Let graphs  Q（C

k

C

s1

C

s2

…

C

sk

）

  Q(P

k

C

s1

C

s2

…

C

sk

) 

  Q(W

k

C

s1

C

s2

…

C

sk

) be obtained from C

k

P

k

W

k

 with every vertex  v

i

(i=1

2

…

k)(The center vertex of   W

k

can not be contained) attached one cycle C

si

（i=1

2

…

k）

respectively. The highest bounds of Merrifield-Simmons indexes of Q（C

k

C

s1

C

s2

…

C

sk

）

Q(W

k

C

s1

C

s2

…

C

sk

)  graphs are determined. The two graphs Q（C

k

C

4

C

4

…

C

4

C

s1+s2+…+sk－4(k－1)

） andQ（W

k

C

4

C

4

…

C

4

C

s1+s2+…+sk－4(k－1)

） which have the largest Merrifield-Simmons index of those graphs are characterized

respectively.