The resistance distance
As the number of possible chemical compounds is so big, their huge part will never be experimentally tested. For this reason, there is a need for mathematical modeling and analysis of certain classes of compounds. To this end, many topological indices are defined and applied in the modeling of chemical and pharmacological properties of molecules. In the present work, we will take a new molecular cyclicity measure into consideration.
There are different possible measures of “cyclicity” of a molecular graph
As a new measure of cyclicity of graphs, the global cyclicity index has less degeneracy than the standard cyclomatic number and has some intuitively appealing features. Since the idea of cyclicity is related to measures of connectivity or complexity [
A
In [
In this subsection, we will compute resistance distances between pairs of adjacent vertices in linear polyacenes. Denote by
A benzenoid chain with
To compute resistance distances between pairs of adjacent vertices in
Let
Now suppose that
Let
In the enumeration of spanning trees of graphs, there is a famous recursion formula, known as the deletioncontraction recurrence. As stated in the following lemma, the recursion formula is applicable to
Let
Combining Lemmas
Let
For computing resistance distances between adjacent vertices in
Consider
Now we are arriving at Theorem
Resistance distances between pairs of adjacent vertices of a linear polyacene could be computed as follows.
For
We first compute resistance distances between adjacent degree two and degree three vertices. For
Now, we compute resistance distances between
In [
All benzenoid chains with the same number of hexagons have the same global cyclicity index.
Now we are ready to give the main result of the paper.
Let
By Lemma
As numerical results, global cyclicity indices of benzenoid chains from












1.2 

13.927 

26.669 

39.39 

52.122 

2.469 

15.2 

27.932 

40.664 

53.395 

3.7419 

16.474 

29.205 

41.937 

54.668 

5.0151 

17.747 

30.478 

43.21 

55.942 

6.2883 

19.02 

31.752 

44.483 

57.215 

7.5614 

20.293 

33.025 

45.756 

58.488 

8.8346 

21.566 

34.298 

47.029 

59.761 

10.108 

22.839 

35.571 

48.303 

61.034 

11.318 

24.113 

36.844 

49.576 

62.307 

12.654 

25.386 

38.117 

50.849 

63.581 
The authors acknowledge the support of the National Natural Science Foundation of China under Grant no. 11371307. Y. Yang acknowledges the support of the National Natural Science Foundation of China under Grant no. 11201404, China Postdoctoral Science Foundation under Grant nos. 2012M521318 and 2013T60662, Special Funds for Postdoctoral Innovative Projects of Shandong Province under Grant no. 201203056, and Shandong Province Higher Educational Science and Technology Program through Grant J12LI05. Y. Wang and Y. Li acknowledge the support of the Natural Science Foundation of Shandong Province under Grant no. ZR2011AM005.