& 0& 0& 0& 1& 0& 0\\ + 1& 0& 0& 0& 0& 1& 0& 0\\ + 1& 1& 0& 0& 0& 0& 0& 0\\ + 0& 1& 1& 0& 0& 0& 0& 0\\ + 1& 0& 0& 0& 0& 0& 1& 0\\ +\end{bmatrix} + ** b. Escribir la matriz de adyacencia de los digrafos G4 y G5. +Nota: La columna representa el destino y la fila el origen +G4 +\begin{bmatrix} + 0& 0& 0& 0& 0& 0& 1\\ + 1& 0& 0& 0& 0& 0& 0\\ + 0& 1& 0& 0& 0& 0& 0\\ + 0& 0& 0& 0& 0& 0& 0\\ + 0& 0& 1& 1& 0& 1& 0\\ + 1& 1& 0& 1& 0& 0& 0\\ + 0& 1& 0& 0& 0& 0& 0\\ +\end{bmatrix} + +G5 +\begin{bmatrix} + 0& 0& 0& 0& 0& 0& 1& 0\\ + 1& 0& 0& 0& 0& 0& 0& 0\\ + 0& 1& 0& 0& 0& 0& 0& 0\\ + 0& 1& 1& 0& 0& 1& 0& 0\\ + 0& 0& 0& 1& 0& 1& 0& 0\\ + 1& 1& 0& 0& 0& 0& 0& 0\\ + 0& 0& 0& 0& 0& 0& 0& 0\\ + 0& 0& 1& 0& 1& 0& 0& 0\\ +\end{bmatrix} * 6 - Un n-cubo es un grafo en el que los vértices se etiquetan con n-uplas de ceros y unos. Una arista conecta dos vértices u y v si las etiquetas de u y v difieren exactamente en un símbolo. ** a. 2-cubo @@ -249,13 +304,31 @@ $\frac{2\cdot{6}}{4}= 3$ \\ El grafo puede ser regular. * 11 - Indicar para qué valores de n es regular: - +Intervalos con números enteros ** a. el grafo completo K_n +\begin{center} +\left [ 3;\infty \right ) +\end{center} ** b. el ciclo de n vértices, C_n +\begin{center} +\left [ 3;\infty \right ) +\end{center} ** c. la rueda de n+1 vértices, W_n +\begin{center} +2 +\end{center} ** d. la estrella con n+1 vértices, S_n +\begin{center} +1 +\end{center} ** e. el n-cubo +\begin{center} +\left [ 1;\infty \right ) +\end{center} ** f. el grafo lineal L_n +\begin{center} +1,2 +\end{center} * 12 - Hallar la matriz de adyacencia de los grafos: K_6 , C_6, W_5, S_5 , L_6. ** K_6 @@ -397,6 +470,9 @@ Tenemos 2 listas una de los vertices visitados y otra para el resultado. \\ Entonces usando el algoritmo de profundidad solo se puede llegar hasta el 9 desde el primer vertice. * 21 - ¿Cuál de los grafos G1, G2 o G3 es un árbol? Indicar los vértices colgantes (hojas).¿Cuántos caminos distintos hay entre cada par de vértices? +G3 es un árbol. +Las hojas son F, E, B, C, D, G +Existe un único camino entre cada par de vértices. * 22 - Hallar árboles recubridores para cada una de las componentes conexas de los grafos del problema 1 #+ATTR_LATEX: :height 10cm [[./ej22.png]] @@ -409,7 +485,14 @@ el arbol recubridor del grafo 6 seria [[./imageej24.png]] -* 27 - Considerar el grafo ponderado con matriz de pesos +* 27 - Dada la siguiente matriz de pesos de un grafo +** a. Dar un árbol recubridor minimal +[[./ej27arbolrecubridor.png]] +** b. Considerando el árbol obtenido en el inciso anterior como un árbol con raíz en el vértice 1, ¿cuál es la altura? +Su altura es de 5 +** c. El grafo descrito por esa matriz, ¿admite un recorrido euleriano? Justificar. +No lo admite, existen más de dos vértices con grado impar (a, b, c, d...) + * 28 - Dada la siguiente matriz de pesos de un grafo, dar un árbol recubridor minimal, e indicar su peso. D= \begin{bmatrix}