You are an expert at summarizing long articles. Proceed to summarize the following text: 
for general notations and concepts in graph theory , we refer to @xcite , @xcite and @xcite . all graphs mentioned in this paper are simple , connected undirected and finite , unless mentioned otherwise . a _ hole _ of a simple connected graph @xmath0 is a chordless cycle @xmath3 , where @xmath4 , in @xmath0 . the _ girth _ of a simple connected graph @xmath0 , denoted by @xmath5 , is the order of the smallest cycle in @xmath0 . the following notions are introduced in @xcite . @xcite a _ primitive hole _ of a graph @xmath0 is a cycle of length @xmath1 in @xmath0 . the number of primitive holes in a given graph @xmath0 is called the _ primitive hole number _ of that graph @xmath0 . the primitive hole number of a graph @xmath0 is denoted by @xmath6 . @xcite the _ primitive degree _ of a vertex @xmath2 of a given graph @xmath0 is the number of primitive holes incident on the vertex @xmath2 and the primitive degree of the vertex @xmath2 in the graph @xmath0 is denoted by @xmath7 . some studies on primitive holes of certain graphs have been made in @xcite . the number of primitive holes in certain standard graph classes , their line graphs and total graphs were determined in this study . some of the major results proved in @xcite are the following . @xcite the number of primitive holes in a complete graph @xmath8 is @xmath9 . @xcite for any subgraph @xmath10 of a graph @xmath0 , we have @xmath11 . moreover , if @xmath0 is a graph on @xmath12 vertices , then @xmath13 . in this paper , we introduce the notion of set - graphs and study certain characteristics of set - graphs and also present a number of interesting results related to graph properties and invariants . a set - graph is defined as follows . [ d - sg ] let @xmath14 be a non - empty set and the @xmath15-th @xmath16-element subset of @xmath17 be denoted by @xmath18 . now consider @xmath19 . the _ set - graph _ corresponding to set @xmath17 , denoted @xmath20 , is defined to be the graph with @xmath21 and @xmath22 , where @xmath23 . it can be noted from the definition of set - graphs that @xmath24 and if @xmath25 is a singleton , then @xmath20 to be the trivial graph . hence , all sets we consider here are non - empty , non - singleton sets . let us now write the vertex set of a set - graph @xmath20 as @xmath26 , where @xmath16 is the cardinality of the subset @xmath27 of @xmath17 corresponding to the vertex @xmath28 . the following result is perhaps obvious , but an important property of set - graphs . if @xmath0 is a set - graph , then @xmath0 has odd number of vertices . let @xmath0 be a set - graph with respect to the set @xmath17 . it is to be noted the number of non - empty subsets of @xmath17 is @xmath29 . since every vertex of @xmath0 corresponds to a non - empty subset of @xmath17 , the number of vertices in @xmath0 must be @xmath29 , an odd integer . consider the set - graph with respect to the set @xmath30 . here we have the subsets of @xmath31 which are @xmath32 . then , the vertices of @xmath33 have the labeling as follows . @xmath34 . figure [ fig-1 ] depicts the above mentioned labeling procedure of the set - graph @xmath33 . [ t - sgdv ] let @xmath20 be a set - graph . then , the vertices @xmath35 of @xmath20 , corresponding to subsets @xmath36 and @xmath37 in @xmath38 of equal cardinality , have the same degree in @xmath20 . that is , @xmath39 . consider the set - graph @xmath40 . we begin by considering the vertices of @xmath0 corresponding to the @xmath12 singleton subsets of @xmath17 . let these vertices be denoted by @xmath41 , where @xmath42 . clearly , for all @xmath43 , we have @xmath44 . hence , by the definition of set - graphs , it follows that no edges are induced amongst the vertices @xmath45 . now , construct all the two element subsets of @xmath17 . now choose two arbitrary vertices @xmath46 and @xmath47 , where @xmath48 . then , here we have the subsets of @xmath17 of the form @xmath49 , for @xmath50 . it can be observed that the subsets of the form @xmath49 and @xmath51 are the elements of @xmath38 , where @xmath52 . moreover , @xmath53 for all @xmath54 . in a similar way , we can extend this argument for the sets @xmath55 and an arbitrary subset of @xmath17 containing the element @xmath56 . that is , the vertex @xmath41 is adjacent to those vertices of @xmath0 whose corresponding sets have @xmath57 elements including the common elements @xmath56 , for @xmath58 . therefore , @xmath59 . since the choice of @xmath15 is arbitrary , we have @xmath60 for all @xmath61 . therefore , the result holds for @xmath62 . now , assume that the result holds for @xmath63 , where @xmath64 is a positive integer . that is , we have @xmath65 for all @xmath66 . next , consider the vertices of @xmath0 corresponding to the @xmath67-element subsets of @xmath17 . let @xmath68 be a @xmath67-element subset of @xmath17 and let @xmath69 be the vertex of @xmath0 corresponding to the set @xmath68 . let @xmath70 be an arbitrary element of the set @xmath68 and let @xmath71 . then , the vertex @xmath69 is adjacent to the vertices of @xmath0 corresponding to the sets containing the element @xmath70 in addition to the vertices of @xmath0 corresponding to the proper subsets of @xmath68 and @xmath72 . hence , the difference between the number of edges incident on @xmath69 and the number of edges incident on the vertex @xmath73 corresponding to the set @xmath72 is equal to the number of subsets of @xmath17 containing the element @xmath70 , other than @xmath68 . this number is a constant for any set of @xmath67-element sets . therefore , @xmath74 for all @xmath75 . that is , the result is true for @xmath76 if it is true for @xmath63 . therefore , the theorem follows by induction .    a question that arouses much interest in this context is what the degree of an arbitrary vertex of a set - graph @xmath20 . the following result provides a solution to this problem . let @xmath0 be a set - graph with respect to a non - empty set @xmath77 and let @xmath78 be an arbitrary vertex of @xmath0 corresponding to an @xmath64-element subset of @xmath17 . then , @xmath79 , where @xmath80 is an indexing set such that @xmath81 and @xmath82 is the collection of subsets of @xmath17 containing the element @xmath83 . let @xmath0 be a set - graph with respect to a non - empty set @xmath17 . without loss of generality , let @xmath84 be a @xmath64-element subset of @xmath17 , say @xmath85 and let @xmath78 be the vertex of @xmath0 corresponding to the set @xmath84 . therefore , the vertex @xmath78 is adjacent to the vertices of @xmath0 which correspond to the subsets of @xmath17 , containing the at least one element of @xmath84 . that is @xmath86 . but , by principle of inclusion and exclusion of sets , we have @xmath87 , where @xmath81 . therefore , @xmath79 , where @xmath81 .    determining the degree of vertices of a set - graph is an important and interesting problem at this time . the following result determines a lower and upper limits for the degree of vertices of a given set - graph . [ t - sgdv1 ] for any vertex @xmath88 of a set - graph @xmath89 , we have @xmath90 . let @xmath89 be a set - graph with respect to a non - empty set @xmath17 . here , we need to consider the following two cases . it is to be noted that the vertices of @xmath0 corresponding to singleton subsets of @xmath0 have the minimum degree in @xmath0 . without loss of generality , let the vertex @xmath88 of @xmath0 corresponds to the set @xmath55 . then , @xmath88 should be adjacent to the vertices of @xmath0 corresponding to the subsets of @xmath17 , other than itself , containing the element @xmath56 . therefore , degree of the vertex @xmath88 is equal to the the number of @xmath57-element subsets of @xmath17 containing the element @xmath56 for @xmath91 . by binomial theorem , the total number of subsets of an @xmath12-element set , containing a particular element is @xmath92 . therefore , the minimum degree of a vertex in @xmath0 is @xmath93 . note that we need to consider @xmath94 of the subsets of @xmath17 only excluding @xmath95 . hence , the final vertex @xmath96 of the graph @xmath89 corresponding to the set @xmath17 in @xmath38 will be adjacent to all its preceding vertices . since @xmath0 has @xmath29 vertices , @xmath97 . no other vertices in @xmath0 can be adjacent to all other vertices of @xmath0 , the vertex @xmath96 has the maximum possible degree in @xmath0 . that is , the maximum degree of a vertex in @xmath0 is @xmath98 . the following results are immediate consequences of the above theorem . for any set - graph @xmath89 , @xmath99 .    from the proof the above theorem , we have @xmath100 and @xmath101 . this completes the proof . [ c - udg ] there exists a unique vertex @xmath96 in a set - graph @xmath20 having degree @xmath102 . the proof follows from case-2 of theorem [ t - sgdv1 ] . the following result indicates the nature of the minimal and maximal degrees of the vertices of a set - graph . the maximal degree of vertex in a set - graph @xmath0 is always an even number and the minimal degree of a vertex in @xmath0 is always an odd number . let @xmath0 be a set - graph with respect to a non - empty set @xmath17 . then , by theorem [ t - sgdv1 ] , the maximum degree of a vertex in a set - graph @xmath0 is @xmath103 , which is always an even number and the minimal degree of a vertex @xmath0 is @xmath104 , which is always an odd number .    we have already proved that the vertex of the set - graph @xmath0 corresponding to the set @xmath17 itself has the maximum degree @xmath105 in @xmath0 . analogous to this result , we propose the following result on the primitive degree of this vertex @xmath96 .    for set - graph @xmath89 , the primitive degree of the vertex corresponding the set @xmath17 is @xmath106 . let @xmath89 be a set - graph with respect to the set @xmath17 . consider the subgraph @xmath107 . by theorem [ t - sgdv ] , we have @xmath108 and hence @xmath109 . furthermore , both ends ends of every edge @xmath110 are adjacent to vertex @xmath96 in @xmath20 . hence , each such edge @xmath111 in @xmath112 corresponds to a primitive hole @xmath113 in @xmath20 on the vertices @xmath114 . hence , @xmath115 .    in the result given below , we describe a recursive formula to determine the number of edges of a set - graph . [ t - sgrd ] for a set - graph @xmath116 we have    1 . @xmath117 2 . @xmath118 . consider the set - graph @xmath20 . to extend it to @xmath116 , we proceed in five steps as explained below .    1 . replicate the vertices of @xmath20 as an _ edgeless _ graph and add the new element @xmath119 as an element to all subsets corresponding to all @xmath120 and label each _ replica vertex _ , @xmath121 . also add the new vertex @xmath122 corresponding to the single element subset @xmath123 . apply the definition of a set - graph to these new vertices . clearly , we obtain the complete graph @xmath124 . each vertex @xmath120 corresponding to the subset @xmath18 can be linked with its replica vertex corresponding to the new subset @xmath125 . we refer to _ parallel linkage _ and exactly @xmath126 such edges are added . 4 .   for the ends of each edge in @xmath127 say @xmath128 and @xmath129 with corresponding subsets say , @xmath130 and @xmath131 we have @xmath132 and @xmath133 . so the edges @xmath134 and @xmath135 , with @xmath136 corresponding to subsets @xmath137 and @xmath138 respectively , exist . hence @xmath139 additional edges are linked . relabel the vertices according to the definition [ d - sg ] to obtain the set - graph @xmath116 . the summation of the edges added through steps ( i ) to ( v ) plus the existing edges of @xmath20 provides the result : @xmath117 . the second part of the theorem is an immediate consequence of the above proof of first part . then , the proof is complete . the following is a result related to the largest complete graphs found in a set - graph . [ p - sgkn ] the set - graph @xmath140 has exactly two largest complete graphs , @xmath141 . consider the set - graph @xmath142 and extend to @xmath20 . from step ( ii ) in the proof of theorem [ t - sgrd ] , we construct a largest complete graph amongst the replica vertices and the new vertex @xmath143 because no vertex of @xmath142 is linked to @xmath144 . however , the erstwhile vertex @xmath145 of @xmath142 is also linked to all the _ replica _ vertices hence , inducing the complete graph @xmath141 . clearly , another largest complete graph does not exist . the primitive hole number of a set - graph is determined recursively in the following theorem .    for a set - graph @xmath146 we find the number of primitive holes through the recursive formula @xmath147 . consider the set - graph @xmath20 whose primitive hole number is denoted by @xmath148 . extending @xmath20 to @xmath116 will only increase the number of primitive holes . what we need here to determine the number of additional primitive holes formed on extending @xmath20 to @xmath116 . this calculation is done as follows . the set of replica vertices together with the vertex @xmath149 induce a complete subgraph @xmath150 and hence an additional @xmath151 primitive holes are added to the extended graph . finally , for each edge @xmath152 the vertices @xmath153 induce a @xmath154 subgraph and the number of primitive holes thus formed is @xmath155 . hence , a further @xmath156 primitive holes are added to the extended graph . this completes the proof . next , we introduce the following notions for a set - graph as follows . let @xmath89 be a set - graph on a non - empty set @xmath17 and let @xmath18 be an arbitrary subset of the set @xmath17 . characteristic function _ of a subset @xmath157 of @xmath17 with respect to @xmath18 , denoted by @xmath158 , is defined as @xmath159     the _ tightness number _ of a subset @xmath160 , denoted @xmath161 is the number of subsets distinct from @xmath130 for which the intersection with @xmath130 is non - empty . hence , @xmath162 . we note that in terms of the definition of a set - graph and for the vertex @xmath88 corresponding to the subset @xmath130 we have , @xmath163 . also , we have that @xmath164 . the next theorem enables us to employ a step - wise recursive formula to determine the tightness number of all non - empty subsets of @xmath165 if the the tightness number of all non - empty subsets of @xmath17 are known . consider a set - graph @xmath166 and its extended set - graph @xmath116 . then we have    1 . @xmath167 , 2 .   for each erstwhile subset @xmath36 with @xmath168 in @xmath20 , we have @xmath169 in @xmath116 , 3 .   for a replica vertex say , @xmath121 representing the new subset @xmath125 we have @xmath170 . let @xmath20 be a set - graph and @xmath116 be its extended graph obtained by introducing a new element , say @xmath119 , to the set @xmath17 . then ,    1 .   to generate the set - graph @xmath116 by extending the set - graph @xmath20 , we initially add the subsets @xmath123 and @xmath125 , for all applicable values of @xmath16 and @xmath15 . clearly , @xmath171 . so , @xmath172 . also , @xmath173 . therefore , we have @xmath174 . 2 .   if @xmath175 in @xmath20 , the subset @xmath18 has non - zero intersections with exactly @xmath64 distinct subsets of @xmath17 . since in the replication , we have @xmath176 together with the subsets @xmath177 , for all @xmath64 , the result follows . the replica vertices are @xmath29 in number and induce a complete graph together with vertex @xmath122 . this partially represents @xmath94 non - zero intersections in respect of any subset say , @xmath176 corresponding to any replica vertex say , @xmath121 . clearly , @xmath178 and @xmath179 , for all @xmath180 . it implies that an additional @xmath67 non - zero intersections exist in respect of @xmath176 . hence , @xmath181 . this completes the proof . let @xmath0 be a given non - trivial finite graph . the _ chromatic number _ , denoted by @xmath182 , of @xmath0 is the minimum @xmath64 for which @xmath0 is @xmath64-colourable . the chromatic number of a set - graph @xmath20 is @xmath183 . it is easy to see that @xmath184 , @xmath185 , @xmath186 ( see figure [ fig-1 ] ) . assume the result holds for the set - graph @xmath187 therefore , we have @xmath188 . now consider the set - graph @xmath189 . from the steps to be followed to extend from @xmath187 to @xmath189 ( see proof of theorem [ t - sgrd ] ) , we have the erstwhile vertices of @xmath187 , and in addition , the replica vertices corresponding to the vertices of @xmath187 and one more vertex @xmath190 . from the proof of proposition [ p - sgkn ] we can notice that the replica vertices and vertex @xmath190 induce a largest complete subgraph , @xmath191 in the extended graph of @xmath20 . we also note that the replica vertices and vertex @xmath192 form a second largest complete subgraph , @xmath191 .    since the vertices @xmath192 and @xmath193 are not adjacent in @xmath20 , both of them have the same colour , say @xmath194 and colour the replica vertices by the colours @xmath195 . since , no other largest complete graph exists it is always possible to find at least one pair of erstwhile - replica vertices which are non - adjacent . hence the erstwhile vertex can carry the colour of such a replica vertex . this can be done in such a way that two adjacent erstwhile vertices do not carry the same colour by using the colours @xmath196 accept for the colour of @xmath96 , exhaustively . so the result @xmath197 follows . hence , the main result follows by induction . an _ independent set _ of graph @xmath0 is a set of mutually non - adjacent vertices of @xmath0 . the _ independence number _ , denoted by @xmath198 , of @xmath0 is the cardinality of a maximal independent set of @xmath0 . the independence number of a set - graph is determined in the following theorem . the independence number of a set - graph @xmath20 is @xmath199 . let @xmath200 be a given set - graph . then , as explained in the proof of theorem [ t - sgdv ] , the vertices @xmath201 , corresponding to the singleton subsets of @xmath17 , are pairwise non - adjacent . hence , the set @xmath202 is an independent set . by the definition [ d - sg ] , we note that any vertex in @xmath203 is adjacent to at least one vertex in @xmath204 . therefore , @xmath204 is the maximal set of mutually non - adjacent vertices and hence is the maximal independent set in @xmath0 . hence , @xmath199 . a _ domianting set _ of a graph is a set of vertices @xmath205 such that every vertex of @xmath0 is either in @xmath205 or is adjacent to at least one vertex in @xmath205 . the _ domination number _ , denoted by @xmath206 , of a graph @xmath0 is the cardinality of the minimal dominating set of @xmath0 . the following discusses the domination number of a set - graph . the domination number of a set - graph@xmath20 is @xmath207 . let @xmath89 be a given set - graph . by corollary [ c - udg ] , the vertex @xmath96 , corresponding to the @xmath12-element set @xmath17 , is the unique vertex in the set - graph @xmath0 that is adjacent to all other vertices of the set - graph @xmath0 . therefore , the singleton set @xmath208 is the minimal set such that every vertex of @xmath0 is adjacent to the unique element in the set @xmath205 . therefore , @xmath209 . another parameter we consider here is the bondage number of a graph @xmath0 , which is denoted by @xmath210 defined as the minimum number of edges to be removed to increase the domination number @xmath206 by @xmath211 . the bondage number of a set - graph @xmath20 is @xmath212 . since @xmath208 is the minimal dominating set of the set - graph @xmath89 , the removal of any edge @xmath213 will increase the domination number by @xmath211 in the reduced graph @xmath214 . another parameter we are going to discuss here is the mcphersion number of undirected graphs . for this , let us now recall the definition mcphersion number , as given in @xcite . @xcite the _ mcpherson recursion _ is a series of _ vertex explosions _ such that on the first iteration a vertex @xmath215 explodes to arc ( directed edges ) to all vertices @xmath216 for which the edge @xmath217 , to obtain the mixed graph @xmath218 . now @xmath218 is considered on the second iteration and a vertex @xmath219 may explode to arc to all vertices @xmath220 if edge @xmath221 and arc @xmath222 or @xmath223 .    the _ mcpherson number _ , denoted by @xmath224 , of a simple connected graph @xmath0 is the minimum number of iterative vertex explosions say @xmath225 , to obtain the mixed graph @xmath226 such that the underlying graph @xmath227 . the mcpherson number of a set - graph @xmath20 is determined in the following theorem .    for a set - graph @xmath20 we have @xmath228 . consider the set - graph @xmath142 which has @xmath93 vertices . on extending the set - graph @xmath20 , the replica vertices , as explained in theorem [ t - sgrd ] , together with vertex @xmath96 induce a complete subgraph and hence no further vertex explosions are required to ensure complete induced by these vertices . however , at least all the erstwhile vertices require vertex explosions to ensure complete connectivity amongst themselves and the replica vertices . hence , @xmath229 . we have discussed particular types of graphs called set - graphs and studied certain characteristics and structural properties of these graphs . the study seems to be promising as it can be extended to certain standard graph classes and certain graphs that are associated with the given graphs . more problems in this area are still open and hence there is a wide scope for further studies . now , we have the notion of mla numbers as follows . finding other number theoretical results for _ mla numbers _ are also challenging problems which seems to be promising . all these facts indicate that there is a wide scope for further research in this area .
a _ primitive hole _ of a graph @xmath0 is a cycle of length @xmath1 in @xmath0 . 
 the number of primitive holes in a given graph @xmath0 is called the primitive hole number of that graph @xmath0 . the primitive degree of a vertex @xmath2 of a given graph @xmath0 is the number of primitive holes incident on the vertex @xmath2 . in this paper , we introduce the notion of set - graphs and study the properties and characteristics of set - graphs . 
 we also check the primitive hole number and primitive degree of set - graphs . 
 interesting introductory results on the nature of order of set - graphs , degree of the vertices corresponding to subsets of equal cardinality , the number of largest complete subgraphs in a set - graph etc . are discussed in this study . 
 a recursive formula to determine the primitive hole number of a set - graph is also derived in this paper . 
 * key words : * set - graphs , primitive hole , primitive degree .    * ms classification * : 05c07 , 05c38 , 05c78 .