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Problem: Find the spanning tree of the minimum weight of the graph. The edge weights are shown in the graph.

6.4.19 Graph theory

2.39 $

Problem: Prove that any \( k \)-connected graph \( G \), built on \( n \geq 2 \) vertices, \( k \geq 2 \), contains a cycle \( C \), length of which is greater than or equal to \( 2 k \).

6.4.20 Graph theory

4.78 $

Problem: Prove that graph \( G \) is edge biconnected if and only if it can be represented as \( G=G_{0} \cup G_{1} \cup \ldots \cup G_{k} \), where \( G_{0} \) is an arbitrary cycle in the graph \( G \), and \( G_{i}, i>0 \), is either a handle, or a closed handle for the subgraph \( G_{0} \cup G_{1} \cup \ldots \cup G_{i-1} \) of graph \( G \).

6.4.21 Graph theory

5.98 $

Problem: Let \( G k \)-vertex connected. We form a new graph \( G^{\prime} \) from \( \mathrm{G} \) by adding a new vertex \( y \) to \( \mathrm{G} \) and at least \( k \) edges from \( y \) in \( k \) different vertices of the graph \( G \). Prove that \( G^{\prime} \) is also \( k \)-connected.

6.4.22 Graph theory

3.59 $

Problem: Using the given weight matrix \( \Omega \) of graph \( G \) find the value of the minimum path and the path itself from vertex \( v_{1} \) to the vertex \( v_{7} \) applying the Bellman-Moore algorithm. \[ \Omega=\left(\begin{array}{ccccccc} - & 6 & \infty & \infty & 12 & \infty & \infty \\ \infty & - & 4 & 10 & \infty & 15 & \infty \\ \infty & \infty & - & 4 & \infty & \infty & \infty \\ \infty & \infty & \infty & - & \infty & \infty & 6 \\ \infty & -8 & 7 & 11 & - & -6 & \infty \\ \infty & \infty & -8 & 7 & 8 & - & 5 \\ \infty & \infty & \infty & \infty & \infty & \infty & - \end{array}\right) \]

6.4.23 Graph theory

4.78 $

Problem: Find the main characteristics of the graph (vertex eccentricities, vertex degrees, adjacency matrix, vertex and edge connectivity numbers, radius, diameter and diametrical chain, density and looseness, chromatic number), represented by the figure. In the case of a disconnected graph, the characteristics should be found for the larger component.

6.4.24 Graph theory

3.59 $

Problem: A weighted graph is defined by a matrix of arc lengths. Plot the graph. Find: a) the minimum weight spanning tree; b) the shortest distance from vertex \( v_{6} \) to the remaining vertices of the graph, using Dijkstra's algorithm. \[ \left[\begin{array}{cccccc} \infty & 2 & 3 & \infty & 1 & \infty \\ 2 & \infty & 1 & 1 & \infty & 4 \\ 3 & 1 & \infty & 5 & \infty & \infty \\ \infty & 1 & 5 & \infty & 4 & 2 \\ 1 & \infty & \infty & 4 & \infty & 3 \\ \infty & 4 & \infty & 2 & 3 & \infty \end{array}\right] \]

6.4.25 Graph theory

5.98 $

Problem: A digraph is defined by an adjacency matrix. It is necessary to: a) draw a graph; b) identify components of strong connectivity; c) replace all arcs with edges and find an Euler chain (or cycle) in the obtained undirected graph. \[ \left(\begin{array}{llllll} 0 & 0 & 0 & 0 & 1 & 1 \\ 0 & 0 & 0 & 0 & 1 & 0 \\ 1 & 1 & 1 & 0 & 0 & 0 \\ 1 & 0 & 1 & 0 & 0 & 1 \\ 0 & 1 & 0 & 0 & 0 & 0 \\ 1 & 0 & 1 & 1 & 1 & 0 \end{array}\right) \]

6.4.26 Graph theory

4.78 $

Problem: Prove the statement by mathematical induction: \( \left(n^{5}-n\right) \) is a multiple of 5 for all natural \( n \).

6.6.15 Combinatorics

0 $

Problem: Nine students must take a test in four subjects: Physics, Algebra, English and History. All tests are scheduled at the same time and each person can take only one test, so students need to be divided into groups. In how many ways can this be done? In how many ways can they take a sit after the test at two completely identical tables (at least one at a table) in order to celebrate the results?

6.6.16 Combinatorics

2.87 $

Problem: Find the coefficients when \[ a=x^{2} \cdot y^{4} \cdot z^{3}, b=x^{2} \cdot y^{3} \cdot z, c=x^{4} \cdot z^{4} \] in the expansion \( \left(5 \cdot x^{2}+3 \cdot y^{2}+2 \cdot z\right)^{6} \).

6.6.17 Combinatorics

1.91 $

Problem: Find the sequence \( \left\{a_{n}\right\} \), that satisfies the recurrence relation \( 2 \cdot a_{n+2}+10 \cdot a_{n+1}+8 \cdot a_{n}=0 \) and the initial conditions \( a_{1}=3, a_{2}=9 \).

6.6.18 Combinatorics

0 $

Problem: How many positive three-digit numbers are there: a) divisible by the numbers 8,22 or 26 ? b) divisible by exactly one of these three numbers?

6.6.19 Combinatorics

2.87 $

Problem: Prove equalities using the properties of set operations and definitions of operations. Illustrate using Euler-Venn diagrams. a) \( A((A \cap B) \backslash C)=(A \backslash B) \cup(A \cap C) \) b) \( U^{2} \backslash(C \times D)=(U \times(U \backslash D)) \cup((U \backslash C) \times U) \).

6.8.7 Set theory

1.44 $

Problem: Calculate the value of the iterated integral: \[ I=\int_{0}^{1} d x \int_{0}^{1} \frac{x d y}{\frac{\left(1+x^{2}+y^{2}\right)^{3}}{2}} \]

9.1.34 Double integrals

1.44 $

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