A Study of Hexagon Star Network with Vertex-Edge-Based Topological Descriptors

<table class="table-group" id="tab1"><tr><td><table class="table"><tr><td class="thead-hr" colspan="3"><hr/></td></tr><tr class="thead"><td class="align_left"><span style="width: 64.5212ptpx;"><svg height="11.5564pt" id="M66" style="vertical-align:-2.26807pt" version="1.1" viewbox="-0.0498162 -9.28833 64.5212 11.5564" width="64.5212pt" xmlns="http://www.w3.org/2000/svg" xmlns:xlink="http://www.w3.org/1999/xlink"><g transform="matrix(.013,0,0,-0.013,0,0)"><path d="M300 -147C201 -63 143 98 143 270S200 602 300 686L282 710C136 610 70 450 70 271V270C70 89 136 -72 282 -170L300 -147Z"></path></g><g transform="matrix(.013,0,0,-0.013,4.498,0)"><path d="M699 659C666 654 622 640 595 610C579 591 569 572 562 530L552 472C530 342 504 307 403 302V529C403 604 411 617 483 622V650H238V622C310 617 318 604 318 529V302C232 308 194 347 172 471L161 531C153 573 145 592 131 609C103 643 59 654 25 659L15 632C62 611 63 583 73 524L85 452C97 383 116 347 155 316C199 282 255 269 318 267V122C318 46 310 33 233 28V0H489V28C412 33 403 46 403 122V267C470 269 518 283 559 315C599 346 618 378 632 450L646 519C658 580 665 608 709 632L699 659Z"></path></g><g transform="matrix(.013,0,0,-0.013,13.927,0)"><path d="M300 -147C201 -63 143 98 143 270S200 602 300 686L282 710C136 610 70 450 70 271V270C70 89 136 -72 282 -170L300 -147Z"></path></g><g transform="matrix(.013,0,0,-0.013,18.425,0)"><path d="M475 507C475 612 440 712 326 712C139 712 23 420 23 215C23 96 58 -12 180 -12C369 -12 475 293 475 507ZM391 522C391 486 387 448 379 394H126C155 538 222 677 310 677C386 677 391 571 391 522ZM373 346C344 193 283 22 189 22C126 22 106 114 106 196C106 243 111 293 118 346H373Z"></path></g><g transform="matrix(.013,0,0,-0.013,24.9,0)"><path d="M275 270C275 450 212 609 64 710L45 686C145 604 203 442 203 270S147 -63 45 -147L64 -170C213 -68 275 89 275 270Z"></path></g><g transform="matrix(.013,0,0,-0.013,29.398,0)"><path d="M95 130C70 130 46 113 46 88C46 72 54 64 59 64C93 55 121 33 121 -3C121 -41 93 -68 44 -88L55 -117C117 -98 186 -56 186 22C186 91 131 130 95 130Z"></path></g><g transform="matrix(.013,0,0,-0.013,34.542,0)"><path d="M699 659C666 654 622 640 595 610C579 591 569 572 562 530L552 472C530 342 504 307 403 302V529C403 604 411 617 483 622V650H238V622C310 617 318 604 318 529V302C232 308 194 347 172 471L161 531C153 573 145 592 131 609C103 643 59 654 25 659L15 632C62 611 63 583 73 524L85 452C97 383 116 347 155 316C199 282 255 269 318 267V122C318 46 310 33 233 28V0H489V28C412 33 403 46 403 122V267C470 269 518 283 559 315C599 346 618 378 632 450L646 519C658 580 665 608 709 632L699 659Z"></path></g><g transform="matrix(.013,0,0,-0.013,43.971,0)"><path d="M300 -147C201 -63 143 98 143 270S200 602 300 686L282 710C136 610 70 450 70 271V270C70 89 136 -72 282 -170L300 -147Z"></path></g><g transform="matrix(.013,0,0,-0.013,48.469,0)"><path d="M501 444C501 643 409 712 321 712C208 712 108 618 108 502C108 412 172 331 293 331C337 331 394 359 420 378C397 154 341 41 258 41C209 41 178 76 178 161C178 232 171 299 114 299C76 299 38 272 23 257L29 232C42 236 54 239 66 239C91 239 106 221 102 149C95 20 158 -12 219 -12C264 -12 314 10 353 47C461 150 501 295 501 444ZM423 443C423 433 422 422 421 412C401 396 370 385 335 385C245 385 180 444 180 537C180 609 212 671 286 671C349 671 423 595 423 443Z"></path></g><g transform="matrix(.013,0,0,-0.013,55.204,0)"><path d="M275 270C275 450 212 609 64 710L45 686C145 604 203 442 203 270S147 -63 45 -147L64 -170C213 -68 275 89 275 270Z"></path></g><g transform="matrix(.013,0,0,-0.013,59.702,0)"><path d="M275 270C275 450 212 609 64 710L45 686C145 604 203 442 203 270S147 -63 45 -147L64 -170C213 -68 275 89 275 270Z"></path></g></svg></span></td><td class="align_center"><span class="nowrap"><svg height="6.1673pt" id="M67" style="vertical-align:-0.2063904pt" version="1.1" viewbox="-0.0498162 -5.96091 11.7652 6.1673" width="11.7652pt" xmlns="http://www.w3.org/2000/svg" xmlns:xlink="http://www.w3.org/1999/xlink"><g transform="matrix(.013,0,0,-0.013,0,0)"><path d="M391 364C391 409 353 448 295 448C249 448 198 426 152 393C65 331 23 225 23 139C23 14 96 -12 146 -12C198 -12 280 9 367 101L351 124C300 78 242 48 194 48C129 48 109 107 109 162V191C208 213 391 266 391 364ZM313 350C313 305 268 261 113 223C132 334 187 381 217 398C227 404 244 405 261 405C290 405 313 385 313 350Z"></path></g><g transform="matrix(.013,0,0,-0.013,5.382,0)"><path d="M457 332C457 394 438 448 414 448C388 448 365 421 365 396C365 386 368 381 376 372C387 359 388 344 388 315C388 150 306 45 220 45C178 45 151 68 151 123C151 139 154 158 158 180C164 212 175 259 197 332C206 361 207 385 207 402C207 434 196 448 174 448C125 448 66 406 23 342L43 319C84 368 110 383 121 383C126 383 128 382 128 376S127 359 122 343C99 268 85 204 76 156C72 137 69 120 69 105C69 25 125 -12 181 -12C314 -12 457 166 457 332Z"></path></g></svg>-</span>Degree</td><td class="align_center">Frequency</td></tr><tr><td class="thead-hr" colspan="3"><hr/></td></tr><tr><td class="align_left"><span style="width: 26.8104ptpx;"><svg height="11.5564pt" id="M68" style="vertical-align:-2.26807pt" version="1.1" viewbox="-0.0498162 -9.28833 26.8104 11.5564" width="26.8104pt" xmlns="http://www.w3.org/2000/svg" xmlns:xlink="http://www.w3.org/1999/xlink"><g transform="matrix(.013,0,0,-0.013,0,0)"><path d="M300 -147C201 -63 143 98 143 270S200 602 300 686L282 710C136 610 70 450 70 271V270C70 89 136 -72 282 -170L300 -147Z"></path></g><g transform="matrix(.013,0,0,-0.013,4.498,0)"><path d="M412 140C382 77 369 73 315 73H129L270 222C362 320 402 379 402 466C402 571 322 635 234 635C177 635 130 609 99 576L42 495L64 475C90 514 133 568 201 568C274 568 318 519 318 435C318 349 255 267 193 193C144 135 87 78 32 23V0H405C417 45 427 89 440 131L412 140Z"></path></g><g transform="matrix(.013,0,0,-0.013,10.738,0)"><path d="M95 130C70 130 46 113 46 88C46 72 54 64 59 64C93 55 121 33 121 -3C121 -41 93 -68 44 -88L55 -117C117 -98 186 -56 186 22C186 91 131 130 95 130Z"></path></g><g transform="matrix(.013,0,0,-0.013,15.881,0)"><path d="M456 178V225H360V632H320C217 496 115 347 20 206V178H280V106C280 40 276 34 189 27V0H445V27C364 34 360 39 360 106V178H456ZM280 225H82C149 335 214 431 278 520H280V225Z"></path></g><g transform="matrix(.013,0,0,-0.013,22.121,0)"><path d="M275 270C275 450 212 609 64 710L45 686C145 604 203 442 203 270S147 -63 45 -147L64 -170C213 -68 275 89 275 270Z"></path></g></svg></span></td><td class="align_center">6</td><td class="align_center"><span style="width: 38.8978ptpx;"><svg height="9.63826pt" id="M69" style="vertical-align:-0.3499298pt" version="1.1" viewbox="-0.0498162 -9.28833 38.8978 9.63826" width="38.8978pt" xmlns="http://www.w3.org/2000/svg" xmlns:xlink="http://www.w3.org/1999/xlink"><g transform="matrix(.013,0,0,-0.013,0,0)"><path d="M249 635C141 635 70 555 70 471C70 401 114 353 179 316C143 294 106 267 90 252C68 231 45 202 45 157C45 50 130 -12 237 -12C322 -12 435 52 435 169C435 256 372 304 303 343C349 374 375 398 383 407C401 429 411 458 411 487C411 569 344 635 249 635ZM238 603C285 603 337 567 337 482C337 422 310 385 276 358C205 393 145 426 145 500C145 552 179 603 238 603ZM248 20C183 20 125 70 125 163C125 218 158 268 206 300C284 261 355 217 355 143C355 66 308 20 248 20Z"></path></g><g transform="matrix(.013,0,0,-0.013,6.24,0)"><path d="M238 681C243 705 239 712 230 712C217 712 156 682 75 674L70 648H105C148 648 153 641 144 598L39 110C18 11 35 -12 55 -12C90 -12 166 36 221 103L205 125C174 93 130 65 118 65C112 65 108 68 114 96L238 681Z"></path></g><g transform="matrix(.013,0,0,-0.013,12.644,0)"><path d="M535 230V280H323V490H265V280H52V230H265V-3H323V230H535Z"></path></g><g transform="matrix(.013,0,0,-0.013,23.18,0)"><path d="M456 178V225H360V632H320C217 496 115 347 20 206V178H280V106C280 40 276 34 189 27V0H445V27C364 34 360 39 360 106V178H456ZM280 225H82C149 335 214 431 278 520H280V225Z"></path></g><g transform="matrix(.013,0,0,-0.013,29.42,0)"><path d="M689 332C689 394 670 448 646 448C620 448 597 421 597 396C597 386 600 381 608 372C619 359 620 334 620 315C620 150 538 45 454 45C414 45 386 67 386 122C386 138 388 158 394 180L457 426L452 432L377 416L315 156C302 100 259 45 216 45C176 45 148 67 148 122C148 133 152 158 156 180C162 212 173 259 194 332C201 357 206 384 206 405C206 430 198 448 174 448C125 448 66 406 23 342L43 319C84 368 110 383 121 383C126 383 128 382 128 377C128 370 127 359 122 343C99 268 84 204 77 156C74 137 70 111 70 104C70 25 125 -12 180 -12C228 -12 276 12 319 50C338 8 378 -12 418 -12C549 -12 689 166 689 332Z"></path></g></svg></span></td></tr><tr><td class="align_left"><span style="width: 26.8104ptpx;"><svg height="11.5564pt" id="M70" style="vertical-align:-2.26807pt" version="1.1" viewbox="-0.0498162 -9.28833 26.8104 11.5564" width="26.8104pt" xmlns="http://www.w3.org/2000/svg" xmlns:xlink="http://www.w3.org/1999/xlink"><g transform="matrix(.013,0,0,-0.013,0,0)"><path d="M300 -147C201 -63 143 98 143 270S200 602 300 686L282 710C136 610 70 450 70 271V270C70 89 136 -72 282 -170L300 -147Z"></path></g><g transform="matrix(.013,0,0,-0.013,4.498,0)"><path d="M456 178V225H360V632H320C217 496 115 347 20 206V178H280V106C280 40 276 34 189 27V0H445V27C364 34 360 39 360 106V178H456ZM280 225H82C149 335 214 431 278 520H280V225Z"></path></g><g transform="matrix(.013,0,0,-0.013,10.738,0)"><path d="M95 130C70 130 46 113 46 88C46 72 54 64 59 64C93 55 121 33 121 -3C121 -41 93 -68 44 -88L55 -117C117 -98 186 -56 186 22C186 91 131 130 95 130Z"></path></g><g transform="matrix(.013,0,0,-0.013,15.881,0)"><path d="M456 178V225H360V632H320C217 496 115 347 20 206V178H280V106C280 40 276 34 189 27V0H445V27C364 34 360 39 360 106V178H456ZM280 225H82C149 335 214 431 278 520H280V225Z"></path></g><g transform="matrix(.013,0,0,-0.013,22.122,0)"><path d="M275 270C275 450 212 609 64 710L45 686C145 604 203 442 203 270S147 -63 45 -147L64 -170C213 -68 275 89 275 270Z"></path></g></svg></span></td><td class="align_center">8</td><td class="align_center"><span style="width: 77.696ptpx;"><svg height="9.63826pt" id="M71" style="vertical-align:-0.3499298pt" version="1.1" viewbox="-0.0498162 -9.28833 77.696 9.63826" width="77.696pt" xmlns="http://www.w3.org/2000/svg" xmlns:xlink="http://www.w3.org/1999/xlink"><g transform="matrix(.013,0,0,-0.013,0,0)"><path d="M384 0V27C293 34 287 42 287 114V635C232 613 172 594 109 583V559L157 557C201 555 205 550 205 499V114C205 42 199 34 109 27V0H384Z"></path></g><g transform="matrix(.013,0,0,-0.013,6.24,0)"><path d="M412 140C382 77 369 73 315 73H129L270 222C362 320 402 379 402 466C402 571 322 635 234 635C177 635 130 609 99 576L42 495L64 475C90 514 133 568 201 568C274 568 318 519 318 435C318 349 255 267 193 193C144 135 87 78 32 23V0H405C417 45 427 89 440 131L412 140Z"></path></g><g transform="matrix(.013,0,0,-0.013,12.48,0)"><path d="M238 681C243 705 239 712 230 712C217 712 156 682 75 674L70 648H105C148 648 153 641 144 598L39 110C18 11 35 -12 55 -12C90 -12 166 36 221 103L205 125C174 93 130 65 118 65C112 65 108 68 114 96L238 681Z"></path></g><g transform="matrix(.013,0,0,-0.013,15.978,0)"><path d="M689 332C689 394 670 448 646 448C620 448 597 421 597 396C597 386 600 381 608 372C619 359 620 334 620 315C620 150 538 45 454 45C414 45 386 67 386 122C386 138 388 158 394 180L457 426L452 432L377 416L315 156C302 100 259 45 216 45C176 45 148 67 148 122C148 133 152 158 156 180C162 212 173 259 194 332C201 357 206 384 206 405C206 430 198 448 174 448C125 448 66 406 23 342L43 319C84 368 110 383 121 383C126 383 128 382 128 377C128 370 127 359 122 343C99 268 84 204 77 156C74 137 70 111 70 104C70 25 125 -12 180 -12C228 -12 276 12 319 50C338 8 378 -12 418 -12C549 -12 689 166 689 332Z"></path></g><g transform="matrix(.013,0,0,-0.013,28.141,0)"><path d="M535 230V280H52V230H535Z"></path></g><g transform="matrix(.013,0,0,-0.013,38.677,0)"><path d="M412 140C382 77 369 73 315 73H129L270 222C362 320 402 379 402 466C402 571 322 635 234 635C177 635 130 609 99 576L42 495L64 475C90 514 133 568 201 568C274 568 318 519 318 435C318 349 255 267 193 193C144 135 87 78 32 23V0H405C417 45 427 89 440 131L412 140Z"></path></g><g transform="matrix(.013,0,0,-0.013,44.917,0)"><path d="M238 681C243 705 239 712 230 712C217 712 156 682 75 674L70 648H105C148 648 153 641 144 598L39 110C18 11 35 -12 55 -12C90 -12 166 36 221 103L205 125C174 93 130 65 118 65C112 65 108 68 114 96L238 681Z"></path></g><g transform="matrix(.013,0,0,-0.013,51.321,0)"><path d="M535 230V280H52V230H535Z"></path></g><g transform="matrix(.013,0,0,-0.013,61.858,0)"><path d="M456 178V225H360V632H320C217 496 115 347 20 206V178H280V106C280 40 276 34 189 27V0H445V27C364 34 360 39 360 106V178H456ZM280 225H82C149 335 214 431 278 520H280V225Z"></path></g><g transform="matrix(.013,0,0,-0.013,68.098,0)"><path d="M689 332C689 394 670 448 646 448C620 448 597 421 597 396C597 386 600 381 608 372C619 359 620 334 620 315C620 150 538 45 454 45C414 45 386 67 386 122C386 138 388 158 394 180L457 426L452 432L377 416L315 156C302 100 259 45 216 45C176 45 148 67 148 122C148 133 152 158 156 180C162 212 173 259 194 332C201 357 206 384 206 405C206 430 198 448 174 448C125 448 66 406 23 342L43 319C84 368 110 383 121 383C126 383 128 382 128 377C128 370 127 359 122 343C99 268 84 204 77 156C74 137 70 111 70 104C70 25 125 -12 180 -12C228 -12 276 12 319 50C338 8 378 -12 418 -12C549 -12 689 166 689 332Z"></path></g></svg></span></td></tr><tr class="table-tr"><td colspan="3"><hr class="tbody-hr"/></td></tr></table></td></tr></table>

<div><span class="nowrap"><svg height="6.1673pt" id="M64" style="vertical-align:-0.2063904pt" version="1.1" viewbox="-0.0498162 -5.96091 11.7652 6.1673" width="11.7652pt" xmlns="http://www.w3.org/2000/svg" xmlns:xlink="http://www.w3.org/1999/xlink"><g transform="matrix(.013,0,0,-0.013,0,0)"><path d="M391 364C391 409 353 448 295 448C249 448 198 426 152 393C65 331 23 225 23 139C23 14 96 -12 146 -12C198 -12 280 9 367 101L351 124C300 78 242 48 194 48C129 48 109 107 109 162V191C208 213 391 266 391 364ZM313 350C313 305 268 261 113 223C132 334 187 381 217 398C227 404 244 405 261 405C290 405 313 385 313 350Z"></path></g><g transform="matrix(.013,0,0,-0.013,5.382,0)"><path d="M457 332C457 394 438 448 414 448C388 448 365 421 365 396C365 386 368 381 376 372C387 359 388 344 388 315C388 150 306 45 220 45C178 45 151 68 151 123C151 139 154 158 158 180C164 212 175 259 197 332C206 361 207 385 207 402C207 434 196 448 174 448C125 448 66 406 23 342L43 319C84 368 110 383 121 383C126 383 128 382 128 376S127 359 122 343C99 268 85 204 76 156C72 137 69 120 69 105C69 25 125 -12 181 -12C314 -12 457 166 457 332Z"></path></g></svg>-</span>Degree of <span class="nowrap"><svg height="12.9866pt" id="M65" style="vertical-align:-4.3507pt" version="1.1" viewbox="-0.0498162 -8.6359 20.6928 12.9866" width="20.6928pt" xmlns="http://www.w3.org/2000/svg" xmlns:xlink="http://www.w3.org/1999/xlink"><g transform="matrix(.013,0,0,-0.013,0,0)"><path d="M673 297H417L411 270C517 261 522 258 506 183L487 92C476 38 428 19 360 19C207 19 123 132 123 287C123 472 243 631 441 631C557 631 617 583 617 469L647 472C650 545 655 604 658 632C625 643 554 666 467 666C201 666 23 502 23 278C23 90 156 -16 339 -16C410 -16 501 6 569 24C566 43 567 70 573 101L589 183C604 259 607 262 667 271L673 297Z"></path></g><g transform="matrix(.0091,0,0,-0.0091,8.827,3.132)"><path d="M238 675C244 702 241 710 231 710C220 710 155 682 74 673L70 643H101C147 643 151 635 142 594L39 111C18 11 36 -12 55 -12C89 -12 170 38 223 103L207 129C176 98 130 68 118 68C113 68 108 72 114 102L238 675Z"></path></g><g transform="matrix(.0091,0,0,-0.0091,11.294,3.132)"><path d="M98 134C72 134 46 117 46 90C46 73 55 65 60 64C95 55 124 32 124 -4C124 -42 95 -68 44 -89L57 -123C122 -104 194 -60 194 22C194 94 136 134 98 134Z"></path></g><g transform="matrix(.0091,0,0,-0.0091,13.451,3.132)"><path d="M698 334C698 397 680 451 655 451C627 451 602 424 602 397C602 389 605 383 614 371C622 360 624 340 624 317C624 161 546 47 458 47C419 47 389 69 389 125C389 142 391 162 397 185L457 430L451 435L380 421L315 164C301 107 266 47 222 47C182 47 153 69 153 126C153 137 157 162 161 185C166 214 179 265 197 334C203 358 208 385 208 407C208 433 201 451 176 451C124 451 64 404 24 348L44 320C86 366 113 382 121 382C126 382 128 381 128 374C128 368 126 356 122 341C102 266 88 204 81 164C77 143 73 121 73 107C73 23 131 -12 187 -12C234 -12 281 10 322 45C343 5 383 -12 422 -12C559 -12 698 174 698 334Z"></path></g></svg>.</span></div>

Complexity

tab1

Table 1

Table 1: A Study of Hexagon Star Network with Vertex-Edge-Based Topological Descriptors 