Two Classes of Topological Indices of Phenylene Molecule Graphs

<div>A linear phenylene chain <svg height="11.6791pt" id="M76" style="vertical-align:-3.29112pt" version="1.1" viewbox="-0.0498162 -8.38798 29.5712 11.6791" width="29.5712pt" 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="M46 650V622C120 617 128 613 128 525V125C128 42 120 34 40 28V0H311V28C221 34 212 39 212 124V281L286 262C297 261 316 261 331 263C429 275 526 338 526 468C526 533 501 579 462 609C422 638 364 650 293 650H46ZM212 559C212 588 215 600 223 606C230 613 251 618 279 618C361 618 430 572 430 464C430 337 350 302 285 302C252 302 225 309 212 314V559Z" id="g121-78"></path><glyph.data ascent="989" descent="-360" horiz-adv-x="563" vert-adv-y="563"></glyph.data></g><g transform="matrix(.013,0,0,-0.013,7.254,0)"><path d="M729 650H468V622C546 615 554 610 554 527V365H213V527C213 610 221 616 297 622V650H38V622C120 616 128 611 128 527V123C128 39 120 34 40 28V0H303V28C221 34 213 40 213 123V322H554V123C554 39 546 34 460 28V0H728V28C647 34 639 39 639 123V527C639 610 647 615 729 622V650Z" id="g121-70"></path><glyph.data ascent="989" descent="-360" horiz-adv-x="766" vert-adv-y="766"></glyph.data></g><g transform="matrix(.013,0,0,-0.013,17.212,0)"><path d="M495 163C480 117 462 85 444 65C421 39 387 34 332 34C290 34 256 36 236 47C218 57 213 77 213 131V526C213 612 222 616 301 622V650H40V622C122 616 128 611 128 526V126C128 41 120 34 36 28V0H489C498 31 519 126 525 157L495 163Z" id="g121-74"></path><glyph.data ascent="989" descent="-360" horiz-adv-x="538" vert-adv-y="538"></glyph.data></g><g transform="matrix(.0091,0,0,-0.0091,24.206,3.132)"><path d="M504 89L488 119C453 85 428 69 416 69C408 69 405 75 412 102L462 304C492 438 460 451 436 451S388 441 356 423C311 397 229 336 170 256H168L189 340C208 418 200 451 177 451C144 451 82 413 24 352L40 322C65 345 98 369 108 369C114 369 119 363 112 335L28 -3L36 -12C54 -3 83 4 112 10C125 73 138 122 153 174C205 258 328 382 376 382C395 382 399 369 384 305L330 93C312 25 323 -12 351 -12C378 -12 439 21 504 89Z" id="g50-111"></path><glyph.data ascent="3443" descent="-2856" horiz-adv-x="512" vert-adv-y="512"></glyph.data></g></svg> and a zigzag phenylene chain <svg height="12.0575pt" id="M77" style="vertical-align:-3.2911pt" version="1.1" viewbox="-0.0498162 -8.7664 30.4194 12.0575" width="30.4194pt" 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="M46 650V622C120 617 128 613 128 525V125C128 42 120 34 40 28V0H311V28C221 34 212 39 212 124V281L286 262C297 261 316 261 331 263C429 275 526 338 526 468C526 533 501 579 462 609C422 638 364 650 293 650H46ZM212 559C212 588 215 600 223 606C230 613 251 618 279 618C361 618 430 572 430 464C430 337 350 302 285 302C252 302 225 309 212 314V559Z" id="g121-78"></path><glyph.data ascent="989" descent="-360" horiz-adv-x="563" vert-adv-y="563"></glyph.data></g><g transform="matrix(.013,0,0,-0.013,7.254,0)"><path d="M729 650H468V622C546 615 554 610 554 527V365H213V527C213 610 221 616 297 622V650H38V622C120 616 128 611 128 527V123C128 39 120 34 40 28V0H303V28C221 34 213 40 213 123V322H554V123C554 39 546 34 460 28V0H728V28C647 34 639 39 639 123V527C639 610 647 615 729 622V650Z" id="g121-70"></path><glyph.data ascent="989" descent="-360" horiz-adv-x="766" vert-adv-y="766"></glyph.data></g><g transform="matrix(.013,0,0,-0.013,17.212,0)"><path d="M561 176C543 127 527 95 506 73C478 43 441 36 316 36C247 36 185 38 153 42C290 245 427 439 566 636L559 650H160C130 650 119 652 107 675H83C81 630 76 555 71 492L100 491C112 535 123 560 135 578C152 605 168 614 276 614H449C310 411 169 213 29 15L37 0H554C565 41 583 134 590 170L561 176Z" id="g121-88"></path><glyph.data ascent="989" descent="-360" horiz-adv-x="603" vert-adv-y="603"></glyph.data></g><g transform="matrix(.0091,0,0,-0.0091,25.051,3.132)"><path d="M504 89L488 119C453 85 428 69 416 69C408 69 405 75 412 102L462 304C492 438 460 451 436 451S388 441 356 423C311 397 229 336 170 256H168L189 340C208 418 200 451 177 451C144 451 82 413 24 352L40 322C65 345 98 369 108 369C114 369 119 363 112 335L28 -3L36 -12C54 -3 83 4 112 10C125 73 138 122 153 174C205 258 328 382 376 382C395 382 399 369 384 305L330 93C312 25 323 -12 351 -12C378 -12 439 21 504 89Z" id="g50-111"></path><glyph.data ascent="3443" descent="-2856" horiz-adv-x="512" vert-adv-y="512"></glyph.data></g></svg>.</div>

Mathematical Problems in Engineering

Two Classes of Topological Indices of Phenylene Molecule Graphs

Figure 1