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G. C. Ying, Y. Y. Meng, B. E. Sagan, and V. R. Vatter [1] found the maximum number of maximal independent sets in connected graphs which contain at most two cycles. In this paper, we give an alternative proof to determine the largest number of maximal independent sets among all connected graphs of order n ≥ 12, which contain at most two cycles. We also characterize the extremal graph achieving this maximum value.

Let

The problem of determining the largest value of

There are results on independent sets in graphs from a different point of view. The Fibonacci number of a graph is the number of independent vertex subsets. The concept of the Fibonacci number of a graph was introduced in [

Jou and Chang [

For a graph

The following results are needed.

Lemma 1. ( [

1)

2) If x is a leaf adjacent to y, then

Lemma 2. ( [

Lemma 3. Let

Then

Proof. The derivative of

So

□Theorem 1. ( [

Furthermore,

is the set of batons, which are the graphs obtained from a basic path P of

Theorem 2. ( [

Furthermore,

where

Theorem 3. ( [

Furthermore,

Theorem 4. ( [

Furthermore,

Theorem 5. ( [

Furthermore,

In this section, we give an alternative proof to determine the largest number of maximal independent sets among all connected graphs of order

Theorem 6. If H is a connected graph of order

Furthermore,

A unicyclic graph is a connected graph having one cycle. The order of a unicyclic graph is at least three. The following lemmas will be needed in the proof of main theorem.

Lemma 4. Suppose that

Proof. Let

Case 1.

By Lemma 2, Theorem 1 and Theorem 5, we have

If the equality holds, then

Hence the equality holds if and only if

Case 2.

By Lemma 3 and since

By Theorem 1 and Theorem 5, we have

If the equality holds, then

By case 1 and case 2, we have that

only if

Lemma 5. Suppose that F is a forest of order

Proof. Let F be a forest of order

n is odd and

have two components. Let

Case 1.

By Lemma 3,

The equalities hold and

Case 2.

Then F has exactly one even component, we assume that

equalities hold and

The following is the proof of Theorem 6.

Proof. Let H be a connected graph of order

Case 1.

Then

that v connects to every component of

Then

Case 2.

Let

Then

Claim.

Suppose that

and

By Theorem 2,

Theorem 5,

Lemma 5,

where

By Claim,

The equalities hold. Then

diction. Thus

some cycle such that

for odd

The authors are grateful to the referees for the helpful comments.

Jou, M.-J. and Lin, J.-J. (2016) An Alternative Proof of the Largest Number of Maximal Independent Sets in Connected Graphs Having at Most Two Cycles. Open Journal of Discrete Mathematics, 6, 227-237. http://dx.doi.org/10.4236/ojdm.2016.64019