From d335d18ead1d5e0f776640bcaa410283fa83446f Mon Sep 17 00:00:00 2001 From: fede Date: Sat, 24 Aug 2024 12:24:49 -0300 Subject: [PATCH] demostracion --- Discreta/entrega1.org | 154 +++++++++++++++++++++++++++++++++++++----- Discreta/entrega1.pdf | Bin 115091 -> 312344 bytes Discreta/grafico6.jpg | Bin 0 -> 8415 bytes 3 files changed, 138 insertions(+), 16 deletions(-) create mode 100644 Discreta/grafico6.jpg diff --git a/Discreta/entrega1.org b/Discreta/entrega1.org index 11fd114..52905fb 100644 --- a/Discreta/entrega1.org +++ b/Discreta/entrega1.org @@ -1,7 +1,13 @@ -#+title: Tp1 +#+title: Trabajo Practico 1 - Matematicas Discretas #+author: Martin, Luca, Roy, Fede -* 1 -** a +#+options: date:nil +#+LATEX_HEADER: \usepackage{titlesec} +#+LATEX_HEADER: \titlelabel{\thetitle.\ } +#+LATEX_HEADER: \renewcommand\thesection{\arabic{section}} +#+LATEX_HEADER: \renewcommand\thesubsection{\alph{subsection}} + +* Para cada uno de los grafos de la figura, +** el grado de cada vértice. Verificar la fórmula que relaciona los grados de vértices con el número de aristas. |-------+------| | Grafo | Cant | |-------+------| @@ -10,12 +16,12 @@ | G3 | 4 | |-------+------| -** b +** identificar los bucles (si existen) - G1 f-f - G2 g-g - G2 h-h -** c +** un ciclo en el grafo |-------+-----------| | Grafo | | |-------+-----------| @@ -24,7 +30,7 @@ | G3 | - | |-------+-----------| -** d +** un camino de a a c de longitud 3, si existe |-------+-------| | Grafo | | |-------+-------| @@ -32,7 +38,7 @@ | G2 | a-b-c | | G3 | a-b-c | |-------+-------| -** e +** un ciclo que contenga a g, de longitud par, si existe. |-------+---------| | Grafo | | |-------+---------| @@ -41,7 +47,7 @@ | G3 | - | |-------+---------| -** f +** los vértices conectados con e |-------+-------| | grafo | | |-------+-------| @@ -50,7 +56,7 @@ | G3 | J | |-------+-------| -** g +** todos los caminos simples de e a g. |-------+-------------| | grafo | | |-------+-------------| @@ -67,7 +73,7 @@ | G3 | g-i-j-e | |-------+-------------| -** h +** la distancia entre b y cada uno de los vértices |-------+---------------+---| | grafo | Trasado | N | |-------+---------------+---| @@ -92,9 +98,13 @@ | | b-j-i-k-a-h-d | 6 | |-------+---------------+---| -* 2 -* 4 -** a +* ¿Cuántas componentes conexas tienen los graos G1, G2 y G3? ¿Cuál es el número máximo de aristas que pueden eliminarse de cada uno, manteniendo el número de componentes conexas? ¿Cuál es el mínimo número de aristas que deben eliminarse para aumentar la cantidad de componentes conexas en cada grafo? +* . +** Escribir la matriz de incidencia de los grafos G1, G2 y G3. +** Escribir la matriz de adyacencia de los grafos G1, G2 y G3. + +* Para los siguientes grafos dirigidos, indicar: +** el grado de entrada y de salida de cada vértice. Verificar la fórmula que relaciona los grados de entrada y salida con el número de aristas. - G4 Entrada = 3 Salida = 3 @@ -102,7 +112,7 @@ - G5 Entrada = 3 Salida = 3 -** b +** un camino dirigido de e a a, si existe |-------+---------| | grafo | Trasado | |-------+---------| @@ -110,7 +120,7 @@ | G5 | e-d-b-a | |-------+---------| -** c +** un ciclo dirigido, si existe |-------+---------| | grafo | Trasado | |-------+---------| @@ -118,7 +128,11 @@ | G5 | - | |-------+---------| -* 6 +* . +** Escribir la matriz de incidencia de los digrafos G4 y G5. +** Escribir la matriz de adyacencia de los digrafos G4 y G5. + +* 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. ** 2-cubo #+begin_src U -- B @@ -132,3 +146,111 @@ U - C - V ** 3-cubo [[./3cubo.png]] + +** ¿Es conexo el n-cubo? Justificar. +Es conexo porque existe un camino entre cualquier par de vertices. + +** Calcular el número de vértices y de aristas del n-cubo. +Para los vertices es: + +\begin{center} +2^n +\end{center} + +y para las aristas + +\begin{center} +n * 2^{n-1} +\end{center} + +Entonces para un n-cubo siendo n = 3 + +\begin{center} +$Vertices = 2^3 = 8$ + +$Aristas = 3 * 2^{3-1} = 12$ +\end{center} + +* Dada una colección de conjuntos, el grafo de intersección se define como el que tiene un vértice por cada conjunto, y una arista entre dos vértices que representan conjuntos de intersección no vacía. Representar el grafo de intersección de los conjuntos. +** A1 = {0,1,2,3,4,5}; A2 = {10, 11, 12}; A3 = {1, 5, 10, 15, 20}; A4 = {3, 12, 19}; A5 = {0, 10, 20, 30, 40}; A6 = N = {números naturales} + + +** B1 = {n \in N: n es divisor de 4 mayor que 1} = {2, 4}; B2 = {n \in N: n es divisor de 6 mayor que 1} = {2, 3, 6}; B3 = {n \in N: n es divisor de 11 mayor que 1} = {11}; B4 = {n \in N: n es divisor de 9 mayor que 1} = {3, 9}; + + +** C1 = {n \in N: n es múltiplo de 4} = {4, 8, 12, ...}; C2 = {n \in N: n es múltiplo de 6} = {6, 12, 18, ...}; C3 = {n \in N: n es múltiplo de 11} = {11, 22, 33, ...}; C4 = {n \in N: n es múltiplo de 9} = {9, 18, 27, ...}; + + +* Dada una lista de sentencias a ejecutar por un programa, un grafo de precedencias es el grafo dirigido donde cada vértice representa una sentencia, y tiene un arco del vértice i al vértice j si la ejecución de j necesariamente debe realizarse después de la sentencia i (porque j depende del resultado de i, o porque j modifica una variable que se usa antes en i). Representar el grafo dirigido correspondiente a cada una de las listas de sentencias: +#+begin_quote +|--------------+--------------| +| S1: x := 0 | S1: x := 0 | +| S2: x := x+1 | S2: y := 1 | +| S3: y := 2 | S3: z := x+1 | +| S4: z := y | S4: w := x+y | +| S5: x := x+2 | S5: v := w+1 | +| S6: y := x+z | S6: v := z+w | +| S7: z := 4 | | +|--------------+--------------| +#+end_quote +~¿Puede hallarse un camino cerrado dirigido en los grafos hallados? Justificar la respuesta.~ + +* Dado el grafo de rutas aéreas G6 +[[./grafico6.jpg]] + +** Dar el número de caminos con 3 tramos entre Buenos Aires y Tucumán. +** ¿Hay algún ciclo de longitud 4 que incluya a Buenos Aires? +** ¿Hay algún ciclo que incluya a Mendoza y Corrientes? ¿En caso afirmativo, dé la longitud de cada ciclo hallado. +** Dé un circuito que incluya Mendoza y Corrientes de longitud 6. +** ¿Qué representa el grado de cada nodo? + +* Indicar si existe un grafo regular con las siguientes características (dar un ejemplo o justificar la no existencia): + +** 7 vértices y 7 aristas +** 7 vértices y 16 aristas +** 3 vértices, 6 aristas, sin lazos +** 4 vértices, 6 aristas, sin lazos + +* Indicar para qué valores de n es regular: + +** el grafo completo K_n +** el ciclo de n vértices, C_n +** la rueda de n+1 vértices, W_n +** la estrella con n+1 vértices, S_n +** el n-cubo +** el grafo lineal L_n + +* Hallar la matriz de adyacencia de los grafos: K_6 , C_6, W_5, S_5 , L_6. + +* Describir las matrices de adyacencia de K_n , C_n, W_n, S_n , Ln dando el valor de cada componente a_{ij} en función de i, j y n. + +* Dada la siguiente matriz de adyacencia de un grafo no dirigido + +. + + +. + +\setcounter{MaxMatrixCols}{12} +\begin{bmatrix} +0 & 0 & 1 & 0 & 1 & 0 & 1 & 0 & 1 & 0 & 0 & 0 \\ +0 & 1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\ +1 & 0 & 0 & 0 & 0 & 0 & 1 & 0 & 1 & 0 & 0 & 1 \\ +0 & 0 & 0 & 0 & 0 & 1 & 0 & 0 & 0 & 1 & 0 & 0 \\ +1 & 0 & 0 & 0 & 0 & 0 & 1 & 1 & 0 & 0 & 0 & 0 \\ +0 & 0 & 0 & 1 & 0 & 1 & 0 & 0 & 0 & 0 & 0 & 0 \\ +1 & 0 & 1 & 0 & 1 & 0 & 0 & 1 & 0 & 0 & 0 & 0 \\ +0 & 0 & 0 & 0 & 1 & 0 & 1 & 0 & 1 & 0 & 1 & 0 \\ +1 & 0 & 1 & 0 & 0 & 0 & 0 & 1 & 0 & 0 & 0 & 0 \\ +0 & 0 & 0 & 1 & 0 & 0 & 0 & 0 & 0 & 1 & 0 & 0 \\ +0 & 0 & 0 & 0 & 0 & 0 & 0 & 1 & 0 & 0 & 0 & 1 \\ +0 & 0 & 1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 1 & 0 \\ +\end{bmatrix} + +Responder (sin graficar el grafo) + +** ¿Cómo se puede determinar la cantidad de aristas en el grafo, a partir de dicha matriz? +** Determinar el grado de cada vértice. +** Describir un algoritmo para determinar todos los vértices conectados con el primer vértice, teniendo como entrada la matriz de adyacencia, y aplicarlo a la matriz dada. + +* Si A es la matriz de adyacencia de un grafo, Ak es una matriz cuyo elemento (i,j) es el número de caminos de longitud k que hay en el grafo entre los vértices i y j. ¿Cómo se puede usar este resultado para probar si un grafo es conexo? diff --git a/Discreta/entrega1.pdf b/Discreta/entrega1.pdf index fd5e70615a6a060c660cb53bdf45098d585d6c3c..a01e6d70103d10699e44b62167cb6a15fccc9802 100644 GIT binary patch delta 294260 zcmZsiQfM3I{4Kb<=l$b{dRV<;-NOhniTZ%k~^PdOTVbeKCIUxf@YF2jwqx^#c zF~Oz-I{}oUqv4=0gzOK}`~b4gS|<&JA37-mR7klGBOo>b#5*AktnoWm8sJglJR%Lh z22;!RX*n|qQ;QxS+te5zs5W2hKOmcu_^X0sXj$|( z@Gme21nCrHJQCQ9068qnak%1sfp}Pi9uUZa;(dINbVofwTMI;C@_f2mFj9Ew&3$0fS&sp^p2_6a+ra6&Nk#bfOyClDJ#iBOV}NG*qI$6*9FiGm z9b<`3G}tW%=zROYe6lRxK9fN~n@5H+xTc19?GzomS#%?7dnQx#JnVjJf?GfCYmVt8 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