发帖
雷达卡
1 Graphs and Weighted Graphs ................................... 1
1.1 Graph Theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2
1.2 Weighted Graph Theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .10
1.3 Heavy Paths and Optimal Cycles ............................ 11
1.4 Tritrees and Cyclic Extremal Graphs . . . . . . . . . . . . . . . . . . . . . . .37
1.5 Weighted Digraphs . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 47
References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 50
2 Connectivity ................................................... 53
2.1 Strength of Connectedness . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .53
2.2 Strong Edges and Strong Paths . . . . . . . . . . . . . . . . . . . . . . . . . . 57
2.3 Partial Blocks . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 61
2.4 Partial Trees .............................................. 66
References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 69
3 More on Connectivity ........................................... 71
3.1 Precisely Weighted Graphs . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 71
3.2 Generalized Menger’s Theorem . . . . . . . . . . . . . . . . . . . . . . . . . . 80
3.3 Generalized Connectivity Parameters . . . . . . . . . . . . . . . . . . . . . . .85
References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 88
4 Cycle Connectivity ............................................. 89
4.1 Strongest Strong Cycles . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .89
4.2 Cycle Connectivity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 91
4.3 Cycle Connectivity of Weighted Graphs . . . . . . . . . . . . . . . . . . . . . 94
4.4 Cyclic Cut Vertices and Cyclic Bridges . . . . . . . . . . . . . . . . . . . . . . .95
4.5 Cyclically Balanced Graphs . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 98
5 Distance and Convexity ......................................... 105
5.1 Weighted Distance . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .105
5.2 Self Centered Weighted Graphs . . . . . . . . . . . . . . . . . . . . . . . . . 111
5.3 The Distance Matrix . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 113
5.4 Weighted Center of Trees ................................... 115
5.5 Complement of a Weighted Graph ........................... 116
5.6 Geodetic Convex Sets . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .117
5.7 Geodetic Blocks . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .119
5.8 Geodetic Boundary and Interior . . . . . . . . . . . . . . . . . . . . . . . . 123
5.9 Monophonic Convex Sets . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 125
5.10 Monophonic Blocks . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .127
5.11 Monophonic Boundary and Interior . . . . . . . . . . . . . . . . . . . . . . . 130
References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 132
6 Degree Sequences and Saturation ................................ 133
6.1 Sequences in a Weighted Graph . . . . . . . . . . . . . . . . . . . . . . . . .133
6.2 Characterization of Partial Blocks . . . . . . . . . . . . . . . . . . . . . . . . 134
6.3 Characterization of Partial Trees ............................. 137
6.4 Vertex and Edge Saturation Counts . . . . . . . . . . . . . . . . . . . . . . . 141
6.5 Saturation ................................................ 146
References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 154
7 Intervals and Gates ............................................. 155
7.1 Distances in Weighted Graphs . . . . . . . . . . . . . . . . . . . . . . . . . . . .155
7.2 Intervals in Weighted Graphs . . . . . . . . . . . . . . . . . . . . . . . . . . . . 161
7.3 Operations on Strong Intervals . ............................. 167
7.4 Strong Gates in Weighted Graphs . . . . . . . . . . . . . . . . . . . . . . . . 168
References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 177
8 Weighted Graphs and Fuzzy Graphs ............................. 179
8.1 Isomorphisms of Different Families of Fuzzy Sets . . . . . . . . . . . . . . 181
8.2 Weighted Graphs and Fuzzy Graphs . . . . . . . . . . . . . . . . . . . . . . . . . 183
8.3 t -Conorm Fuzzy Graphs . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .187
8.4 Nonstandard Analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 190
8.5 Nonstandard Weighted Graphs . . . . . . . . . . . . . . . . . . . . . . . . . . . .194
References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 195
9 Fuzzy Results from Crisp Results ................................ 197
9.1 Derivation of Fuzzy Results from Crisp Results . . . . . . . . . . . . . . . . 197
9.2 Fuzzy Results from Crisp Results Continued . . . . . . . . . . . . . . . . . . 200
9.3 Fuzzifiable Operations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 201
9.4 Fuzzification of Fuzzifiable Operations . . . . . . . . . . . . . . . . . . . . . . .
203
9.5 ( A) and ( A) Generate the Same Variety of Algebras . . . . . . . 204
9.6 Undirected Power Graphs of Semigroups . . . . . . . . . . . . . . . . . . . . . 206
9.7 Fuzzy Subgraphs of Undirected Power Graphs . . . . . . . . . . . . . . . . . 207
References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 208
Bibliography ...................................................... 209
Index ............................................................. 213
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