From my current basic organic chemistry knowledge, I suppose it is 1,1-dimethylcyclohexane-4-methylhexane. I considered one of the hexane ring as parent chain. But I'm looking for the correct answer with appropriate explanation for it.
The relevant sections on ring assemblies of two identical cyclic systems in the current version of Nomenclature of Organic Chemistry – IUPAC Recommendations and Preferred Names 2013 (Blue Book) actually read as follows:
P-28.2.1 Ring assemblies with a single bond junction
Assemblies of two identical cyclic systems joined by a single bond are named by one of two methods:
(1) by placing the prefix ‘bi’ (see P-14.2.3) before the name of the corresponding parent hydride enclosed in parentheses, if necessary. Parentheses are used to avoid confusion with von Baeyer names;
(2) by placing the prefix ‘bi’ (see P-14.2.3) before the name of the corresponding substituent group (for names of substituent groups, see P-29), enclosed in parentheses, if necessary.
Each cyclic system is numbered in the traditional way, one with unprimed locants, the other with primed locants. Lowest possible locants must be used to denote the positions of attachment. These locants must be cited in preferred IUPAC names; they can be omitted in general nomenclature when no ambiguity would result.
P-18.104.22.168 Preferred IUPAC names for assemblies of two or more identical cyclic systems joined by a single bond are formed using the names of parent hydrides rather than the names of substituent groups, (…). For two-component assemblies, locants of one ring are unprimed; locants of the second ring are primed. Locants, including the locants 1 and 1', are necessary in preferred IUPAC names to indicate points of attachment of rings or ring systems (see P-28.2.1 ).
Therefore, the preferred IUPAC name for the compound that is given in the question is formed according to Rule (1) as 4,4'-dimethyl-1,1'-bi(cyclohexane).
The corresponding name according to Rule (2) is 4,4'-dimethyl-1,1'-bi(cyclohexyl). This name may be used in general nomenclature.
Chemspider claims 4,4'-Dimethyl-1,1'-bi(cyclohexyl) was the [ACD/IUPAC Name].
The parent structure are 2 linked cyclohexane rings.
User @orthocresol found
P-22.214.171.124 of the Blue Book, "Preferred IUPAC names ... are formed using the names of parent hydrides rather than the names of substituent groups"; hence cyclohexane rather than cyclohexyl)
The outdated 1993 version had this rule
R-126.96.36.199 Assemblies of two identical cyclic systems are named in one of two ways:
(a) by placing the prefix 'bi-' before the name of the corresponding substituent group (see R-2.5) enclosed in parentheses, if necessary (additive operation); or
(b) by placing the prefix 'bi-' before the name of the corresponding parent hydride enclosed in parentheses, if necessary (conjunctive operation).
According to that rule, both bi(cyclohexyl) (a) and bi(cyclohexane) (b) were valid. However, I do not have access to the to the current version of the blue book at the moment and do not know, whether this rule still exists that way. Would be great, if someone could comment on this. I will continue using bi(cyclohexane).
Regarding parentheses, user @orthocresol found
P-28.2.1 "Parentheses are used to avoid confusion with von Baeyer names."
or, as described in an Wikipedia article:
Fused and bridged bicyclic compounds get the prefix bicyclo
Since the rings are not fused, the carbon atoms are numbered separately in both rings, i.e. 1 to 6 in one ring, and 1' to 6' in the other.
The numbering starts in each ring at the carbon atom which is linked to the highest molecular group, i.e. the C that links to the other ring. This gives the 1,1'-bi(cyclohexane) part.
Using this numbering, you end up at the C4 and C4' as the number for the carbon atoms that are linked with a methyl-group each. Thus, the name is prefixed with 4,4'-dimethyl-.
Putting this all together, a correct IUPAC name should be 4,4'-dimethyl-bi(cyclohexane).
According to PubChem the IUPAC name is 1-methyl-4-(4-methylcyclohexyl)cyclohexane.
In this case, they ignored the ring assembly and used only one cyclohexane ring as a parent and treated the other one as a substituent.