Which Of The Following Graphs Are Identical

Introduction

When a textbook or an exam presents several drawings of graphs and asks “which of the following graphs are identical?So conversely, two graphs that share obvious features such as the same number of vertices or the same degree sequence can still be fundamentally different. Also, at first glance, two pictures may look different because the vertices are placed in distinct positions or the edges are drawn with different curves, yet they can represent exactly the same abstract structure. ”, it is really testing the concept of graph isomorphism—the formal notion that two graphs are the same up to a relabeling of their vertices. Understanding how to decide whether graphs are identical is essential not only for discrete mathematics courses but also for applications in chemistry (molecular graphs), computer science (network topology), and sociology (social‑network analysis).

In this article we will unpack what it means for graphs to be identical, walk through a reliable step‑by‑step procedure for comparing them, illustrate the ideas with concrete examples, discuss the underlying theory, highlight common pitfalls, and answer frequently asked questions. In real terms, by the end, you should feel confident tackling any “which graphs are identical? ” problem that comes your way.

Detailed Explanation

What Does “Identical” Mean in Graph Theory?

In everyday language, two objects are identical if you can superimpose them perfectly. In graph theory, the analogous idea is isomorphism. Two simple undirected graphs (G=(V_G,E_G)) and (H=(V_H,E_H)) are said to be identical (or isomorphic) if there exists a bijection

[ \phi : V_G \longrightarrow V_H ]

such that for any two vertices (u,v\in V_G),

[ {u,v}\in E_G \iff {\phi(u),\phi(v)}\in E_H . ]

In plain words: you can rename the vertices of (G) using the mapping them one‑to‑one onto the vertices of (H) and every edge of (G) becomes an edge of (H) and vice‑versa. The geometric layout of the drawing is irrelevant; only the adjacency relationships matter But it adds up..

Several graph invariants—properties that are preserved under any isomorphism—are useful first checks. If two graphs differ in any invariant, they cannot be identical. Common invariants include:

• Order – the number of vertices (|V|).
• Size – the number of edges (|E|).
• Degree sequence – the multiset of vertex degrees, usually listed in non‑increasing order.
• Connectedness – whether the graph is one piece or splits into components.
• Number of cycles of a given length (e.g., triangles, 4‑cycles).
• Spectral properties – eigenvalues of the adjacency or Laplacian matrix.

If all easy invariants match, the graphs might be identical, but further inspection is required because invariants are not always sufficient to guarantee isomorphism (different graphs can share the same degree sequence, for instance).

Step‑by‑Step or Concept Breakdown

Below is a practical workflow you can follow when faced with a set of candidate graphs. Each step builds on the previous one, allowing you to eliminate non‑matches quickly and focus on the harder cases.

1. Count Vertices and Edges

Write down (|V|) and (|E|) for each graph. If any graph has a different order or size, discard it immediately.

2. Compare Degree Sequences

List the degree of each vertex, sort the list descending, and compare the sequences. A mismatch guarantees non‑isomorphism.

Tip: If the sequences are long, you can also compute the degree‑frequency histogram (how many vertices have degree 0, 1, 2, …) – this is sometimes easier to spot differences And that's really what it comes down to..

3. Examine Connectivity and Components

Determine whether each graph is connected. If one graph is connected and another is not, they cannot be identical. For disconnected graphs, compare the multiset of component sizes (e.Practically speaking, g. Here's the thing — , one component of 4 vertices, two isolated vertices, etc. ) Nothing fancy..

4. Look for Small Subgraph Patterns

Count occurrences of easy‑to‑spot subgraphs: triangles (3‑cycles), 4‑cycles, paths of length 2, stars (K_{1,3}), etc. These counts are invariants and often break ties when degree sequences match.

5. Try to Construct an Explicit Bijection (When Needed)

If the previous steps leave you with a small handful of candidates, attempt to build a vertex mapping (\phi).

Strategy:

• Start by mapping vertices of unique degree (e.g., the only vertex of degree 5) to each other.
• Propagate constraints: if (u) maps to (u') and (v) is adjacent to (u), then (\phi(v)) must be a neighbor of (u').
• Use backtracking if you reach a contradiction.

For graphs with up to about 10 vertices, this manual process is usually manageable; larger instances rely on computer algorithms (e.Day to day, g. , NAUTY, Bliss).

6. Verify the Mapping

Once you have a candidate bijection, check every edge of (G) to see if its image is an edge of (H) and vice‑versa. If the check passes, the graphs are identical; otherwise, they are not.

7. Use Matrix Invariants as a Final Sanity Check

Compute the adjacency matrix (A_G) and (A_H). Because of that, if there exists a permutation matrix (P) such that (P A_G P^T = A_H), the graphs are isomorphic. In practice, you can compare the characteristic polynomial or the sorted list of eigenvalues; a mismatch proves non‑isomorphism, while a match only suggests further testing And that's really what it comes down to. Less friction, more output..

Some disagree here. Fair enough.

Following this flowchart will let you answer “which of the following graphs are identical?” questions be solved efficiently and with confidence That alone is useful..

Real Examples

Example 1: Graphs That Look Different but Are Identical

Consider the two drawings below (des

To smoothly continue the article and conclude it properly, we can focus on the final example and wrap up the discussion with a summary of the isomorphism-checking process and its importance in graph theory.

Real Examples (continued)

Example 1: Graphs That Look Different but Are Identical

Consider the two drawings below (described in abstract terms for clarity):

• Graph G: A cycle of 5 vertices (a pentagon), with one additional vertex connected to two non-adjacent vertices of the cycle.
• Graph H: A different drawing of the same graph, where the additional vertex is placed outside the cycle but still connected to the same two vertices.

At first glance, the two graphs may appear different due to their embeddings in the plane. Still, isomorphism is not about how the graph is drawn, but about whether there exists a bijection between the vertex sets that preserves adjacency And that's really what it comes down to..

Let’s walk through the isomorphism-checking steps:

Step 1: Count Vertices and Edges

• Both graphs have 6 vertices and 7 edges.
• ✅ Passed.

Step 2: Compare Degree Sequences

• Graph G:

  • One vertex of degree 4 (the one connected to two cycle vertices and two others)
  • Two vertices of degree 2 (the ones adjacent to the degree-4 vertex)
  • Three vertices of degree 2 (the rest of the cycle)
  • Degree sequence: [4, 2, 2, 2, 2, 2]

• Graph H:

  • Same structure, just redrawn.
  • Degree sequence: [4, 2, 2, 2, 2, 2]

• ✅ Degree sequences match.

Step 3: Examine Connectivity

• Both graphs are connected.
• ✅ Passed.

Step 4: Look for Small Subgraph Patterns

• Both graphs contain:
  • One 5-cycle
  • One path of length 2 involving the degree-4 vertex
  • No triangles
• ✅ Subgraph counts match.

Step 5: Try to Construct an Explicit Bijection

Let’s label the vertices of Graph G as follows:

• $ v_1 $: degree 4 (central vertex)
• $ v_2, v_3 $: neighbors of $ v_1 $ on the cycle
• $ v_4, v_5 $: the other two neighbors of $ v_1 $
• $ v_6 $: the sixth vertex, not connected to $ v_1 $

Now, in Graph H, we can relabel the vertices accordingly to match the adjacency:

• Map $ v_1 \to v'_1 $
• Map $ v_2 \to v'_2 $, $ v_3 \to v'_3 $, $ v_4 \to v'_4 $, $ v_5 \to v'_5 $, $ v_6 \to v'_6 $

After checking all edges, we find that the adjacency is preserved Small thing, real impact. Turns out it matters..

• ✅ A valid bijection exists.

Step 6: Verify the Mapping

We double-check that for every edge $ (u, v) $ in $ G $, the pair $ (\phi(u), \phi(v)) $ is an edge in $ H $, and vice versa.

• ✅ All edges match under the mapping.

Step 7: Use Matrix Invariants

• The adjacency matrices of both graphs are permutation-equivalent.
• Their characteristic polynomials and eigenvalues match.
• ✅ Final confirmation of isomorphism.

Conclusion

Through a systematic application of isomorphism-checking techniques—degree sequence comparison, connectivity analysis, subgraph counting, explicit mapping, and matrix invariants—we have confidently determined that Graph G and Graph H are isomorphic, despite their different visual representations.

This process underscores the importance of not judging graphs by their drawings, but rather by their combinatorial structure. Isomorphism is a powerful concept in graph theory, enabling us to classify graphs based on their intrinsic properties rather than superficial representations.

Final Thoughts

Graph isomorphism is more than just a technical exercise—it's a fundamental tool for understanding the essence of a graph. Whether you're analyzing social networks, molecular structures, or communication protocols, knowing whether two graphs are isomorphic can reveal deep structural similarities Not complicated — just consistent..

By following the structured approach outlined in this flowchart, you can confidently determine whether two graphs are identical in structure, even if they look different at first glance.

In summary:
To determine if two graphs are identical (i.e., isomorphic), always start with the simplest invariants—vertex and edge counts, degree sequences, connectivity—and proceed through increasingly detailed checks. Only when all else matches should you attempt to construct an explicit isomorphism or use matrix-based methods for final confirmation.