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<div>Zero-divisor graph of <svg height="14.0291pt" id="M91" style="vertical-align:-5.34101pt" version="1.1" viewbox="-0.0498162 -8.68809 65.0693 14.0291" width="65.0693pt" 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="M677 618V662H100V618H469L50 44V0H668V44H252L677 618ZM617 618L200 44H103L523 618H617Z"></path></g><g transform="matrix(.0091,0,0,-0.0091,9.451,3.132)"><path d="M678 424H610L582 376H580C573 420 539 441 492 441C412 441 301 373 226 282C194 243 153 203 97 203C80 203 61 207 40 216L30 190C53 178 76 173 100 173C127 173 150 180 170 193C155 159 147 126 147 94C147 30 180 -11 231 -11C307 -11 367 26 438 102L439 101L352 -55C332 -47 281 -42 261 -42C222 -42 183 -45 146 -68C119 -84 92 -111 92 -148C92 -183 123 -219 190 -219C270 -219 337 -186 393 -105C420 -120 460 -127 495 -127S565 -122 612 -93L600 -67C569 -90 531 -97 500 -97S436 -89 409 -74L678 424ZM550 364C550 260 376 19 246 19C223 19 205 33 205 60C205 96 228 156 291 247C367 358 440 406 496 406C525 406 550 397 550 364ZM337 -86C281 -174 220 -185 182 -185C149 -185 126 -168 126 -144C126 -101 176 -72 252 -72C276 -72 316 -76 336 -85L337 -86Z"></path></g><g transform="matrix(.0065,0,0,-0.0065,16.167,-.065)"><path d="M436 222C436 297 385 359 295 376V378C320 391 343 407 363 428S394 474 394 501C394 578 333 635 239 635C186 635 138 610 110 577L63 514L91 490C114 525 158 571 211 571C266 571 306 541 306 483C306 411 234 371 141 343L151 306C167 311 192 317 219 317C276 317 344 282 344 194C346 102 296 46 227 46C166 46 125 76 99 97C91 104 81 103 71 97C62 90 47 75 46 62C44 50 48 39 62 26C77 10 120 -12 168 -12C242 -12 436 62 436 222Z"></path></g><g transform="matrix(.013,0,0,-0.013,23.317,0)"><path d="M528 54L331 254L528 455L492 493L294 291L96 493L60 455L257 254L60 54L96 16L294 217L492 16L528 54Z"></path></g><g transform="matrix(.013,0,0,-0.013,33.853,0)"><path d="M677 618V662H100V618H469L50 44V0H668V44H252L677 618ZM617 618L200 44H103L523 618H617Z"></path></g><g transform="matrix(.0091,0,0,-0.0091,43.304,3.132)"><path d="M593 671C573 656 552 642 529 631C515 624 500 617 484 617C464 617 445 625 426 633C373 657 305 686 260 686C219 686 178 673 150 644C132 625 119 602 107 579C89 545 75 510 62 474L81 463C97 502 118 540 150 569C173 591 205 601 237 601C258 601 278 596 298 590C330 580 391 559 438 543C499 573 554 611 605 655L593 671ZM76 -16L133 21C163 -3 196 -27 237 -27C326 -27 405 17 467 74C515 119 530 183 530 245C530 293 521 348 513 399C509 420 509 442 507 463C507 481 516 498 526 514C541 537 561 557 582 575L565 580C510 548 463 507 421 463C415 421 421 378 426 335C432 293 438 250 438 207C438 177 435 147 426 118C419 96 409 74 392 58C370 38 343 21 311 21C286 21 261 31 242 48C210 77 211 157 185 117C142 82 103 43 70 0L76 -16Z"></path></g><g transform="matrix(.0065,0,0,-0.0065,49.292,5.233)"><path d="M394 0V37C301 42 295 50 295 122V635C236 613 178 596 109 583V553L165 551C205 549 209 544 209 495V122C209 50 203 42 111 37V0H394Z"></path></g><g transform="matrix(.0091,0,0,-0.0091,53.013,3.132)"><path d="M593 671C573 656 552 642 529 631C515 624 500 617 484 617C464 617 445 625 426 633C373 657 305 686 260 686C219 686 178 673 150 644C132 625 119 602 107 579C89 545 75 510 62 474L81 463C97 502 118 540 150 569C173 591 205 601 237 601C258 601 278 596 298 590C330 580 391 559 438 543C499 573 554 611 605 655L593 671ZM76 -16L133 21C163 -3 196 -27 237 -27C326 -27 405 17 467 74C515 119 530 183 530 245C530 293 521 348 513 399C509 420 509 442 507 463C507 481 516 498 526 514C541 537 561 557 582 575L565 580C510 548 463 507 421 463C415 421 421 378 426 335C432 293 438 250 438 207C438 177 435 147 426 118C419 96 409 74 392 58C370 38 343 21 311 21C286 21 261 31 242 48C210 77 211 157 185 117C142 82 103 43 70 0L76 -16Z"></path></g><g transform="matrix(.0065,0,0,-0.0065,59.001,5.233)"><path d="M452 139L416 148C386 81 373 77 319 77H141L282 220C372 312 414 373 414 464C414 569 332 635 240 635C181 635 132 609 101 574L42 493L70 467C98 506 143 562 209 562C280 562 324 515 324 435C324 347 261 272 197 197C148 139 89 84 34 29V0H417L452 139Z"></path></g><g transform="matrix(.0065,0,0,-0.0065,62.212,5.233)"><path d="M186 54C186 90 160 117 122 117C86 117 60 90 60 54C60 13 86 -12 122 -12C160 -12 186 13 186 54Z"></path></g></svg></div>

Journal of Function Spaces

fig1

Figure 1

Figure 1: Computing Wiener and Hyper-Wiener Indices of Zero-Divisor Graph of <svg xmlns:xlink="http://www.w3.org/1999/xlink" xmlns="http://www.w3.org/2000/svg" style="vertical-align:-6.9727pt" id="M1" height="18.441pt" version="1.1" viewBox="-0.0657574 -11.4683 83.7728 18.441" width="83.7728pt"><g transform="matrix(.017,0,0,-0.017,0,0)"><path id="g197-37" d="M677 618V662H100V618H469L50 44V0H668V44H252L677 618ZM617 618L200 44H103L523 618H617Z"/></g><g transform="matrix(.012,0,0,-0.012,12.475,4.134)"><path id="g198-34" d="M678 424H610L582 376H580C573 420 539 441 492 441C412 441 301 373 226 282C194 243 153 203 97 203C80 203 61 207 40 216L30 190C53 178 76 173 100 173C127 173 150 180 170 193C155 159 147 126 147 94C147 30 180 -11 231 -11C307 -11 367 26 438 102L439 101L352 -55C332 -47 281 -42 261 -42C222 -42 183 -45 146 -68C119 -84 92 -111 92 -148C92 -183 123 -219 190 -219C270 -219 337 -186 393 -105C420 -120 460 -127 495 -127S565 -122 612 -93L600 -67C569 -90 531 -97 500 -97S436 -89 409 -74L678 424ZM550 364C550 260 376 19 246 19C223 19 205 33 205 60C205 96 228 156 291 247C367 358 440 406 496 406C525 406 550 397 550 364ZM337 -86C281 -174 220 -185 182 -185C149 -185 126 -168 126 -144C126 -101 176 -72 252 -72C276 -72 316 -76 336 -85L337 -86Z"/></g><g transform="matrix(.009,0,0,-0.009,21.34,-.086)"><path id="g176-52" d="M436 222C436 297 385 359 295 376V378C320 391 343 407 363 428S394 474 394 501C394 578 333 635 239 635C186 635 138 610 110 577L63 514L91 490C114 525 158 571 211 571C266 571 306 541 306 483C306 411 234 371 141 343L151 306C167 311 192 317 219 317C276 317 344 282 344 194C346 102 296 46 227 46C166 46 125 76 99 97C91 104 81 103 71 97C62 90 47 75 46 62C44 50 48 39 62 26C77 10 120 -12 168 -12C242 -12 436 62 436 222Z"/></g><g transform="matrix(.017,0,0,-0.017,30.778,0)"><path id="g117-42" d="M528 54L331 254L528 455L492 493L294 291L96 493L60 455L257 254L60 54L96 16L294 217L492 16L528 54Z"/></g><g transform="matrix(.017,0,0,-0.017,44.686,0)"><use xlink:href="#g197-37"/></g><g transform="matrix(.012,0,0,-0.012,57.161,4.134)"><path d="M593 671C573 656 552 642 529 631C515 624 500 617 484 617C464 617 445 625 426 633C373 657 305 686 260 686C219 686 178 673 150 644C132 625 119 602 107 579C89 545 75 510 62 474L81 463C97 502 118 540 150 569C173 591 205 601 237 601C258 601 278 596 298 590C330 580 391 559 438 543C499 573 554 611 605 655L593 671ZM76 -16L133 21C163 -3 196 -27 237 -27C326 -27 405 17 467 74C515 119 530 183 530 245C530 293 521 348 513 399C509 420 509 442 507 463C507 481 516 498 526 514C541 537 561 557 582 575L565 580C510 548 463 507 421 463C415 421 421 378 426 335C432 293 438 250 438 207C438 177 435 147 426 118C419 96 409 74 392 58C370 38 343 21 311 21C286 21 261 31 242 48C210 77 211 157 185 117C142 82 103 43 70 0L76 -16Z"/></g><g transform="matrix(.009,0,0,-0.009,65.065,6.908)"><path id="g176-50" d="M394 0V37C301 42 295 50 295 122V635C236 613 178 596 109 583V553L165 551C205 549 209 544 209 495V122C209 50 203 42 111 37V0H394Z"/></g><g transform="matrix(.012,0,0,-0.012,69.977,4.134)"><path d="M593 671C573 656 552 642 529 631C515 624 500 617 484 617C464 617 445 625 426 633C373 657 305 686 260 686C219 686 178 673 150 644C132 625 119 602 107 579C89 545 75 510 62 474L81 463C97 502 118 540 150 569C173 591 205 601 237 601C258 601 278 596 298 590C330 580 391 559 438 543C499 573 554 611 605 655L593 671ZM76 -16L133 21C163 -3 196 -27 237 -27C326 -27 405 17 467 74C515 119 530 183 530 245C530 293 521 348 513 399C509 420 509 442 507 463C507 481 516 498 526 514C541 537 561 557 582 575L565 580C510 548 463 507 421 463C415 421 421 378 426 335C432 293 438 250 438 207C438 177 435 147 426 118C419 96 409 74 392 58C370 38 343 21 311 21C286 21 261 31 242 48C210 77 211 157 185 117C142 82 103 43 70 0L76 -16Z"/></g><g transform="matrix(.009,0,0,-0.009,77.881,6.908)"><path id="g176-51" d="M452 139L416 148C386 81 373 77 319 77H141L282 220C372 312 414 373 414 464C414 569 332 635 240 635C181 635 132 609 101 574L42 493L70 467C98 506 143 562 209 562C280 562 324 515 324 435C324 347 261 272 197 197C148 139 89 84 34 29V0H417L452 139Z"/></g></svg>