Merge pull request #1617 from Deepika14145/main

Handshaking Lemma and Interesting Tree Properties

Graph Algorithms/Handshaking Lemma and Interesting Tree Properties/README.md

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+-----
+
+# Handshaking Lemma and Interesting Tree Properties
+
+## Table of Contents
+- [Introduction](#introduction)
+- [Handshaking Lemma](#handshaking-lemma)
+  - [Formula and Explanation](#formula-and-explanation)
+  - [Example](#example)
+- [Interesting Tree Properties](#interesting-tree-properties)
+  - [Properties of Trees](#properties-of-trees)
+  - [Tree Formulas](#tree-formulas)
+- [Applications](#applications)
+- [Summary](#summary)
+
+---
+
+## Introduction
+The **Handshaking Lemma** and various **Tree Properties** form the foundation for understanding graphs and trees in computer science and mathematics. They’re widely used in network theory, data structures, and algorithm design.
+
+---
+
+## Handshaking Lemma
+The **Handshaking Lemma** is a key concept in graph theory that relates the total number of edges in a graph to the sum of vertex degrees.
+
+### Formula and Explanation
+- **Formula**: The Handshaking Lemma states that in any undirected graph:
+
+  \[
+  2 \times \text{number of edges} = \sum \text{(degrees of all vertices)}
+  \]
+
+- This lemma tells us that the sum of the degrees of all vertices in a graph is always even.
+
+### Example
+Consider a simple undirected graph with three vertices A, B, and C, and two edges (A-B, B-C):
+
+- **Vertices**: A, B, C
+- **Edges**: (A-B), (B-C)
+- **Degrees**:
+  - Vertex A: 1
+  - Vertex B: 2
+  - Vertex C: 1
+- **Application of Handshaking Lemma**:
+
+  \[
+  2 \times \text{number of edges} = 1 + 2 + 1 = 4
+  \]
+
+  Hence, this satisfies the lemma since we have two edges and the sum of degrees is 4.
+
+---
+
+## Interesting Tree Properties
+Trees are a specific type of graph with unique properties. Here’s a breakdown of key properties and useful formulas related to trees.
+
+### Properties of Trees
+1. **Acyclic Connected Graph**: A tree is an undirected, connected graph with no cycles.
+2. **N Nodes and N-1 Edges**: A tree with `N` nodes has exactly `N-1` edges.
+3. **Unique Path**: Between any two nodes in a tree, there is exactly one path.
+4. **Degree of Leaf Nodes**: Leaf nodes (end nodes) have a degree of 1.
+
+### Tree Formulas
+- **Number of Edges**: `Edges = Nodes - 1`
+- **Sum of Degrees**:
+
+  \[
+  \sum \text{(degrees of all nodes)} = 2 \times (\text{number of edges})
+  \]
+
+- **Rooted Tree Properties** (for rooted trees):
+  - Height of a tree: Maximum number of edges from root to any leaf.
+  - Total nodes at depth `d`: `2^d` for a binary tree.
+
+### Example
+Consider a tree with 4 nodes: A, B, C, D and edges A-B, A-C, A-D.
+
+- **Properties Check**:
+  - **Number of nodes**: 4
+  - **Number of edges**: 3 (satisfies `N-1` rule)
+  - **Degree sum**: Degree(A) = 3, Degree(B) = 1, Degree(C) = 1, Degree(D) = 1 → Sum = 6 (which is `2 x edges`)
+
+---
+
+## Applications
+The Handshaking Lemma and Tree Properties are applied in:
+
+1. **Network Analysis**: Used to analyze and design communication and computer networks.
+2. **Data Structures**: Essential in binary trees, binary search trees, heaps, etc.
+3. **Algorithm Design**: Helps in developing algorithms for shortest path, spanning tree, and network flow.
+4. **Social Network Theory**: Handshaking Lemma can model friend relationships and connectivity.
+5. **Distributed Systems**: Tree properties are used to design and manage hierarchies in distributed databases.
+
+---
+
+## Summary
+- **Handshaking Lemma**: Relates total degrees to the number of edges in any undirected graph.
+- **Tree Properties**: Trees have a unique structure, are acyclic, and have `N-1` edges if they have `N` nodes.
+- **Formulas**: Useful for calculations involving edges, degrees, and paths.
+
+This knowledge is crucial in **graph theory**, **data structures**, and **network algorithms**, providing the mathematical foundation to solve many complex problems efficiently.
+
+---

Graph Algorithms/Handshaking Lemma and Interesting Tree Properties/program.c

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+#include <stdio.h>
+#include <stdbool.h>
+
+#define MAX_NODES 100
+
+// Structure for Graph using Adjacency Matrix
+typedef struct {
+    int adjMatrix[MAX_NODES][MAX_NODES];
+    int nodes;
+    int edges;
+} Graph;
+
+// Initialize the graph with zero edges
+void initGraph(Graph* g, int nodes) {
+    g->nodes = nodes;
+    g->edges = 0;
+    for (int i = 0; i < nodes; i++) {
+        for (int j = 0; j < nodes; j++) {
+            g->adjMatrix[i][j] = 0;
+        }
+    }
+}
+
+// Add an undirected edge between two nodes
+void addEdge(Graph* g, int u, int v) {
+    if (u >= g->nodes || v >= g->nodes) {
+        printf("Invalid nodes\n");
+        return;
+    }
+    g->adjMatrix[u][v] = 1;
+    g->adjMatrix[v][u] = 1;
+    g->edges++;
+}
+
+// Calculate the sum of degrees to verify Handshaking Lemma
+bool verifyHandshakingLemma(Graph* g) {
+    int degreeSum = 0;
+    for (int i = 0; i < g->nodes; i++) {
+        int degree = 0;
+        for (int j = 0; j < g->nodes; j++) {
+            if (g->adjMatrix[i][j] == 1)
+                degree++;
+        }
+        degreeSum += degree;
+    }
+    return degreeSum == 2 * g->edges;
+}
+
+// Check if the structure is a tree
+bool isTree(Graph* g) {
+    // A tree with n nodes has n-1 edges
+    if (g->edges != g->nodes - 1)
+        return false;
+
+    // A connected and acyclic graph with n-1 edges is a tree
+    // This is a simple approach, assuming a connected graph without cycles
+    return true;
+}
+
+int main() {
+    int nodes = 4;
+    Graph g;
+    initGraph(&g, nodes);
+
+    // Add edges
+    addEdge(&g, 0, 1);
+    addEdge(&g, 0, 2);
+    addEdge(&g, 0, 3);
+
+    printf("Total nodes: %d\n", g.nodes);
+    printf("Total edges: %d\n", g.edges);
+
+    // Check Handshaking Lemma
+    if (verifyHandshakingLemma(&g)) {
+        printf("Handshaking Lemma holds: Sum of degrees = 2 * Number of edges\n");
+    } else {
+        printf("Handshaking Lemma does not hold.\n");
+    }
+
+    // Check if the graph is a tree
+    if (isTree(&g)) {
+        printf("The graph is a tree.\n");
+    } else {
+        printf("The graph is not a tree.\n");
+    }
+
+    return 0;
+}