Figure 3: A connected graph in Case 2.

3. Multisimilar Equivalence Relation In this section, we investigate another equivalence relation on a vertex set of a connected graph. First, we need some additional definitions and notations. Let be a collection of multisets. For an integer , we define where . Let be a set of vertices of a connected graph and let and be vertices of . A multisimilar relation with respect to on a vertex set is defined by if there is an integer such that An integer satisfying (12 ) is called a multisimilar constant of or simply a multisimilar constant . Clearly, is an equivalence relation on . For each vertex in , let denote the multisimilar equivalence class of with respect to . Then where is a multisimilar constant. Observe that if , then there is a multisimilar constant with a property that, for every vertex , there is a corresponding vertex such that With this observation, we may as well say that if and only if there are multisimilar constant and a bijective function on defined as The function is called a multisimilar function of or a multisimilar function if there is no ambiguity. Consequently, it is not surprising that an inverse function is also multisimilar function of with a multisimilar constant .

To illustrate these concepts, consider a vertex in a connected graph of Figure 1 and the set . There is only one vertex in such that is related to by a multisimilar relation with a multisimilar constant ; that is, Therefore, . Thus, a multisimilar function of is defined by Moreover, there is another multisimilar function of ; that is, The example just described shows an important point that the multisimilar function of any two vertices in the same multisimilar equivalence class with respect to a set is not necessarily unique.

More generally, for a vertex and a set of vertices of a connected graph , let , where and is a repetition number of type for each with . If , where , then it follows by (13 ) and (14 ) that, for each type of , there is a corresponding type of such that their repetition numbers are equal. Therefore, we may assume that , where . For each integer with , let and . Then the types of partition into sets . On the other hand, is also partitioned into sets depending on the types of . Hence, the multisimilar function of has the property that, for every vertex , there is a vertex such that where . Indeed, there are distinct multisimilar functions of . These observations yield the following result.

Theorem 6. Let be a set of vertices of a connected graph and let and be vertices of such that . Suppose that , where and is a repetition number of type for each with . Then (i) for some integers with ,(ii) there is a multisimilar function of such that , where and for each with , (iii) there are distinct multisimilar functions of .

By Theorem 6 , the following result is obtained.

Corollary 7. Let be a set of vertices of a connected graph and let and be vertices of such that with a multisimilar constant . Then (i) if and are the maximum elements of and , respectively, then , (ii) if and are the minimum elements of and , respectively, then .

Proof. Suppose that . Let and , where and . Since and are the maximum elements of and , respectively, there are vertices and in such that and . It follows by Theorem 6 that there is a multisimilar function of such that . Then , where is a multisimilar constant. Thus, (i) holds. For (ii), the statement may be proven in the same way as (i), and therefore such proof is omitted.

Next, we are prepared to establish the upper bound for the cardinality of a multisimilar equivalence class of a vertex in a connected graph. To show this, let us present a useful proposition as follows.

Proposition 8. Let be a set of vertices of a connected graph and let and be vertices of such that . Then and have the same minimum (or maximum) element if and only if .

Proof. If , then the minimum (and maximum) elements of and are the same. For the converse, assume that and are the minimum elements of and , respectively, such that . Since , there is a multisimilar constant such that By Corollary 7 (ii), it follows that . Thus, . Hence, . Similarly, if and have the same maximum element, then .

Theorem 9. If is a multiresolving set of a connected graph , then the cardinality of multisimilar equivalence class of each vertex of with respect to is at most .

Proof. Assume, to the contrary, that there is a vertex of such that has the cardinality at least . Since the minimum elements of multirepresentations of vertices in with respect to have at most distinct values, there are at least two vertices and in having the same value of the minimum element of and . It follows by Proposition 8 that , contradicting the fact that is a multiresolving set of .

We can show that the upper bound in Theorem 9 is sharp. Consider the path . We have that and the set is a multiresolving set of . Thus, contains all vertices of , and so .

In the last result, we describe the properties of a multisimilar equivalence classes with respect to a set of vertices.

Theorem 10. Let and be vertices of a connected graph and let be a set of vertices of . Then (i) if , then for all and , (ii) if for all , then is a multiresolving set of .

Proof. (i) Assume, to the contrary, that there exist two distinct vertices and such that . Then there are multisimilar constants and such that and . Therefore, Thus, Hence, belongs to , which is a contradiction. (ii) Assume, to the contrary, that is not a multiresolving set of . Then there exist two distinct vertices and such that . Hence, belongs to , producing a contradiction.

4. Final Remarks The complete graph is only one graph that its dimension is but not so for multidimensions. It follows by [10 , 11 ] that the multidimension of complete graph is not defined. Thus, (5 ) leads us to the conjecture: If is a connected graph such that is defined, then .

Data Availability No data sharing was used to support this study as no datasets were generated or analyzed during the current study. Other data sources are referenced throughout the paper.

Conflicts of Interest The authors declare that they have no conflicts of interest.

Acknowledgments This work was funded by the Faculty of Science, Srinakharinwirot University.

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