# IGNOU MMTE-001 Solved Assignment 2023 | M.Sc. MACS

Solved By – Narendra Kr. Sharma – M.Sc (Mathematics Honors) – Delhi University

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## IGNOU MMTE-001 Assignment Question Paper 2023

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MMTE-001 Question Paper 2023
1.State whether the following statements are true or false. Justify your answers with a short proof or a counterexample
i) Every tree has a perfect matching.
ii) Every 2-connected bipartite graph is Hamiltonian.
iii) For some nonnegative integers ${d}_{1},{d}_{2},\dots ,{d}_{n}$${d}_{1},{d}_{2},\dots ,{d}_{n}$d_(1),d_(2),dots,d_(n)d_1, d_2, \ldots, d_n${d}_{1},{d}_{2},\dots ,{d}_{n}$ the sequence $\left(max\left\{{d}_{1},n\right\},{d}_{2},{d}_{3},\dots ,{d}_{n}\right)$$\left(max\left\{{d}_{1},n\right\},{d}_{2},{d}_{3},\dots ,{d}_{n}\right)$(max{d_(1),n},d_(2),d_(3),dots,d_(n))\left(\max \left\{d_1, n\right\}, d_2, d_3, \ldots, d_n\right)$\left(max\left\{{d}_{1},n\right\},{d}_{2},{d}_{3},\dots ,{d}_{n}\right)$ is graphic.
iv) Grötzsch graph is Eulerian.
v) ${K}_{4}$${K}_{4}$K_(4)K_4${K}_{4}$, as a plane graph, is self-dual.
vi) The vertex-covering number of an odd cycle is 1 more than its independence number.
vii) The complement of a disconnected graph is connected.
viii) If $\kappa \left(G\right)<{\kappa }^{\mathrm{\prime }}\left(G\right)$$\kappa \left(G\right)<{\kappa }^{\mathrm{\prime }}\left(G\right)$kappa(G) < kappa^(‘)(G)\kappa(G)<\kappa^{\prime}(G)$\kappa \left(G\right)<{\kappa }^{\mathrm{\prime }}\left(G\right)$, then $\delta \left(G\right)\ge 4$$\delta \left(G\right)\ge 4$delta(G) >= 4\delta(G) \geq 4$\delta \left(G\right)\ge 4$.
ix) The line graph of the Petersen graph has 30 edges.
x) There exists a complete binary tree on 15 vertices.
2 (a) Does there exist a 3 -edge-colourable graph on 10 vertices and having 20 edges? Justify.
(b) Prove or disprove that the height of an $n$$n$nn$n$-vertex complete $k$$k$kk$k$-ary tree is at least ${\mathrm{log}}_{k}\left(n+1\right)-1$${\mathrm{log}}_{k}\left(n+1\right)-1$log _(k)(n+1)-1\log _k(n+1)-1${\mathrm{log}}_{k}\left(n+1\right)-1$.
(c) Find a minimum-weigh spanning tree in the following graph.
3(a) An $n$$n$nn$n$-vertex forest with $n/2$$n/2$n//2n / 2$n/2$ edges has exactly $n/2$$n/2$n//2n / 2$n/2$ trees as its components. True or false? Justify.
(b) Draw the complement of the following graph.
Is the complement Hamiltonian? Justify your answer.
(c) Verify the König Egarváry Theorem for the following graph.
4(a) Draw a diagram, as nice as possible, of the line graph of the Petersen graph. Write the number of vertices, the number of edges, the minimum and maximum degrees of it.
(b) There exists a self-complementary graph on 2023 vertices. True of false? Justify your answer.
(c) Every 3-colourable graph contains an odd cycle. True or false? Justify.
5(a) If $G$$G$GG$G$ is a $k$$k$kk$k$-connected graph having $n$$n$nn$n$ vertices, what is the minimum size of $G$$G$GG$G$ ? Justify.
(b) Draw the dual of the following plane graph.
Does the dual have any cut-vertex? Justify.
(c) Define a flow on the following network, having value at least 5.
6(a) Check whether the sequence $\left(4,4,4,3,2,2,1,1,1\right)$$\left(4,4,4,3,2,2,1,1,1\right)$(4,4,4,3,2,2,1,1,1)(4,4,4,3,2,2,1,1,1)$\left(4,4,4,3,2,2,1,1,1\right)$ is graphic or not. If yes, draw a graph realising this degree sequence.
(b) Let $G$$G$GG$G$ be a graph having no isolated vertex and no induced subgraph with exactly two edges. Show that $G$$G$GG$G$ is a complete graph.
7(a) Draw an (8,15)-graph $G$$G$GG$G$ with ${\chi }^{\mathrm{\prime }}\left(G\right)=5$${\chi }^{\mathrm{\prime }}\left(G\right)=5$chi^(‘)(G)=5\chi^{\prime}(G)=5${\chi }^{\mathrm{\prime }}\left(G\right)=5$.
(b) Explain the difference between a maximal and a maximum matching, with the help of an example.
(c) Find the thickness of the line graph of ${K}_{4}$${K}_{4}$K_(4)K_4${K}_{4}$.
8(a) Starting with the cycle $\left({v}_{1},{v}_{2},{v}_{3},{v}_{4},{v}_{5},{v}_{1}\right)$$\left({v}_{1},{v}_{2},{v}_{3},{v}_{4},{v}_{5},{v}_{1}\right)$(v_(1),v_(2),v_(3),v_(4),v_(5),v_(1))\left(v_1, v_2, v_3, v_4, v_5, v_1\right)$\left({v}_{1},{v}_{2},{v}_{3},{v}_{4},{v}_{5},{v}_{1}\right)$ in the following weighted ${K}_{5}$${K}_{5}$K_(5)K_5${K}_{5}$ perform the reduction step twice to get a Hamiltonian cycle with smaller weight.
(b) Let $G$$G$GG$G$ be a planar graph with at least 11 vertices. Show that $\overline{G}$$\overline{G}$bar(G)\bar{G}$\overline{G}$ is nonplanar.
1. (a) Let $G$$G$GG$G$ and $H$$H$HH$H$ be any graphs such that $L\left(G\right)\cong L\left(H\right)$$L\left(G\right)\cong L\left(H\right)$L(G)~=L(H)L(G) \cong L(H)$L\left(G\right)\cong L\left(H\right)$. Is it necessary that $G\cong H$$G\cong H$G~=HG \cong H$G\cong H$ ? Justify.
(b) For the graph given in Q. 3(b), find the number of $\left({v}_{2},{v}_{5}\right)$$\left({v}_{2},{v}_{5}\right)$(v_(2),v_(5))\left(v_2, v_5\right)$\left({v}_{2},{v}_{5}\right)$-walks of length 3.
$$2\:sin\:\theta \:sin\:\phi =-cos\:\left(\theta +\phi \right)+cos\:\left(\theta -\phi \right)$$

## MMTE-001 Sample Solution 2023

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MMTE-001 Solved Assignment 2023

### Question:-01

1.State whether the following statements are true or false. Justify your answers with a short proof or a counterexample
i) Every tree has a perfect matching.
Answer:
let’s address the statement:
“Every tree has a perfect matching.”
This statement is false.
Justification:
A perfect matching in a graph is a set of edges such that every vertex of the graph is adjacent to exactly one edge in the set, and no two edges in the set share a common vertex.
For a tree to have a perfect matching, it must have an even number of vertices. This is because each edge in the matching covers exactly two vertices, so if there’s an odd number of vertices, at least one vertex will be left unmatched.
Counterexample:
Consider a simple tree with 3 vertices and 2 edges (a path of length 2). This tree does not have a perfect matching because one of the three vertices will always be left unmatched.
Thus, the statement “Every tree has a perfect matching” is false.
ii) Every 2-connected bipartite graph is Hamiltonian.
Answer:
The statement is:
“Every 2-connected bipartite graph is Hamiltonian.”
This statement is false.
Justification:
A graph is Hamiltonian if it has a Hamiltonian cycle, which is a cycle that visits every vertex exactly once and returns to the starting vertex. A bipartite graph is one whose vertices can be divided into two disjoint sets such that no two vertices within the same set are adjacent.
For a bipartite graph to be Hamiltonian, it must satisfy the following condition: both sets of vertices (let’s call them A and B) must have the same number of vertices. This is because a Hamiltonian cycle in a bipartite graph will alternate between vertices in A and B. If one set has more vertices than the other, it’s impossible to form a Hamiltonian cycle.
Counterexample:
Consider a bipartite graph with sets A and B where $|A|=3$$|A|=3$|A|=3|A| = 3$|A|=3$ and $|B|=2$$|B|=2$|B|=2|B| = 2$|B|=2$. The graph is 2-connected, but it cannot have a Hamiltonian cycle because the two sets have different numbers of vertices. Specifically, imagine three vertices on the left (set A) and two on the right (set B). Even if every vertex is connected to every other vertex across the sets, you can’t form a cycle that visits every vertex exactly once and returns to the starting vertex.
Thus, the statement “Every 2-connected bipartite graph is Hamiltonian” is false.
iii) For some nonnegative integers ${d}_{1},{d}_{2},\dots ,{d}_{n}$${d}_{1},{d}_{2},\dots ,{d}_{n}$d_(1),d_(2),dots,d_(n)d_1, d_2, \ldots, d_n${d}_{1},{d}_{2},\dots ,{d}_{n}$ the sequence $\left(max\left\{{d}_{1},n\right\},{d}_{2},{d}_{3},\dots ,{d}_{n}\right)$$\left(max\left\{{d}_{1},n\right\},{d}_{2},{d}_{3},\dots ,{d}_{n}\right)$(max{d_(1),n},d_(2),d_(3),dots,d_(n))\left(\max \left\{d_1, n\right\}, d_2, d_3, \ldots, d_n\right)$\left(max\left\{{d}_{1},n\right\},{d}_{2},{d}_{3},\dots ,{d}_{n}\right)$ is graphic.
Answer:
The statement is:
“For some nonnegative integers ${d}_{1},{d}_{2},\dots ,{d}_{n}$${d}_{1},{d}_{2},\dots ,{d}_{n}$d_(1),d_(2),dots,d_(n)d_1, d_2, \ldots, d_n${d}_{1},{d}_{2},\dots ,{d}_{n}$ the sequence $\left(max\left\{{d}_{1},n\right\},{d}_{2},{d}_{3},\dots ,{d}_{n}\right)$$\left(max\left\{{d}_{1},n\right\},{d}_{2},{d}_{3},\dots ,{d}_{n}\right)$(max{d_(1),n},d_(2),d_(3),dots,d_(n))\left(\max \left\{d_1, n\right\}, d_2, d_3, \ldots, d_n\right)$\left(max\left\{{d}_{1},n\right\},{d}_{2},{d}_{3},\dots ,{d}_{n}\right)$ is graphic.”
To determine if this statement is true or false, we need to understand what it means for a sequence to be graphic. A sequence of nonnegative integers is said to be graphic if there exists a simple, undirected graph with no loops or multiple edges such that the degrees of its vertices correspond to the numbers in the sequence.
The statement is suggesting that for some sequences of degrees, if we replace ${d}_{1}$${d}_{1}$d_(1)d_1${d}_{1}$ with the maximum of ${d}_{1}$${d}_{1}$d_(1)d_1${d}_{1}$ and $n$$n$nn$n$, the resulting sequence is still graphic.
This statement is false.
Justification:
Consider the sequence ${d}_{1},{d}_{2},\dots ,{d}_{n}$${d}_{1},{d}_{2},\dots ,{d}_{n}$d_(1),d_(2),dots,d_(n)d_1, d_2, \ldots, d_n${d}_{1},{d}_{2},\dots ,{d}_{n}$ where ${d}_{1}=0$${d}_{1}=0$d_(1)=0d_1 = 0${d}_{1}=0$ and all other ${d}_{i}$${d}_{i}$d_(i)d_i${d}_{i}$ values are 1. This sequence is graphic because it represents a star graph with $n-1$$n-1$n-1n-1$n-1$ edges and $n$$n$nn$n$ vertices.
Now, let’s replace ${d}_{1}$${d}_{1}$d_(1)d_1${d}_{1}$ with the maximum of ${d}_{1}$${d}_{1}$d_(1)d_1${d}_{1}$ and $n$$n$nn$n$. Since ${d}_{1}=0$${d}_{1}=0$d_(1)=0d_1 = 0${d}_{1}=0$, the new ${d}_{1}$${d}_{1}$d_(1)d_1${d}_{1}$ value will be $n$$n$nn$n$. The new sequence becomes $n,1,1,\dots ,1$$n,1,1,\dots ,1$n,1,1,dots,1n, 1, 1, \ldots, 1$n,1,1,\dots ,1$. This sequence is not graphic. The reason is that the first vertex is supposed to have $n$$n$nn$n$ edges connected to it, but there are only $n-1$$n-1$n-1n-1$n-1$ other vertices available, each of which can only provide one edge.
Thus, the statement “For some nonnegative integers ${d}_{1},{d}_{2},\dots ,{d}_{n}$${d}_{1},{d}_{2},\dots ,{d}_{n}$d_(1),d_(2),dots,d_(n)d_1, d_2, \ldots, d_n${d}_{1},{d}_{2},\dots ,{d}_{n}$ the sequence $\left(max\left\{{d}_{1},n\right\},{d}_{2},{d}_{3},\dots ,{d}_{n}\right)$$\left(max\left\{{d}_{1},n\right\},{d}_{2},{d}_{3},\dots ,{d}_{n}\right)$(max{d_(1),n},d_(2),d_(3),dots,d_(n))\left(\max \left\{d_1, n\right\}, d_2, d_3, \ldots, d_n\right)$\left(max\left\{{d}_{1},n\right\},{d}_{2},{d}_{3},\dots ,{d}_{n}\right)$ is graphic” is false.
iv) Grötzsch graph is Eulerian.
Answer:
The statement is:
“Grötzsch graph is Eulerian.”
A graph is Eulerian if it contains an Eulerian cycle, which is a cycle that visits every edge exactly once and returns to the starting vertex. For a graph to be Eulerian, every vertex must have an even degree.
The Grötzsch graph is a triangle-free graph with 11 vertices and 20 edges.
To determine if the Grötzsch graph is Eulerian, we need to check the degrees of its vertices.
Justification:
The Grötzsch graph is known to have the following degree sequence: 5, 5, 5, 5, 4, 4, 4, 4, 4, 4, 4.
From the degree sequence, it’s clear that there are four vertices of degree 5, which is odd. Since an Eulerian graph must have all vertices of even degree, the Grötzsch graph cannot be Eulerian.
Thus, the statement “Grötzsch graph is Eulerian” is false.
v) ${K}_{4}$${K}_{4}$K_(4)K_4${K}_{4}$, as a plane graph, is self-dual.
Answer:
The statement is:
${K}_{4}$${K}_{4}$K_(4)K_4${K}_{4}$, as a plane graph, is self-dual.”
A graph is self-dual if its planar embedding is isomorphic to its dual graph. The dual graph of a planar graph has a vertex for each face of the original graph and an edge for each edge in the original graph that separates the corresponding faces.
Justification:
The graph ${K}_{4}$${K}_{4}$K_(4)K_4${K}_{4}$ is the complete graph on 4 vertices. When drawn in the plane without any edges crossing, ${K}_{4}$${K}_{4}$K_(4)K_4${K}_{4}$ divides the plane into 4 regions: 3 triangular regions and 1 outer region.
The dual graph of this planar embedding of ${K}_{4}$${K}_{4}$K_(4)K_4${K}_{4}$ will have 4 vertices (one for each region). Since each triangular region in ${K}_{4}$${K}_{4}$K_(4)K_4${K}_{4}$ is bounded by 3 edges, each vertex in the dual graph will have degree 3. This means the dual graph is also ${K}_{4}$${K}_{4}$K_(4)K_4${K}_{4}$.
Thus, the planar embedding of ${K}_{4}$${K}_{4}$K_(4)K_4${K}_{4}$ is isomorphic to its dual, making ${K}_{4}$${K}_{4}$K_(4)K_4${K}_{4}$ self-dual when considered as a plane graph.
Therefore, the statement “${\mathbit{K}}_{\mathbf{4}}$${\mathbit{K}}_{\mathbf{4}}$K_(4)\boldsymbol{K_4}${\mathbit{K}}_{\mathbf{4}}$, as a plane graph, is self-dual” is true.
$$c=a\:cos\:B+b\:cos\:A$$

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$$sin\:3\theta =3\:sin\:\theta -4\:sin^3\:\theta$$

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