Extremal Trees with respect to Number of <svg xmlns:xlink="http://www.w3.org/1999/xlink" xmlns="http://www.w3.org/2000/svg" style="vertical-align:-2.8905pt" id="M1" height="15.1511pt" version="1.1" viewBox="-0.0657574 -12.2606 69.2183 15.1511" width="69.2183pt"><g transform="matrix(.017,0,0,-0.017,0,0)"><path id="g50-41" d="M309 -142C211 -61 151 99 151 271S210 601 309 683L288 710C141 611 75 451 75 272V271C75 90 141 -72 288 -169L309 -142Z"/></g><g transform="matrix(.017,0,0,-0.017,6.143,0)"><path id="g50-66" d="M722 32C646 41 636 48 622 116L544 496L509 667L472 658L153 123C106 45 93 42 27 32L18 0H259L268 32C178 40 172 48 204 107L272 233H509C523 180 533 137 540 96C548 50 546 39 466 32L457 0H714L722 32ZM502 283H299L448 547H450L502 283Z"/></g><g transform="matrix(.017,0,0,-0.017,18.339,0)"><path id="g50-45" 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"/></g><g transform="matrix(.017,0,0,-0.017,25.316,0)"><path id="g50-67" d="M598 511C598 627 497 650 401 650H141L131 618C219 611 222 606 210 538L130 119C117 43 108 39 25 32L18 0H236C324 0 396 5 456 30C531 59 585 119 585 193C585 285 520 331 428 351V353C529 373 598 423 598 511ZM502 508C502 418 431 368 330 368H263L298 559C303 584 308 595 317 600C329 608 345 612 369 612C428 612 502 595 502 508ZM483 197C483 95 404 39 309 39C225 39 206 58 219 139L256 330H317C395 330 483 301 483 197Z"/></g><g transform="matrix(.017,0,0,-0.017,35.75,0)"><use xlink:href="#g50-45"/></g><g transform="matrix(.017,0,0,-0.017,42.728,0)"><path id="g50-51" d="M414 144C384 79 371 75 317 75H135L276 221C367 316 408 376 408 465C408 570 327 635 237 635C179 635 131 609 100 575L42 494L67 471C94 510 138 565 205 565C277 565 321 517 321 435C321 348 258 270 195 195C146 137 88 81 33 26V0H411C423 44 433 88 446 135L414 144Z"/></g><g transform="matrix(.017,0,0,-0.017,51.084,0)"><path id="g50-68" d="M690 629C653 643 577 667 483 667C225 667 24 528 24 292C24 92 174 -17 367 -17C448 -17 534 1 570 10C592 34 642 131 658 174C648 180 639 186 629 192C562 89 499 21 382 21C223 21 130 138 130 298C130 467 255 627 462 627C582 627 643 589 650 462L686 466C689 517 688 580 690 629Z"/></g><g transform="matrix(.017,0,0,-0.017,62.703,0)"><path id="g50-42" d="M283 271C283 451 220 610 70 710L48 683C146 603 207 443 207 271S149 -59 48 -142L70 -169C220 -67 283 90 283 271Z"/></g></svg>-Edge Colourings

The first terms of <svg height="12.449pt" id="M26" style="vertical-align:-3.16067pt" version="1.1" viewbox="-0.0498162 -9.28833 21.4157 12.449" width="21.4157pt" 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 -169V-136C225 -126 216 -88 216 -42C216 19 230 84 230 152C230 207 206 255 147 268V272C206 286 230 333 230 388C230 453 216 523 216 586C216 630 225 667 300 677V710C239 710 151 696 151 579C152 514 160 438 160 373C160 338 157 293 68 287V253C157 247 160 202 160 168C159 106 151 32 151 -41C151 -157 239 -169 300 -169ZM181 710H180H181ZM181 -186H180H181Z" id="g50-124"></path><glyph.data ascent="3443" descent="-2856" horiz-adv-x="360" vert-adv-y="360"></glyph.data></g><g transform="matrix(.013,0,0,-0.013,4.68,0)"><path d="M622 479C622 589 545 650 400 650H146L140 618C229 609 230 602 215 527L135 132C117 43 112 41 27 32L18 0H305L313 32C213 40 205 44 222 132L251 284L324 262H365C484 269 622 333 622 479ZM528 477C528 356 420 305 344 305C300 305 276 314 259 321L303 554C311 597 313 600 331 606C345 611 361 614 379 614C463 614 528 580 528 477Z" id="g50-81"></path><glyph.data ascent="3443" descent="-2856" horiz-adv-x="640" vert-adv-y="640"></glyph.data></g><g transform="matrix(.0091,0,0,-0.0091,11.401,3.001)"><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><g transform="matrix(.013,0,0,-0.013,16.576,0)"><path d="M292 253V287C203 293 200 339 200 376C201 435 209 507 209 582C209 697 123 710 60 710V677C133 667 144 631 144 586C144 523 130 452 130 388C130 334 154 286 214 272V268C154 254 130 206 130 152C130 89 144 17 144 -45C144 -89 133 -126 60 -136V-169C121 -169 209 -154 209 -42C208 33 200 106 200 164C200 201 203 247 292 253Z" id="g50-126"></path><glyph.data ascent="3443" descent="-2856" horiz-adv-x="360" vert-adv-y="360"></glyph.data></g></svg> and <svg height="12.449pt" id="M27" style="vertical-align:-3.16067pt" version="1.1" viewbox="-0.0498162 -9.28833 19.5758 12.449" width="19.5758pt" 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 -169V-136C225 -126 216 -88 216 -42C216 19 230 84 230 152C230 207 206 255 147 268V272C206 286 230 333 230 388C230 453 216 523 216 586C216 630 225 667 300 677V710C239 710 151 696 151 579C152 514 160 438 160 373C160 338 157 293 68 287V253C157 247 160 202 160 168C159 106 151 32 151 -41C151 -157 239 -169 300 -169ZM181 710H180H181ZM181 -186H180H181Z" id="g50-124"></path><glyph.data ascent="3443" descent="-2856" horiz-adv-x="360" vert-adv-y="360"></glyph.data></g><g transform="matrix(.013,0,0,-0.013,4.68,0)"><path d="M357 391C357 416 325 451 270 451C241 451 179 431 146 400C109 365 97 334 97 307C97 248 143 213 196 180C250 146 261 123 261 101C261 75 237 40 190 40C153 40 110 72 86 109C81 116 68 119 55 112C37 102 24 87 24 66C24 30 85 -12 137 -12C229 -12 331 62 331 140C331 189 304 218 236 260C197 284 164 309 164 346C164 381 194 400 220 400C259 400 282 380 304 350C310 342 321 339 330 344C347 353 357 372 357 391Z" id="g50-116"></path><glyph.data ascent="3443" descent="-2856" horiz-adv-x="381" vert-adv-y="381"></glyph.data></g><g transform="matrix(.0091,0,0,-0.0091,9.568,3.001)"><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><g transform="matrix(.013,0,0,-0.013,14.743,0)"><path d="M292 253V287C203 293 200 339 200 376C201 435 209 507 209 582C209 697 123 710 60 710V677C133 667 144 631 144 586C144 523 130 452 130 388C130 334 154 286 214 272V268C154 254 130 206 130 152C130 89 144 17 144 -45C144 -89 133 -126 60 -136V-169C121 -169 209 -154 209 -42C208 33 200 106 200 164C200 201 203 247 292 253Z" id="g50-126"></path><glyph.data ascent="3443" descent="-2856" horiz-adv-x="360" vert-adv-y="360"></glyph.data></g></svg>.

Journal of Applied Mathematics

tab1

Table 1

Table 1: Extremal Trees with respect to Number of <svg xmlns:xlink="http://www.w3.org/1999/xlink" xmlns="http://www.w3.org/2000/svg" style="vertical-align:-2.8905pt" id="M1" height="15.1511pt" version="1.1" viewBox="-0.0657574 -12.2606 69.2183 15.1511" width="69.2183pt"><g transform="matrix(.017,0,0,-0.017,0,0)"><path id="g50-41" d="M309 -142C211 -61 151 99 151 271S210 601 309 683L288 710C141 611 75 451 75 272V271C75 90 141 -72 288 -169L309 -142Z"/></g><g transform="matrix(.017,0,0,-0.017,6.143,0)"><path id="g50-66" d="M722 32C646 41 636 48 622 116L544 496L509 667L472 658L153 123C106 45 93 42 27 32L18 0H259L268 32C178 40 172 48 204 107L272 233H509C523 180 533 137 540 96C548 50 546 39 466 32L457 0H714L722 32ZM502 283H299L448 547H450L502 283Z"/></g><g transform="matrix(.017,0,0,-0.017,18.339,0)"><path id="g50-45" 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"/></g><g transform="matrix(.017,0,0,-0.017,25.316,0)"><path id="g50-67" d="M598 511C598 627 497 650 401 650H141L131 618C219 611 222 606 210 538L130 119C117 43 108 39 25 32L18 0H236C324 0 396 5 456 30C531 59 585 119 585 193C585 285 520 331 428 351V353C529 373 598 423 598 511ZM502 508C502 418 431 368 330 368H263L298 559C303 584 308 595 317 600C329 608 345 612 369 612C428 612 502 595 502 508ZM483 197C483 95 404 39 309 39C225 39 206 58 219 139L256 330H317C395 330 483 301 483 197Z"/></g><g transform="matrix(.017,0,0,-0.017,35.75,0)"><use xlink:href="#g50-45"/></g><g transform="matrix(.017,0,0,-0.017,42.728,0)"><path id="g50-51" d="M414 144C384 79 371 75 317 75H135L276 221C367 316 408 376 408 465C408 570 327 635 237 635C179 635 131 609 100 575L42 494L67 471C94 510 138 565 205 565C277 565 321 517 321 435C321 348 258 270 195 195C146 137 88 81 33 26V0H411C423 44 433 88 446 135L414 144Z"/></g><g transform="matrix(.017,0,0,-0.017,51.084,0)"><path id="g50-68" d="M690 629C653 643 577 667 483 667C225 667 24 528 24 292C24 92 174 -17 367 -17C448 -17 534 1 570 10C592 34 642 131 658 174C648 180 639 186 629 192C562 89 499 21 382 21C223 21 130 138 130 298C130 467 255 627 462 627C582 627 643 589 650 462L686 466C689 517 688 580 690 629Z"/></g><g transform="matrix(.017,0,0,-0.017,62.703,0)"><path id="g50-42" d="M283 271C283 451 220 610 70 710L48 683C146 603 207 443 207 271S149 -59 48 -142L70 -169C220 -67 283 90 283 271Z"/></g></svg>-Edge Colourings 

Journal of Applied Mathematics

Extremal Trees with respect to Number of -Edge Colourings

Table 1