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##  Appendix A Candidates catalog

Table LABEL:tab:selected_catalog lists the 91 candidates selected by our
method as LAEs, containing only the sources with morph_prob_star ¡ 0.1, as
explained in Sect. 4.10. This entire catalog is representative of what can be
directly obtained of any J-PAS data set. The sources used in the Ly$\alpha$ LF
estimation are marked in blue. As explained in Sect. 4.11, the sub-sample of
objects used for the Ly$\alpha$ LF are chosen after a visual inspection and
cross-matches with SDSS DR16 and HETDEX Source Catalog 1.

The description of each column is the following:

  * •

Field: The sub-field of miniJPAS or J-NEP in which the source was detected by
SExtractor.

  * •

tile_id: The ID identifying the selection tile of miniJPAS or J-NEP.

  * •

number: Unique ID of each object within each tile in the dual-mode catalogs of
miniJPAS or J-NEP.

  * •

RA, DEC: Right ascension and declination.

  * •

$z_{\mathrm{NB}}$: Ly$\alpha$ redshift of the central wavelength of the line
detection. The provided error is inferred from the FWHM of the NB.

  * •

Selected NB: Name of the NB of the Ly$\alpha$ line detection.

  * •

SDSS spCL, SDSS $z_{\mathrm{spec}}$: Spectroscopic class and redshift of the
counterpart in SDSS DR16.

  * •

HETDEX spCl, HETDEX $z_{\mathrm{spec}}$: Spectroscopic class and redshift of
the counterpart in HETDEX Source Catalog 1.

  * •

VI class: Class assigned in the visual inspection (see Sect. 4.11)

  * •

$\log_{10}L_{\mathrm{Ly}\alpha}$: Estimated Ly$\alpha$ luminosity.

  * •

EW${}_{0}^{\mathrm{Ly}\alpha}$: Estimated rest-frame EW of the Ly$\alpha$
line.

  * •

$P^{\mathrm{2D}}$: Purity extracted from the 2D correction maps according to
the source $L_{\mathrm{Ly}\alpha}$ and $r$ magnitude (see Sect. 4.4). This
value can be seen as the probability of each source to be a true positive LAE.

Table 5: Catalog of sources selected as LAEs. Field | tile_id | number | RA | DEC |  $z_{\text{NB}}$ $(\pm 0.04)$ | Sel. NB |  SDSS SpCl |  SDSS $z_{\text{spec}}$ |  HETDEX SpCl |  HETDEX $z_{\mathrm{spec}}$ | VI class |  $\log_{10}L_{\mathrm{Ly}\alpha}$ (erg s-1) |  EW${}_{0}^{\mathrm{Ly}\alpha}$ (Å) | $P(\text{LAE})$
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---
AEGIS001 | 2241 | 468 | 14h15m42.2s | 52°22′7.0″ | 2.11 | J0378 | QSO | 1.00 | agn | 1.00 | Cont. QSO | 43.69${}^{+0.11}_{-0.14}$ | 62$\pm$10 | 0.00
AEGIS001 | 2241 | 9344 | 14h17m38.8s | 52°23′33.1″ | 2.11 | J0378 | QSO | 2.16 | agn | 2.16 | QSO LAE | 44.11${}^{+0.06}_{-0.07}$ | 184$\pm$13 | 1.00
AEGIS001 | 2241 | 11097 | 14h18m42.3s | 52°36′43.9″ | 2.11 | J0378 | QSO | 2.12 | - | - | - | 43.63${}^{+0.12}_{-0.16}$ | 93$\pm$16 | 0.98
AEGIS001 | 2241 | 13090 | 14h16m58.4s | 52°48′6.3″ | 2.11 | J0378 | QSO | 1.96 | - | - | Cont. QSO | 43.73${}^{+0.10}_{-0.13}$ | 37$\pm$6 | 0.00
AEGIS001 | 2241 | 14254 | 14h18m2.0s | 52°35′14.9″ | 2.11 | J0378 | QSO | 1.49 | - | - | Cont. QSO | 43.92${}^{+0.07}_{-0.09}$ | 37$\pm$4 | 0.00
AEGIS001 | 2241 | 17351 | 14h18m32.8s | 52°23′50.0″ | 2.11 | J0378 | QSO | 2.04 | - | - | - | 43.83${}^{+0.08}_{-0.10}$ | 42$\pm$5 | 0.87
AEGIS001 | 2241 | 18457 | 14h17m36.0s | 52°30′29.8″ | 2.11 | J0378 | QSO | 0.99 | - | - | Cont. QSO | 43.62${}^{+0.12}_{-0.17}$ | 43$\pm$8 | 0.00
AEGIS001 | 2241 | 19989 | 14h19m11.4s | 52°32′34.7″ | 2.11 | J0378 | QSO | 1.41 | - | - | Cont. QSO | 43.44${}^{+0.16}_{-0.26}$ | 73$\pm$19 | 0.00
AEGIS001 | 2241 | 13288 | 14h17m37.3s | 52°42′36.7″ | 2.21 | J0390 | QSO | 0.15 | - | - | Cont. QSO | 43.10${}^{+0.14}_{-0.21}$ | 42$\pm$9 | 0.00
AEGIS001 | 2241 | 14038 | 14h18m42.9s | 52°29′19.0″ | 2.21 | J0390 | QSO | 1.08 | - | - | Cont. QSO | 43.07${}^{+0.15}_{-0.23}$ | 55$\pm$13 | 0.00
AEGIS001 | 2241 | 15333 | 14h18m13.1s | 52°31′10.1″ | 2.21 | J0390 | - | - | - | - | Low-$z$ Gal. | 43.04${}^{+0.16}_{-0.25}$ | 74$\pm$18 | 0.00
AEGIS001 | 2241 | 15615 | 14h17m23.9s | 52°38′6.0″ | 2.21 | J0390 | QSO | 1.52 | - | - | Cont. QSO | 43.79${}^{+0.05}_{-0.06}$ | 50$\pm$3 | 0.00
AEGIS001 | 2241 | 15867 | 14h18m16.2s | 52°29′40.7″ | 2.29 | J0400 | QSO | 1.61 | - | - | Cont. QSO | 44.10${}^{+0.05}_{-0.05}$ | 44$\pm$2 | 0.00
AEGIS001 | 2241 | 4536 | 14h16m19.6s | 52°24′58.8″ | 2.46 | J0420 | - | - | - | - | Low-$z$ Gal. | 43.47${}^{+0.15}_{-0.23}$ | 66$\pm$16 | 0.00
AEGIS001 | 2241 | 8524 | 14h16m25.5s | 52°32′42.6″ | 2.54 | J0430 | - | - | agn | 2.56 | QSO LAE | 43.93${}^{+0.05}_{-0.05}$ | 71$\pm$4 | 0.99
AEGIS001 | 2241 | 20770 | 14h18m23.3s | 52°40′4.6″ | 2.54 | J0430 | QSO | 1.77 | - | - | Cont. QSO | 44.13${}^{+0.04}_{-0.05}$ | 91$\pm$3 | 0.00
AEGIS001 | 2241 | 4481 | 14h17m23.1s | 52°15′14.7″ | 2.62 | J0440 | QSO | 2.59 | - | - | QSO LAE | 44.16${}^{+0.05}_{-0.05}$ | 43$\pm$2 | 1.00
AEGIS001 | 2241 | 2717 | 14h16m50.0s | 52°16′40.2″ | 2.70 | J0450 | - | - | - | - | Low-$z$ Gal. | 43.79${}^{+0.14}_{-0.21}$ | 53$\pm$13 | 0.00
AEGIS001 | 2241 | 7775 | 14h16m9.0s | 52°33′23.1″ | 2.79 | J0460 | - | - | - | - | - | 43.23${}^{+0.14}_{-0.20}$ | 96$\pm$21 | 0.95
AEGIS001 | 2241 | 14404 | 14h17m35.3s | 52°38′51.4″ | 2.79 | J0460 | QSO | 1.96 | - | - | - | 44.43${}^{+0.04}_{-0.04}$ | 66$\pm$1 | 0.97
AEGIS001 | 2241 | 15255 | 14h17m47.3s | 52°35′10.6″ | 2.79 | J0460 | - | - | - | - | QSO LAE | 43.57${}^{+0.08}_{-0.09}$ | 81$\pm$8 | 0.91
AEGIS001 | 2241 | 15524 | 14h16m51.6s | 52°43′12.4″ | 2.79 | J0460 | - | - | - | - | - | 43.19${}^{+0.15}_{-0.23}$ | 53$\pm$13 | 0.83
AEGIS001 | 2241 | 6762 | 14h17m42.0s | 52°17′7.3″ | 2.87 | J0470 | - | - | agn | 2.06 | - | 43.44${}^{+0.13}_{-0.19}$ | 90$\pm$18 | 0.89
AEGIS001 | 2241 | 20626 | 14h18m40.6s | 52°37′16.4″ | 2.87 | J0470 | - | - | - | - | - | 43.49${}^{+0.12}_{-0.16}$ | 49$\pm$9 | 0.84
AEGIS001 | 2241 | 4983 | 14h16m15.3s | 52°26′26.5″ | 2.95 | J0480 | - | - | - | - | - | 43.52${}^{+0.13}_{-0.18}$ | 47$\pm$9 | 0.57
AEGIS001 | 2241 | 20297 | 14h18m9.7s | 52°43′0.2″ | 3.19 | J0510 | QSO | 3.19 | - | - | QSO LAE | 44.17${}^{+0.04}_{-0.05}$ | 73$\pm$4 | 0.98
AEGIS001 | 2241 | 5837 | 14h16m11.9s | 52°28′44.5″ | 3.28 | J0520 | QSO | 3.25 | - | - | QSO LAE | 43.95${}^{+0.12}_{-0.17}$ | 68$\pm$13 | 1.00
AEGIS001 | 2241 | 14553 | 14h17m55.3s | 52°35′32.8″ | 3.36 | J0530 | - | - | - | - | - | 43.47${}^{+0.12}_{-0.16}$ | 164$\pm$27 | 0.91
AEGIS001 | 2241 | 9742 | 14h16m25.3s | 52°35′38.3″ | 3.60 | J0560 | - | - | - | - | - | 44.07${}^{+0.08}_{-0.09}$ | 377$\pm$38 | 0.98
AEGIS001 | 2241 | 13746 | 14h17m24.2s | 52°41′50.8″ | 3.69 | J0570 | - | - | - | - | - | 44.02${}^{+0.08}_{-0.10}$ | 88$\pm$10 | 0.97
AEGIS002 | 2243 | 2395 | 14h18m18.5s | 52°43′56.1″ | 2.11 | J0378 | QSO | 2.14 | - | - | QSO LAE | 43.92${}^{+0.06}_{-0.08}$ | 131$\pm$12 | 1.00
AEGIS002 | 2243 | 5085 | 14h19m18.1s | 52°41′58.4″ | 2.11 | J0378 | QSO | 2.03 | - | - | QSO LAE | 43.99${}^{+0.06}_{-0.07}$ | 64$\pm$5 | 0.98
AEGIS002 | 2243 | 4295 | 14h18m14.3s | 52°49′13.1″ | 2.21 | J0390 | - | - | - | - | - | 43.14${}^{+0.17}_{-0.29}$ | 67$\pm$19 | 0.89
AEGIS002 | 2243 | 7958 | 14h20m16.8s | 52°42′2.8″ | 2.21 | J0390 | - | - | - | - | - | 43.29${}^{+0.13}_{-0.18}$ | 150$\pm$29 | 1.00
AEGIS002 | 2243 | 7718 | 14h19m29.0s | 52°49′59.3″ | 2.29 | J0400 | - | - | agn | 2.31 | QSO LAE | 43.60${}^{+0.10}_{-0.13}$ | 162$\pm$23 | 1.00
AEGIS002 | 2243 | 15119 | 14h20m16.0s | 52°51′1.3″ | 2.29 | J0400 | QSO | 1.62 | - | - | Cont. QSO | 43.78${}^{+0.07}_{-0.09}$ | 96$\pm$10 | 0.00
AEGIS002 | 2243 | 12352 | 14h19m52.8s | 53°2′4.2″ | 2.70 | J0450 | QSO | 1.90 | agn | 1.90 | - | 44.29${}^{+0.06}_{-0.08}$ | 52$\pm$4 | 1.00
AEGIS002 | 2243 | 14558 | 14h19m26.8s | 53°0′3.5″ | 2.95 | J0480 | - | - | - | - | - | 43.75${}^{+0.13}_{-0.19}$ | 59$\pm$13 | 0.97
AEGIS002 | 2243 | 4755 | 14h19m23.1s | 52°40′13.4″ | 3.03 | J0490 | - | - | - | - | - | 43.85${}^{+0.12}_{-0.17}$ | 50$\pm$10 | 0.99
AEGIS002 | 2243 | 5404 | 14h18m13.4s | 52°52′40.0″ | 3.28 | J0520 | QSO | 3.29 | - | - | QSO LAE | 44.40${}^{+0.06}_{-0.07}$ | 140$\pm$9 | 1.00
AEGIS002 | 2243 | 14873 | 14h19m35.6s | 52°57′10.7″ | 3.28 | J0520 | QSO | 3.22 | agn | 3.22 | QSO LAE | 44.54${}^{+0.05}_{-0.05}$ | 83$\pm$4 | 1.00
AEGIS002 | 2243 | 673 | 14h19m14.3s | 52°30′50.3″ | 3.43 | J0540 | - | - | - | - | - | 43.86${}^{+0.10}_{-0.12}$ | 134$\pm$20 | 0.95
AEGIS002 | 2243 | 5769 | 14h17m56.9s | 52°56′15.4″ | 3.43 | J0540 | - | - | - | - | - | 43.75${}^{+0.12}_{-0.16}$ | 63$\pm$11 | 0.90
AEGIS002 | 2243 | 15610 | 14h19m43.6s | 52°54′31.3″ | 3.43 | J0540 | QSO | 3.44 | agn | 3.44 | QSO LAE | 44.52${}^{+0.04}_{-0.04}$ | 191$\pm$6 | 0.99
AEGIS003 | 2406 | 6731 | 14h22m5.7s | 53°10′54.1″ | 2.21 | J0390 | QSO | 1.45 | - | - | Cont. QSO | 43.30${}^{+0.13}_{-0.20}$ | 39$\pm$8 | 0.00
AEGIS003 | 2406 | 1224 | 14h21m18.0s | 52°53′46.0″ | 2.29 | J0400 | QSO | 2.31 | - | - | QSO LAE | 44.36${}^{+0.05}_{-0.05}$ | 109$\pm$4 | 1.00
AEGIS003 | 2406 | 1482 | 14h20m6.0s | 53°5′9.4″ | 2.46 | J0420 | - | - | - | - | - | 43.69${}^{+0.14}_{-0.21}$ | 67$\pm$16 | 0.99
AEGIS003 | 2406 | 7049 | 14h20m43.7s | 53°22′6.3″ | 2.46 | J0420 | QSO | 1.72 | - | - | Cont. QSO | 44.45${}^{+0.05}_{-0.06}$ | 54$\pm$2 | 0.00
AEGIS003 | 2406 | 11608 | 14h23m6.1s | 53°15′29.0″ | 2.46 | J0420 | QSO | 2.46 | - | - | QSO LAE | 45.00${}^{+0.04}_{-0.05}$ | 71$\pm$1 | 1.00
AEGIS003 | 2406 | 6169 | 14h20m46.4s | 53°12′24.5″ | 2.54 | J0430 | - | - | - | - | QSO LAE | 43.86${}^{+0.08}_{-0.10}$ | 90$\pm$11 | 0.99
AEGIS003 | 2406 | 14091 | 14h21m32.5s | 53°22′44.2″ | 2.54 | J0430 | - | - | - | - | - | 43.56${}^{+0.13}_{-0.20}$ | 81$\pm$17 | 0.99
AEGIS003 | 2406 | 4342 | 14h20m10.5s | 53°12′23.9″ | 2.62 | J0440 | QSO | 2.59 | - | - | QSO LAE | 44.31${}^{+0.04}_{-0.05}$ | 56$\pm$2 | 0.99
AEGIS003 | 2406 | 4658 | 14h20m25.6s | 53°11′5.2″ | 2.62 | J0440 | QSO | 1.85 | - | - | Cont. QSO | 43.50${}^{+0.12}_{-0.17}$ | 62$\pm$11 | 0.00
AEGIS003 | 2406 | 3657 | 14h20m18.8s | 53°9′11.3″ | 2.79 | J0460 | - | - | - | - | QSO LAE | 43.74${}^{+0.07}_{-0.08}$ | 114$\pm$11 | 0.92
AEGIS003 | 2406 | 8977 | 14h21m13.3s | 53°12′18.6″ | 2.79 | J0460 | QSO | 1.95 | - | - | - | 44.07${}^{+0.04}_{-0.05}$ | 55$\pm$3 | 0.97
AEGIS003 | 2406 | 11219 | 14h21m60.0s | 53°26′9.2″ | 2.79 | J0460 | QSO | 2.73 | - | - | QSO LAE | 44.08${}^{+0.05}_{-0.05}$ | 47$\pm$3 | 0.99
AEGIS003 | 2406 | 14869 | 14h21m36.5s | 53°20′14.2″ | 2.87 | J0470 | QSO | 2.02 | agn | 2.02 | QSO LAE | 43.89${}^{+0.07}_{-0.09}$ | 53$\pm$6 | 0.95
AEGIS003 | 2406 | 4964 | 14h21m54.8s | 52°58′41.9″ | 2.95 | J0480 | - | - | - | - | - | 43.84${}^{+0.07}_{-0.09}$ | 54$\pm$6 | 0.91
AEGIS003 | 2406 | 12752 | 14h21m47.1s | 53°24′5.8″ | 3.03 | J0490 | QSO | 3.04 | - | - | QSO LAE | 44.40${}^{+0.06}_{-0.06}$ | 66$\pm$4 | 0.99
AEGIS004 | 2470 | 8781 | 14h15m51.6s | 52°0′25.6″ | 2.21 | J0390 | QSO | 1.51 | - | - | Cont. QSO | 44.48${}^{+0.05}_{-0.05}$ | 41$\pm$1 | 0.00
AEGIS004 | 2470 | 2363 | 14h15m11.8s | 51°52′55.8″ | 2.29 | J0400 | QSO | 2.31 | - | - | QSO LAE | 44.37${}^{+0.05}_{-0.05}$ | 57$\pm$2 | 0.98
AEGIS004 | 2470 | 4455 | 14h13m47.9s | 52°12′5.0″ | 2.38 | J0410 | QSO | 2.35 | - | - | QSO LAE | 44.01${}^{+0.07}_{-0.09}$ | 97$\pm$11 | 1.00
AEGIS004 | 2470 | 9749 | 14h13m47.7s | 52°16′46.3″ | 2.38 | J0410 | QSO | 1.67 | - | - | Cont. QSO | 43.64${}^{+0.16}_{-0.25}$ | 45$\pm$12 | 0.00
AEGIS004 | 2470 | 13007 | 14h14m59.3s | 52°24′25.0″ | 2.38 | J0410 | - | - | - | - | QSO LAE | 43.77${}^{+0.11}_{-0.15}$ | 127$\pm$23 | 0.99
AEGIS004 | 2470 | 3723 | 14h14m28.2s | 52°3′47.2″ | 2.70 | J0450 | QSO | 2.69 | - | - | QSO LAE | 43.97${}^{+0.10}_{-0.13}$ | 87$\pm$15 | 0.99
AEGIS004 | 2470 | 13064 | 14h15m56.9s | 52°16′7.2″ | 2.70 | J0450 | QSO | 1.39 | - | - | Cont. QSO | 43.87${}^{+0.12}_{-0.17}$ | 47$\pm$10 | 0.00
AEGIS004 | 2470 | 12623 | 14h14m50.9s | 52°26′40.4″ | 2.87 | J0470 | - | - | - | - | QSO LAE | 43.52${}^{+0.12}_{-0.17}$ | 44$\pm$8 | 0.89
AEGIS004 | 2470 | 15095 | 14h16m17.8s | 52°7′18.3″ | 2.95 | J0480 | - | - | - | - | - | 43.77${}^{+0.09}_{-0.11}$ | 49$\pm$7 | 0.88
AEGIS004 | 2470 | 6481 | 14h15m42.7s | 52°9′27.2″ | 3.19 | J0510 | - | - | - | - | QSO LAE | 44.01${}^{+0.05}_{-0.06}$ | 257$\pm$17 | 0.98
J-NEP | 2520 | 3222 | 17h24m40.4s | 65°35′0.2″ | 2.11 | J0378 | - | - | - | - | - | 43.58${}^{+0.16}_{-0.27}$ | 65$\pm$17 | 0.94
J-NEP | 2520 | 6815 | 17h23m43.8s | 65°40′32.8″ | 2.11 | J0378 | - | - | - | - | Cont. QSO | 43.56${}^{+0.17}_{-0.29}$ | 60$\pm$17 | 0.00
J-NEP | 2520 | 9243 | 17h20m45.2s | 65°43′42.0″ | 2.11 | J0378 | - | - | - | - | Cont. QSO | 43.66${}^{+0.14}_{-0.22}$ | 50$\pm$12 | 0.00
J-NEP | 2520 | 12771 | 17h23m14.1s | 65°47′46.2″ | 2.11 | J0378 | QSO | 1.44 | - | - | Cont. QSO | 45.09${}^{+0.05}_{-0.06}$ | 58$\pm$1 | 0.00
J-NEP | 2520 | 22098 | 17h24m13.6s | 65°59′55.7″ | 2.11 | J0378 | - | - | - | - | Cont. QSO | 44.29${}^{+0.06}_{-0.07}$ | 37$\pm$2 | 0.00
J-NEP | 2520 | 10356 | 17h20m59.5s | 65°44′57.8″ | 2.29 | J0400 | - | - | - | - | QSO LAE | 43.72${}^{+0.07}_{-0.09}$ | 109$\pm$10 | 0.99
J-NEP | 2520 | 12539 | 17h22m8.9s | 65°47′43.1″ | 2.29 | J0400 | - | - | - | - | QSO LAE | 44.09${}^{+0.05}_{-0.06}$ | 100$\pm$4 | 1.00
J-NEP | 2520 | 15690 | 17h24m13.3s | 65°55′28.8″ | 2.29 | J0400 | - | - | - | - | - | 43.30${}^{+0.15}_{-0.23}$ | 39$\pm$9 | 0.93
J-NEP | 2520 | 2518 | 17h24m19.5s | 65°33′59.7″ | 2.38 | J0410 | - | - | - | - | - | 43.55${}^{+0.08}_{-0.10}$ | 66$\pm$7 | 0.98
J-NEP | 2520 | 5302 | 17h22m51.4s | 65°38′20.5″ | 2.38 | J0410 | - | - | - | - | - | 43.33${}^{+0.13}_{-0.19}$ | 71$\pm$14 | 0.97
J-NEP | 2520 | 10708 | 17h20m33.0s | 65°45′18.7″ | 2.46 | J0420 | - | - | - | - | - | 44.13${}^{+0.05}_{-0.06}$ | 42$\pm$2 | 0.98
J-NEP | 2520 | 6012 | 17h22m28.2s | 65°39′23.5″ | 2.62 | J0440 | - | - | - | - | - | 43.22${}^{+0.15}_{-0.23}$ | 43$\pm$10 | 0.93
J-NEP | 2520 | 12726 | 17h24m38.0s | 65°47′48.6″ | 2.62 | J0440 | - | - | - | - | - | 43.23${}^{+0.15}_{-0.23}$ | 61$\pm$14 | 0.96
J-NEP | 2520 | 19539 | 17h23m9.4s | 65°50′13.7″ | 2.62 | J0440 | - | - | - | - | - | 43.18${}^{+0.16}_{-0.25}$ | 60$\pm$16 | 0.91
J-NEP | 2520 | 1652 | 17h21m13.3s | 65°32′45.2″ | 2.70 | J0450 | - | - | - | - | QSO LAE | 43.85${}^{+0.09}_{-0.12}$ | 67$\pm$10 | 0.99
J-NEP | 2520 | 4903 | 17h21m22.6s | 65°37′45.3″ | 2.70 | J0450 | - | - | - | - | - | 44.22${}^{+0.05}_{-0.06}$ | 57$\pm$3 | 0.99
J-NEP | 2520 | 5247 | 17h23m26.1s | 65°38′16.3″ | 2.70 | J0450 | - | - | - | - | - | 44.20${}^{+0.05}_{-0.06}$ | 60$\pm$4 | 0.99
J-NEP | 2520 | 6520 | 17h24m11.7s | 65°40′4.7″ | 2.70 | J0450 | - | - | - | - | - | 43.66${}^{+0.12}_{-0.17}$ | 85$\pm$16 | 0.99
J-NEP | 2520 | 6636 | 17h22m39.9s | 65°40′17.1″ | 2.87 | J0470 | - | - | - | - | QSO LAE | 43.55${}^{+0.10}_{-0.14}$ | 48$\pm$7 | 0.92
J-NEP | 2520 | 14925 | 17h21m10.7s | 65°56′27.4″ | 2.87 | J0470 | - | - | - | - | QSO LAE | 44.48${}^{+0.04}_{-0.04}$ | 43$\pm$1 | 0.98
J-NEP | 2520 | 8395 | 17h23m55.8s | 65°42′46.0″ | 2.95 | J0480 | - | - | - | - | QSO LAE | 43.76${}^{+0.06}_{-0.08}$ | 203$\pm$18 | 0.86
J-NEP | 2520 | 14697 | 17h22m17.8s | 66°0′3.9″ | 3.60 | J0560 | - | - | - | - | - | 44.08${}^{+0.08}_{-0.09}$ | 127$\pm$13 | 1.00

##  Appendix B 2D purity and number count corrections

In Fig. 16 we show example representations of the 2D correction maps described
in Sect. 4.4, for $2.3<z<2.8$. These maps are computed for every redshift
interval used in this work, for every field of miniJPAS and J-NEP. The non-
colored regions in Fig. 16 correspond to combinations of parameters
incompatible with the LAEs of our mock, thus the candidates laying in this
areas of the 2D map are assigned $P^{\mathrm{2D}}=w^{\mathrm{2D}}=0$.

(a) AEGIS001

(b) AEGIS002

(c) AEGIS003

(d) AEGIS004

(e) J-NEP

Figure 16: 2D maps of the purity and number count correction for the four
miniJPAS fields and J-NEP in the redshift interval $z=2.3$–$2.8$. The maps are
computed for every field in miniJPAS and J-NEP and every interval of redshift
used in this work, only one $z$ interval is shown for brevity. The black
squares and error bars represent the LAE candidates of each field in the
chosen interval of redshift.

##  Appendix C Purity calibration

Figure 17: Effects of varying the sample purity on the estimated Ly$\alpha$
LF. Top: Colored squares show different realizations of our Ly$\alpha$ LF in
the range $z=2.0-3.8$, obtained after scaling the number of true positives in
the purity computation by a factor $\rho=0.1,0.5,1.0,1.5,2.0$. We also show
the mock QSO LF for each $\rho$ (solid lines). We show the uncertainties of
the LF with $\rho=1$. The variation of the LF is inside the $1\sigma$ errors
for all the shown values of $\rho$. Bottom: Estimated purity for the different
values of $\rho$.

As introduced in Sect. 3, the corrections of our LF are likely to be biased by
the underlying luminosity distribution we impose to our mock data. The number
count correction estimate does not depend on the parameter space distribution
of the mocks, as it is only a measure on how likely is to select a source as a
function of its intrinsic $L_{\mathrm{Ly}\alpha}$ and EW. Nonetheless, the
purity estimate is strongly dependent on the relative abundances of the
objects from the target population in relation to that of the contaminants.

To check the robustness of our corrections in the QSO regime, we alter the
purity estimate by introducing a factor $\rho$ to the true positive count. We
recalculate the 2D purity as

$p^{\mathrm{2D}}=\frac{\rho\cdot\text{TP}}{\rho\cdot\text{TP}+\text{FP}},$
(19)

and compute the Ly$\alpha$ LF of miniJPAS&J-NEP, using this new definition of
the purity, for different values of $\rho$. Since the effect of
increasing/decreasing the value of $\rho$ is equivalent to
increasing/decreasing the mock QSO number density, this procedure tests the
effect of under/overestimating the purity of our selected sample. We estimate
the Ly$\alpha$ LF for different values of $\rho$ using the whole selected
sample of 127 candidates (see Sec. 4.6). The upper panel of Fig. 17 shows the
recomputed LFs for different values of $\rho$ (as detailed by the plot label).
Solid lines show the resulting $L_{\mathrm{Ly}\alpha}$ distribution of our QSO
mock, for each $\rho$ value. The comparison between the different LF
realizations (colored squares and error-bars) show that the change produced
onto the LF by varying $\rho$ is small, especially when compared to the
$1\sigma$ uncertainties for the $\rho=1$ realization (black error bars). In
addition, the value of $\rho$ that produces an input LF comparable to the
estimated is close to $\rho\sim 1$. With the eventual release of larger
catalog of J-PAS, it will be possible to accurately calibrate the purity of
the LAE candidate sample.

As already discussed in Sect. 5.3, our LAE candidate sample is expected to be
vastly dominated by AGN. Consequently, any change in the prior LF used for the
construction of the SFG mock produces a minimum change onto our final results.
Although we predict a non-zero contribution of SFG in our sample, the
observational data is insufficient for calibrating the purity at the SFG
regime ($\log_{10}(L_{\mathrm{Ly}\alpha}/\mathrm{erg\,s}^{-1})\lesssim 43.5$)
with the current selection method.


# Counting Lattice Points that appear as algebraic invariants of
Cameron–Walker Graphs

Sara Faridi, Iresha Madduwe Hewalage  Department of Mathematics
Dalhousie University, Halifax
Canada

###### Abstract.

In 2021, Hibi et. al. studied lattice points in $\mathbb{N}^{2}$ that appear
as $(\operatorname{depth}R/I,\dim R/I)$ when $I$ is the edge ideal of a graph
on $n$ vertices, and showed these points lie between two convex polytopes.
When restricting to the class of Cameron–Walker graphs, they showed that these
pairs do not form a convex lattice polytope. In this paper, for the edge ideal
$I$ of a Cameron–Walker graph on $n$ vertices, we find how many points in
$\mathbb{N}^{2}$ appear as $(\operatorname{depth}(R/I),\dim(R/I))$, and how
many points in $\mathbb{N}^{4}$ appear as

$(\operatorname{depth}(R/I),\operatorname{reg}(R/I),\dim(R/I),\operatorname{degh}(R/I)).$

##  1\.  Introduction

Combinatorial commutative algebra studies the problems in commutative algebra
using the tools and techniques in combinatorics of geometric structures.
Monomial ideals play an important role in studying the relationship between
commutative algebra and combinatorics. Let $R=K[x_{1},\ldots,x_{n}]$ be the
polynomial ring in $n$ variables over a field $K$. For a finite simple graph
$G$ on vertex set $[n]=\\{1,\ldots,n\\}$ and the edge set $E(G)$ we define the
edge ideal of $G$, denoted by $I(G)$, as the monomial ideal of $R$, generated
by those monomials $x_{i}x_{j}$ with $\\{i,j\\}\in E(G)$. Edge ideals of
finite simple graphs have been studied by many authors from a viewpoint of
commutative algebra. In the present paper, we are interested in the
homological invariants

$\displaystyle\operatorname{depth}(G)=\operatorname{depth}(R/I(G)),$
$\displaystyle\dim(G)=\dim(R/I(G)),$
$\displaystyle\operatorname{reg}(G)=\operatorname{reg}(R/I(G)),$
$\displaystyle\operatorname{degh}(G)=\operatorname{degh}(R/I(G)).$

Katzman [7] found that these homological invariants do not depend on the
characteristic of the field $K$, if our graph has at most 10 vertices. Hibi,
Kanno, Kimura, Matsuda and Van Tuyl, have shown [5, 6], that when we
restricted to the family of Cameron–Walker graphs we can represent these
homological invariants in terms of the combinatorics of the graphs only. i.e.
these homological invariants are characteristic–free on the class of
Cameron–Walker graphs. Cameron–Walker graphs are named after the 2005 paper
[2] of Cameron and Walker in which the authors characterized the class of
graphs with the property that the maximum matching number equals the maximum
induced matching number. As a result of their work, they classified these
graphs into 3 types; star graphs, star triangles and finite graphs consisting
of a connected bipartite graph with two vertex partition sets such that there
is at least one leaf edge attached to each vertex in one vertex set and that
there may be possibly some triangles attached to the vertices in the other
vertex set. Hibi, Higashitani, Kimura, and O’Keefe ([4]), named the graphs of
the last type Cameron–Walker (CW) graphs. The relationships among the
homological invariants of the edge ideals of Cameron–Walker graphs have been
studied by many authors. In [6] authors studied the possible tuples
$(\operatorname{reg}(G),\operatorname{degh}(G))$ for all graphs $G$ on a fixed
number of vertices. They found that such lattice points lie between two convex
lattice polytopes. Moreover, it was shown that if we restricted to the class
of Cameron–Walker graphs the set of all such lattice points form a convex
lattice polytope. Inspired by this result, the authors in paper [5], studied
the ordered pairs

(1) $(\operatorname{depth}(G),\dim(G))$

for all the finite simple graphs $G$ with a fixed number of vertices and they
showed the set of all such points always lie between two convex lattice
polytopes. However, when they restricted to the family of Cameron–Walker
graphs the set of all possible lattice points of the form in (1) do not form a
convex lattice polytope. Moreover, they completely determined all the possible
pairs of the form (1) arising from Cameron–Walker graphs $G$. Moving from this
point in paper [5, Theorem 4.4] they found all the tuples

(2)
$(\operatorname{depth}(G),\operatorname{reg}(G),\dim(G),\operatorname{degh}(G))$

of the edge ideals of Cameron–Walker graphs $G$.

For Cameron–Walker graphs on $n$ vertices we define

$\operatorname{\operatorname{CW}_{\operatorname{depth},\dim}}(n)=\\{(\operatorname{depth}(G),\dim(G))\mid
G\mbox{ Cameron--Walker graph}\\}$

and

$\operatorname{\operatorname{CW}_{\operatorname{depth},\operatorname{reg},\dim,\operatorname{degh}}}(n)=\\{(\operatorname{depth}(G),\operatorname{reg}(G),\dim(G),\operatorname{degh}(G))\mid
G\mbox{ graph}\\}.$

Our goal is to find the size of the two sets
$\operatorname{\operatorname{CW}_{\operatorname{depth},\dim}}(n)$ and
$\operatorname{\operatorname{CW}_{\operatorname{depth},\operatorname{reg},\dim,\operatorname{degh}}}(n)$
without finding all the possible tuples for each set similar to the work done
in [6] for the pair, regularity, and the degree of the h–polynomial of
Cameron–Walker graphs of order $n\geq 5$..

We answer this question in this paper; we compute the number of elements
$(a,b)$ where $a=\operatorname{depth}(G)$ and $b=\dim(G)$ for the set
$\operatorname{\operatorname{CW}_{\operatorname{depth},\dim}}(n)$. Next we try
to compare the number of integer points in
$\operatorname{\operatorname{CW}_{\operatorname{depth},\dim}}(n)$ to the
number of possible pairs $(a,b)$ in
$\operatorname{\operatorname{Graph}_{\operatorname{depth},\dim}}(n)$. To do
this we will find

${\lim_{n\rightarrow\infty}\frac{|\operatorname{\operatorname{CW}_{\operatorname{depth},\dim}}(n)|}{|\operatorname{\operatorname{Graph}_{\operatorname{depth},\dim}}(n)|}}$

assuming it exists. Finally, we determine the size of the set of tuples

$(\operatorname{depth}(G),\operatorname{reg}(G),\dim(G),\operatorname{degh}(G))$

arising from Cameron–Walker graphs on fixed number of vertices.

##  2\. Background

A graph is a mathematical structure consisting of a pair $G=(V,E)$, where $V$
is a set of points called vertices, and $E$ is a collection of unordered pairs
of vertices, whose elements are called edges. A loop is an edge that connects
a vertex to itself. If a graph has more than one edge joining some pair of
vertices then these edges are called multiple edges. A simple graph is a graph
without loops or multiple edges. All graphs in this paper are simple graphs.
For a simple graph, two vertices are said to be adjacent if there is an edge
connecting them. A path is a sequence of distinct vertices with the property
that the consecutive vertices are adjacent, i.e. each vertex $x_{i}$ in the
sequence is adjacent by an edge to the vertex $x_{i+1}$ next to it. A path on
$n$ vertices is denoted by $P_{n}$. A graph is said to be connected if any two
of its vertices are joined by a path. In a simple graph, the degree of a
vertex $v$, denoted $\deg(v)$, is the number of edges meeting it, or in other
words, incident to it.

A complete graph is a graph in which each pair of distinct vertices is
connected by an edge. A complete graph on $n$ vertices is denoted by $K_{n}$.
The complete graph $K_{3}$ is called a triangle. A bipartite graph is a graph
whose vertices can be divided into two sets, $U$ and $W$, such that every edge
connects a vertex in $U$ to one in $W$. i.e. there is no edge that connects
vertices of same set. A complete bipartite graph is a bipartite graph such
that every vertex in one set is adjacent to every vertex of other set. If the
two sets have $m$ and $n$ vertices then we denote the complete bipartite graph
by $K_{m,n}$.

Let $G=(V,E)$ be a graph. A matching is a set of edges in $E$ no two of which
share a common vertex. An induced matching $M$ is a matching such that no pair
of edges of $M$ are joined by an edge in the graph $G$. A matching $M$ is said
to be maximum if for any other matching $M^{\prime},\ |M|\geq|M^{\prime}|$.
Here $|M|$ is the size of the matching $M$. The maximum size of a matching in
$G$ is denoted by $\operatorname{m}(G)$, and the maximum size of an induced
matching in $G$ is denoted by $\operatorname{im}(G)$.

######  Example 2.1.

The edge set $\\{af,bc,ed\\}$ in Figure 1 represents a maximum matching and
$\\{af,cd\\}$ gives a maximum induced matching.

$\bullet$$a$$\bullet$$b$$\bullet$$c$$\bullet$$f$$\bullet$$e$$\bullet$$d$
Figure 1.  A simple graph

A leaf in a graph is a vertex of degree 1, that is, a vertex which meets only
one edge. A leaf edge or a pendant edge is an edge which meets a leaf. A
pendant triangle in a graph is a triangle in which exactly two vertices have
degree 2 and the third vertex is of degree greater than 2. The pendant
triangle is attached to the graph at the vertex with degree greater than 2.

Cameron and Walker [2] proved the following statement:

######  Theorem 2.2.

[2, Theorem 1] Let $G$ be a connected graph. Then
$\operatorname{m}(G)=\operatorname{im}(G)$ if and only if $G$ is a star or a
star triangle, or consists of a connected bipartite graph $B$ with vertex
partition $\\{u_{1},\dots,u_{m}\\}\cup\\{v_{1},\dots,v_{n}\\}$ such that there
is at least one leaf edge (and possibly more) attached to each vertex $u_{i}$
of $B$, and possibly some pendant triangles attached to a vertex $v_{j}$ of
$B$.

In other words, a finite connected simple graph $G$ satisfies
$\operatorname{m}(G)=\operatorname{im}(G)$ if and only if $G$ is one of the
following types of graphs:

  1. (1)

a star , i.e. a complete bipartite graph $K_{1,n}$ as in Figure 2;

$\bullet$$a$$\bullet$$b$$\bullet$$g$$\bullet$$c$$\bullet$$f$$\bullet$$d$ |  | $\bullet$$a$$\bullet$$b$$\bullet$$c$$\bullet$$d$$\bullet$$g$$\bullet$$f$$\bullet$$e$
---|---|---
Figure 2. A star graph and a star triangle

  2. (2)

a star triangle, a graph consisting of a set of triangles whose pairwise
intersections are one single vertex as in Figure 2; or

  3. (3)

a finite graph consisting of a connected bipartite graph with two vertex
partition sets such that there is at least one leaf edge attached to each
vertex in one vertex set and that there may be possibly some pendant triangles
attached to the vertices in the other vertex set, as in Figure 3.

$\bullet$$a$$\bullet$$b$$\bullet$$c$$\bullet$$d$$\bullet$$e$$\bullet$$f$$\bullet$$g$$\bullet$$h$$\bullet$$i$$\bullet$$j$$\bullet$$k$$\bullet$$l$$\bullet$$m$$\bullet$$n$$\bullet$$p$$\bullet$$q$$\bullet$$r$$\bullet$$s$$\bullet$$t$
Figure 3. A Cameron–Walker Graph

Hibi et al [4] defined any graph of type (3) to be a Cameron-Walker graph.
i.e. a Cameron–Walker graph is a finite connected simple graph $G$ such that
$\operatorname{m}(G)=\operatorname{im}(G)$ and $G$ is neither a star graph nor
a star triangle.

If $R$ is a ring, the height of a prime ideal $\mathfrak{p}$ is defined as the
supremum of all $n$ where there is a chain
$\mathfrak{p}_{0}\subset...\mathfrak{p}_{(n-1)}\subset\mathfrak{p}_{n}=\mathfrak{p}$
where all $\mathfrak{p}_{i}$ are distinct prime ideals. The Krull dimension or
dimension of $R$ is defined as the supremum of all the heights of all its
prime ideals. Let $M$ be a maximal ideal of $R$. A regular sequence is a
sequence $a_{1}$, …, $a_{n}\in M$ such that all $a_{i}$ are not zero-divisors
in $R/(a_{1},\ldots,a_{i-1})$. The depth of a ring $R$ is the number of
elements in some maximal regular sequence. Suppose $G$ is a graph with
vertices $\\{x_{1},\ldots,x_{n}\\}$. Let $S=$ $K[x_{1},\ldots,x_{n}]$ denote
the polynomial ring in $n$ variables over a field $K$. The edge ideal of $G$,
denoted $I(G)$, is the ideal of $K[x_{1},\ldots,x_{n}]$ with generators
specified as follows: $x_{i}x_{j}$ is a generator of $I(G)$ if and only if
$x_{i}$ is adjacent to $x_{j}$ in $G$.

######  Example 2.3.

The edge ideal of the graph in Figure 1 is $I(G)=(ab,bc,cd,de,ef,fa,bf,ce)$.

##  3\. On the set
${\operatorname{\operatorname{Graph}_{\operatorname{depth},\dim}}(n)}$

Let $\operatorname{Graph}(n)$ be the set of all finite simple graphs having
$n$ vertices and let
$\operatorname{\operatorname{Graph}_{\operatorname{depth},\dim}}(n)$ denote
the set of all possible pairs $(\operatorname{depth}(G),\dim(G))$ arising from
the elements $G$ in $\operatorname{Graph}(n)$. In other words

$\operatorname{\operatorname{Graph}_{\operatorname{depth},\dim}}(n)=\\{(\operatorname{depth}(G),\dim(G)\mid
G\in\operatorname{Graph}(n)\\}.$

A natural question is: what are the possible pairs of
$(a,b)\in\operatorname{\operatorname{Graph}_{\operatorname{depth},\dim}}(n)$
where $a,b\in\mathbb{N}$? It is not difficult to compute the pairs $(a,b)$ for
the simple graphs with small $n$.

######  Example 3.1.

[5, Example 2.1] A connected graph $G$ with three vertices is either a path on
three vertices or a triangle. If $G$ is a path, we have the edge ideal
$I(G)=(x_{1}x_{2},x_{2}x_{3})\subset K[x_{1},x_{2},x_{3}]$ and

$\operatorname{depth}(G)=1,\quad\dim(G)=2.$

If $G$ is a triangle, then $I(G)=(x_{1}x_{2},x_{2}x_{3},x_{2}x_{3})$ and
$\operatorname{depth}(G)=\dim(G)=1$.

While finding
$\operatorname{\operatorname{Graph}_{\operatorname{depth},\dim}}(n)$ is easy
for $n\leq 3$, for larger $n$, determining the elements of this set becomes
more complicated. The authors in [5] give lattice boundaries for the points in
this set.

######  Theorem 3.2.

[5, Theorem 2.9] If $G$ is any simple graph on $n\geq 3$ vertices, then

$C^{-}(n)\subseteq\operatorname{\operatorname{Graph}_{\operatorname{depth},\dim}}(n)\subseteq
C^{+}(n)$

where

${C^{-}(n)=\\{(1,n-1)\\}\cup\left\\{(a,b)\in\mathbb{N}^{2}\mid 1\leq a\leq
b\leq n-2,\
a\leq\left\lfloor{\frac{n}{2}}\right\rfloor\right\\}\subseteq\mathbb{N}^{2}}$

and

$C^{+}(n)=\left\\{(a,b)\in\mathbb{N}^{2}\mid 1\leq a\leq b\leq
n-1\right\\}\subseteq\mathbb{N}^{2}.$

In other words the set
$\operatorname{\operatorname{Graph}_{\operatorname{depth},\dim}}(n)$ is
sandwiched by the convex lattice polytopes $C^{-}(n)$ and $C^{+}(n)$. This
calculates precise bounds for the elements of
$\operatorname{\operatorname{Graph}_{\operatorname{depth},\dim}}(n)$ for
$n\geq 3$. When restricted to the class of Cameron–Walker graphs, they were
able to give a full characterization of the pairs $(a,b)$ that appear as
$\operatorname{depth}(G)$ and $\dim(G)$ of Cameron–Walker graphs.

######  Theorem 3.3.

[5, Theorem 3.15] Let
$\operatorname{\operatorname{CW}_{\operatorname{depth},\dim}}(n)$ denote the
set of all possible pairs of depth and dimension of edge ideals of
Cameron–Walker Graphs with $n$ vertices. Then for $n\geq 5$

(3) $\operatorname{CW}_{2,\dim}(n)=\begin{cases}\\{(2,n-2),(2,n-3)\\},&n\mbox{
even}\\\ \\{(2,n-2),(2,n-3),\left(2,\frac{n-1}{2}\right)\\}&n\mbox{
odd}\end{cases}$

and

(4)
$\displaystyle\operatorname{\operatorname{CW}_{\operatorname{depth},\dim}}(n)$
$\displaystyle=\underbrace{\operatorname{CW}_{2,\dim(n)}}_{A}\cup\underbrace{\left\\{(b,b)\in\mathbb{N}^{2}\
|\ \frac{n}{3}<b<\frac{n}{2}\right\\}}_{B}$
$\displaystyle\cup\underbrace{\left\\{(a,b)\in\mathbb{N}^{2}\mid 3\leq
a\leq\left\lfloor{\frac{n-1}{2}}\right\rfloor,\max\left\\{a,\frac{n-a}{2}\right\\}<b\leq
n-a\right\\}}_{C}.$

We now determine the size of the set
$\operatorname{\operatorname{CW}_{\operatorname{depth},\dim}}(n)$ based on the
description of the elements. This will allow one to find how many
Cameron–Walker graphs there are of a fixed dimension and depth, over a given
number of vertices. To do so, we determine the size of each of the sets $A$,
$B$ and $C$ in (4) and their overlaps.

######  Lemma 3.4.

For the sets, $A$, $B$ and $C$ are defined as in (4), $A\cap C=B\cap
C=\emptyset$, and

$A\cap B=\begin{cases}\\{(2,2)\\}&n=5\\\
\emptyset&\mbox{otherwise.}\end{cases}$

###### Proof.

If $(b,b)\in A\cap B$, then in (3) we must have $2=n-2$ or $2=n-3$ or
$2=\frac{n-1}{2}$. So $n=4$ or $n=5$. But there is no CW graphs with $n=4$, so
$n=5$, and in this case $(2,2)$ is the only point in $A\cap B$.

If $(b,b)\in B\cap C$, then by the definition of $C$,
$\max\left\\{b,\frac{n-b}{2}\right\\}\lneq b$. Therefore, we have $b\lneq b$,
which is a contradiction.

If $(a,b)\in A\cap C$, then $a\geq 3$ and $a=2$, which is not possible. Thus,
$A\cap C=\emptyset$.

∎

######  Lemma 3.5 (The size of $A$).

If $A$ is as in Theorem 3.3 and $n\geq 5$, then

(5) $|A|=\begin{cases}2&n\mbox{ is even or }n=5\\\
3&\mbox{otherwise}.\end{cases}$

###### Proof.

By (3) in Theorem 3.3, it is easy to see that for each $n>5$ we have two
elements in $\mbox{CW}_{\mbox{2,dim}}(n)$ if $n$ is even and we have three
elements if $n$ is odd. When $n=5$, we have the two elements: $(2,3)$ and
$(2,2)$ in $\mbox{CW}_{\mbox{2,dim}}(n)$. Our claim now follows. ∎

######  Lemma 3.6 (The size of $B$).

With $B$ is as in Theorem 3.3, and $n\geq 5$, let $n=6k+i$ where $k\geq 0$ and
$0\leq i\leq 5$. Then

(6) $\left|B\right|=\begin{cases}k-1&i=0\\\ k&1\leq i\leq 4\\\
k+1&i=5.\end{cases}$

###### Proof.

By definition

$B=\left\\{(b,b)\in\mathbb{N}^{2}\mid\frac{n}{3}<b<\frac{n}{2}\right\\}=\left\\{(b,b)\in\mathbb{N}^{2}\mid
2k+\frac{i}{3}<b<3k+\frac{i}{2}\right\\}.$

The options for $b$ are then

$b\in\begin{cases}\\{2k+1,2k+2,\ldots,3k-1\\}&i=0\\\
\\{2k+1,2k+2,\ldots,3k\\}&i=1,2\\\ \\{2k+2,2k+3,\ldots,3k+1\\}&i=3,4\\\
\\{2k+2,2k+3,\ldots,3k+2\\}&i=5\\\ \end{cases}$

which proves our claim. ∎

######  Lemma 3.7 (The size of $C$).

With $C$ is as in Theorem 3.3, $|C|=0$ when $n=5$. When $n>5$, let $n=6k+i$
where $k>0$ and $0\leq i\leq 5$. Then

(7) $|C|=\begin{cases}6k^{2}-7k+1&\ i=0\\\ 6k^{2}-5k&\ i=1\\\ 6k^{2}-3k-1&\
i=2\\\ 6k^{2}-k-1&\ i=3\\\ 6k^{2}+k-1&\ i=4\\\ 6k^{2}+3k-1&\ i=5.\end{cases}$

###### Proof.

Recall that $C=\left\\{(a,b)\in\mathbb{N}^{2}\ \Big{|}\ 3\leq
a\leq\left\lfloor{\frac{n-1}{2}}\right\rfloor,\quad\max\left\\{a,\frac{n-a}{2}\right\\}<b\leq
n-a\right\\}.$

We fix an integer $a$ with satisfying inequality $3\leq
a\leq\left\lfloor{\frac{n-1}{2}}\right\rfloor$. Then we have two possible
scenarios

$a\leq\frac{n-a}{2}\quad\mbox{or}\quad a>\frac{n-a}{2}.$

We consider each case separately.

  1. (1)

When ${a\leq\frac{n-a}{2}}$ (or equivalently $a\leq\frac{n}{3}$) we are
looking for all the possible $b$’s satisfying the inequality

$\frac{n-a}{2}<b\leq n-a.$

    1. (a)

If $n-a$ is odd then the number of $b$’s satisfying the above inequality is

$\frac{n-a-1}{2}+1=\frac{n-a+1}{2}=\left\lceil\frac{n-a}{2}\right\rceil.$

    2. (b)

If $n-a$ is even then the number of $b$’s satisfying the above inequality is

${\frac{n-a}{2}=\left\lceil\frac{n-a}{2}\right\rceil}.$

  2. (2)

When ${a>\frac{n-a}{2}}$ (or equivalently $a>\frac{n}{3}$), the number of
$b$’s satisfying $a<b\leq n-a$ is $n-2a$.

Putting (1) and (2) together, and observing that since $n\geq 6$ we must have
$\frac{n}{3}<\frac{n-1}{2}$, we conclude that

(8)
$\left|C\right|=\sum_{a=3}^{\left\lfloor{\frac{n}{3}}\right\rfloor}\left\lceil\frac{n-a}{2}\right\rceil+\sum_{a={\left\lfloor{\frac{n}{3}}\right\rfloor}+1}^{\left\lfloor{\frac{n-1}{2}}\right\rfloor}(n-2a).$

Mathematica [11] computes this sum to be (7). ∎

######  Theorem 3.8 (The size of
$\operatorname{\operatorname{CW}_{\operatorname{depth},\dim}}(n)$).

Let $n$ be an integer. If $n>5$ let $n=6k+i$ where $k\geq 1$ and $0\leq i\leq
5$.

$|\operatorname{\operatorname{CW}_{\operatorname{depth},\dim}}(n)|=\begin{cases}0&n<5\\\
2&n=5\\\ 6k^{2}-6k+2&i=0\\\ 6k^{2}-4k+3&i=1\\\ 6k^{2}-2k+1&i=2\\\
6k^{2}+2&i=3\\\ 6k^{2}+2k+1&i=4\\\ 6k^{2}+4k+3&i=5.\end{cases}$

###### Proof.

There is no Cameron–Walker graphs with less than $5$ vertices, so assume
$n\geq 5$. By Theorem 3.3, using elementary set theory and Lemma 3.4, we have

$\left|\operatorname{\operatorname{CW}_{\operatorname{depth},\dim}}(n)\right|=\begin{cases}{|A|+|B|+|C|-|A\cap
B|}&\mbox{if}\ n=5\\\ {|A|+|B|+|C|}&\mbox{if}\ n>5\\\ \end{cases}$

Hence, using (5), (6), and (7); when $n=6k$ (or when $n>5$),

$\left|\operatorname{\operatorname{CW}_{\operatorname{depth},\dim}}(n)\right|=2+(k-1)+(k-1)(6k-1)=6k^{2}-6k+2.$

Similarly, we can compute the other cases. Moreover, when $n=5$,

$\left|\operatorname{\operatorname{CW}_{\operatorname{depth},\dim}}(n)\right|=2+1+0-1=2.$

∎

The following statement is a direct consequence of Theorem 3.8.

######  Corollary 3.9.

$\lim_{k\rightarrow\infty}\frac{\left|\operatorname{\operatorname{CW}_{\operatorname{depth},\dim}}(n)\right|}{n^{2}}=\frac{1}{6}.$

Let

$\displaystyle\operatorname{\operatorname{CW}_{RD}}(n)$ $\displaystyle=$
$\displaystyle\\{(\operatorname{reg}(G),\operatorname{degh}(G))\mid
G\in\operatorname{CW}(n)\\}$
$\displaystyle\operatorname{\operatorname{G}_{RD}}(n)$ $\displaystyle=$
$\displaystyle\\{(\operatorname{reg}(G),\operatorname{degh}(G))\mid
G\in\operatorname{Graph}(n)\\}$

where $\operatorname{reg}(G)$ and $\operatorname{degh}(G)$ are the regularity
and the degree of the $h-$polynomial of the edge ideals of $G$, respectively.
The authors in [5] gave a precise description for the size of the set
$\operatorname{\operatorname{CW}_{RD}}(n)$ when $n\geq 5$.

It would be nice to compare the number of integer points in
$\operatorname{\operatorname{CW}_{\operatorname{depth},\dim}}(n)$ to the
number of integer points in
$\operatorname{\operatorname{Graph}_{\operatorname{depth},\dim}}(n)$ similar
to the comparison described between the
$\operatorname{\operatorname{CW}_{RD}}(n)$ and
$\operatorname{\operatorname{G}_{RD}}(n)$ in [5]. Then one might be able to
find the percentage of lattice points recognized by the Cameron–Walker graphs.

Thus it would be interesting if we can answer the following question.

######  Question.

What is the value of

${\lim_{n\rightarrow\infty}\frac{\left|\operatorname{\operatorname{CW}_{\operatorname{depth},\dim}}(n)\right|}{\left|\operatorname{\operatorname{Graph}_{\operatorname{depth},\dim}}(n)\right|}}?$

In [5], the authors asked a similar question for the
$\operatorname{\operatorname{CW}_{RD}}(n)$. Since we can only bound
$\operatorname{\operatorname{Graph}_{\operatorname{depth},\dim}}(n)$, it is
not clear enough whether the this limit exists or not. If we want to show that
this limit exists, it is enough to show that

${\frac{|\operatorname{\operatorname{CW}_{\operatorname{depth},\dim}}(n)|}{|\operatorname{\operatorname{Graph}_{\operatorname{depth},\dim}}(n)|}}$

is increasing for all $n$ as its value is always $\leq 1.$ The authors in [5]
observed, we can use the monotone convergence theorem to show that the limit
exists. Computational evidence shows that this fraction is indeed increasing
and the limit exists.

######  Theorem 3.10.

Suppose
${\lim_{n\rightarrow\infty}\frac{\left|\operatorname{\operatorname{CW}_{\operatorname{depth},\dim}}(n)\right|}{\left|\operatorname{\operatorname{Graph}_{\operatorname{depth},\dim}}(n)\right|}}$
exists. Then

$\frac{1}{3}\leq{\lim_{n\rightarrow\infty}\frac{|\operatorname{\operatorname{CW}_{\operatorname{depth},\dim}}(n)|}{|\operatorname{\operatorname{Graph}_{\operatorname{depth},\dim}}(n)|}}\leq\frac{4}{9}.$

###### Proof.

Setting $n=6k+i$ for $k>0$ and $0\leq i\leq 5$, we have
$(n-3)^{2}=36k^{2}+12k(i-3)+(i-3)^{2}$. On the other hand, for each value of
$i$, by Theorem 3.8 there is an integer $\epsilon_{i}\in\\{0,1,2,3\\}$ such
that

$\left|\operatorname{\operatorname{CW}_{\operatorname{depth},\dim}}(n)\right|=6k^{2}+2(i-3)k+\epsilon_{i}.$

Putting all this together, we have

$\left|\operatorname{\operatorname{CW}_{\operatorname{depth},\dim}}(n)\right|=\frac{1}{6}(n-3)^{2}+\epsilon_{i}-\frac{1}{6}(i-3)^{2}.$

Using Theorem 3.8 again by setting $i=0,\ldots,5$, we observe that

$\epsilon_{i}-\frac{1}{6}(i-3)^{2}\in\left\\{2-\frac{9}{6},3-\frac{4}{6},1-\frac{1}{6},2,1-\frac{1}{6},3-\frac{4}{6}\right\\}=\left\\{\frac{1}{2},\frac{7}{3},\frac{5}{6},2\right\\}.$

We have therefore shown that if $n>5$, for some $u$ such that
${\frac{1}{2}\leq u\leq\frac{7}{3}}$ we have

(9)
$\frac{1}{6}(n-3)^{2}+\frac{1}{2}\leq\left|\operatorname{\operatorname{CW}_{\operatorname{depth},\dim}}(n)\right|\leq\frac{1}{6}(n-3)^{2}+\frac{7}{3}$

Now using Theorem 3.2, we can get an upper and a lower bound for the
${|\operatorname{\operatorname{Graph}_{\operatorname{depth},\dim}}(n)|}$. i.e.

${\left|C^{-}(n)\right|\leq\left|\operatorname{\operatorname{Graph}_{\operatorname{depth},\dim}}(n)\right|\leq\left|C^{+}(n)\right|.}$

Now recall from Theorem 3.2, that

${C^{-}(n)=\left\\{(1,n-1)\right\\}\cup\left\\{\left(a,b\right)\in\mathbb{N}^{2}\mid
a\leq b,1\leq a\leq\left\lfloor{\frac{n}{2}}\right\rfloor,1\leq b\leq
n-2\right\\}\subseteq\mathbb{N}^{2}}.$

Let

${\beta=\left\\{(a,b)\in\mathbb{N}^{2}\ |\ 1\leq
a\leq\left\lfloor{\frac{n}{2}}\right\rfloor,\ a\leq b\leq
n-2\right\\}\subseteq\mathbb{N}^{2}}.$

As ${\left\lfloor{\frac{n}{2}}\right\rfloor\leq n-2}$ for all $n\geq 4$, the
number of possibilities for $b$ in the set $\beta$ can be found by the
inequality $a\leq b\leq n-2$. Therefore,

$\left|\beta\right|=\sum_{a=1}^{\left\lfloor{\frac{n}{2}}\right\rfloor}(n-a-1).$

Now, recall that $n=6k+i$. So, Mathematica [11] computes $|\beta|$ to be,

$|\beta|=\begin{cases}\frac{9}{2}k(3k-1)&i=0\\\ \frac{3}{2}k(9k-1)&i=1\\\
\frac{9}{2}k(3k+1)&i=2\\\ \frac{1}{2}(3k+1)(9k+2)&i=3\\\
\frac{3}{2}(3k+2)(3k+1)&i=4\\\
\frac{1}{2}(3k+2)(9k+5)&i=5\end{cases}\quad\mbox{and}\quad|C^{-}(n)|=\begin{cases}\frac{1}{2}(27k^{2}-9k)+1&i=0\\\
\frac{1}{2}(27k^{2}-3k)+1&i=1\\\ \frac{1}{2}(27k^{2}+9k)+1&i=2\\\
\frac{1}{2}(27k^{2}+15k+2)+1&i=3\\\ \frac{1}{2}(27k^{2}+27k+6)+1&i=4\\\
\frac{1}{2}(27k^{2}+33k+10)+1&i=5.\end{cases}$

Since $|C^{-}(n)|$ is the lower bound for
${|\operatorname{\operatorname{Graph}_{\operatorname{depth},\dim}}(n)|,}$
combining this with the upper bound for
${|\operatorname{\operatorname{CW}_{\operatorname{depth},\dim}}(n)|}$ in (9)
gives

$\lim_{k\rightarrow\infty}\frac{|\operatorname{\operatorname{CW}_{\operatorname{depth},\dim}}(n)|}{|\operatorname{\operatorname{Graph}_{\operatorname{depth},\dim}}(n)|}\leq\lim_{k\rightarrow\infty}\frac{6k^{2}}{\frac{1}{2}27k^{2}}=\frac{4}{9}.$

Similarly by Theorem 3.2 an upper bound for
$|\operatorname{\operatorname{Graph}_{\operatorname{depth},\dim}}(n)|$ is

$\displaystyle|C^{+}(n)|=$
$\displaystyle\sum_{a=1}^{n-1}(n-a)=n(n-1)-\frac{(n-1)n}{2}=\frac{n}{2}(n-1).$

Now, for $n>5$ let $n=6k+i.$ So, we have

$\displaystyle|C^{+}(n)|=$
$\displaystyle\sum_{a=1}^{6k+i-1}(6k+i-a)=\frac{6k+i}{2}(6k+i-1).$

Thus combining the lower bound of
$|\operatorname{\operatorname{Graph}_{\operatorname{depth},\dim}}(n)|$ with
$|C^{+}(n)|$, we get the lower bound for

$\lim_{k\rightarrow\infty}\frac{|\operatorname{\operatorname{CW}_{\operatorname{depth},\dim}}(n)|}{|\operatorname{\operatorname{Graph}_{\operatorname{depth},\dim}}(n)|}\geq\lim_{k\rightarrow\infty}\frac{6k^{2}}{\frac{1}{2}36k^{2}}=\frac{1}{3}$

∎

##  4\. the Size of
$\operatorname{\operatorname{CW}_{\operatorname{depth},\operatorname{reg},\dim,\operatorname{degh}}}(n)$

In [6] the authors determined the size of the set
$\operatorname{CW_{\operatorname{reg},\operatorname{degh}}}(n)$ , which is the
set of all possible pairs of $(\operatorname{reg}(G),\operatorname{degh}(G))$
arising from all the Cameron–Walker graphs with $n$ vertices. Also, we already
determined the size of the set
$\operatorname{\operatorname{CW}_{\operatorname{depth},\dim}}(n)$. Hibi et al
[5] gave a precise description for the set
$\operatorname{\operatorname{CW}_{\operatorname{depth},\operatorname{reg},\dim,\operatorname{degh}}}(n)$.
Based on their characterization we determine the size of the set
$\operatorname{\operatorname{CW}_{\operatorname{depth},\operatorname{reg},\dim,\operatorname{degh}}}(n)$
through this section.

######  Theorem 4.1.

[5, Theorem 4.4] Let $n\geq 5$ be an integer. Then

$\displaystyle\operatorname{\operatorname{CW}_{\operatorname{depth},\operatorname{reg},\dim,\operatorname{degh}}}(n)=$
$\displaystyle\underbrace{\operatorname{\operatorname{CW}_{2,\operatorname{reg},\dim,\operatorname{degh}}}(n)}_{A}$
$\displaystyle\cup\underbrace{\left\\{(a,d,d,d)\in\mathbb{N}^{4}\
\biggr{\rvert}\ 3\leq a\leq d\leq\left\lfloor\frac{n-1}{2}\right\rfloor,\
n<a+2d\right\\}}_{B}$
$\displaystyle\cup\underbrace{\\{(a,a,d,d)\in\mathbb{N}^{4}\ |\ 3\leq a<d\leq
n-a,\ n\leq 2a+d-1\\}}_{C}$
$\displaystyle\cup\underbrace{\\{(a,r,d,d)\in\mathbb{N}^{4}\ |\ 3\leq
a<r<d<n-r,\ n+2\leq a+r+d\\}}_{D}$

where

(10) $A=\begin{cases}\\{(2,2,n-2,n-2),(2,2,n-3,n-3)\\}&n\quad\mbox{even}\\\
\left\\{(2,2,n-2,n-2),(2,2,n-3,n-3),\left(2,\frac{n-1}{2},\frac{n-1}{2},\frac{n-1}{2}\right)\right\\}&n\quad\mbox{odd}.\end{cases}$

######  Lemma 4.2.

The sets $A,B,C$ and $D$ defined in Theorem 4.1 are pairwise disjoint.

###### Proof.

Since $a\geq 3$ in $B,C$ and $D$, we immediately see that $A\cap B=A\cap
C=A\cap D=\emptyset$. In $C$, we have
$\operatorname{depth}(G)=\operatorname{reg}(G)$. So consider the case
$\operatorname{depth}(G)=\operatorname{reg}(G)$ in $B$. Then we have the tuple
$(a,a,a,a)$. However, $(a,a,a,a)\notin C$ as we have the additional condition
$a<d$. Therefore, $B\cap C=\emptyset$.

As $\operatorname{reg}(G)<\dim(G)$ in $D$, $B\cap D=\emptyset$ and using the
same reasoning $\operatorname{depth}(G)<\operatorname{reg}(G)$ in $D$, finally
we get $C\cap D=\emptyset$. ∎

######  Lemma 4.3 (The size of $A$).

If $A$ is as in the statement of Theorem 4.1 and $n\geq 5$, then

$|A|=\begin{cases}2&n\quad\mbox{even}\quad\mbox{or}\quad n=5\\\
3&n\quad\mbox{odd}.\end{cases}$

###### Proof.

By (10), it is easy to see that for each $n>5$ we have two elements in
$\operatorname{\operatorname{CW}_{2,\operatorname{reg},\dim,\operatorname{degh}}}(n)$
if $n$ is even and we have three elements if $n$ is odd. When $n=5$, we have
the two elements: $(2,2,3,3)$ and $(2,2,2,2)$ in
$\operatorname{\operatorname{CW}_{2,\operatorname{reg},\dim,\operatorname{degh}}}(n)$.
This settles our claim. ∎

######  Lemma 4.4 (The size of $B$).

With $B$ is as in the statement of Theorem 4.1 and $n\geq 5$, let $n=6k+i$
where $k\geq 0$ and $0\leq i\leq 5$. Then

$\displaystyle|B|=\begin{cases}\frac{1}{2}(3k^{2}-3k)&i=0\\\
\frac{1}{2}(3k^{2}+k-2)&i=1\\\ \frac{1}{2}(3k^{2}-k)&i=2\\\
\frac{1}{2}(3k^{2}+3k-2)&i=3\\\ \frac{1}{2}(3k^{2}+k)&i=4\\\
\frac{1}{2}(3k^{2}+5k)&i=5.\end{cases}$

###### Proof.

Recall that $B=\left\\{(a,d,d,d)\in\mathbb{N}^{4}\ \biggr{\rvert}\ 3\leq a\leq
d\leq\left\lfloor\frac{n-1}{2}\right\rfloor,\ n<a+2d\right\\}$. Let $a$ and
$d$ be integers such that

$3\leq
a\quad\mbox{and}\quad\max\\{a-1,\frac{n-a}{2}\\}<d\leq\left\lfloor\frac{n-1}{2}\right\rfloor.$

Then we have two scenarios:

  1. (1)

when $a-1<\frac{n-a}{2}$, or equivalently $a\leq\frac{n}{3}$, we have
$\frac{n-a}{2}<d\leq\left\lfloor\frac{n-1}{2}\right\rfloor$. Thus, there are
$\left\lfloor\frac{n-1}{2}\right\rfloor-\left\lfloor\frac{n-a}{2}\right\rfloor$
possibilities for $d$.

  2. (2)

When $a-1\geq\frac{n-a}{2}$, or equivalently $a>\frac{n}{3}$, the number of
$d$’s satisfying ${(a,d,d,d)\in B}$ is given by ${a\leq
d\leq\left\lfloor\frac{n-1}{2}\right\rfloor}$. In other words the number of
possible $d$’s is $\left\lfloor\frac{n-1}{2}\right\rfloor-a+1$.

$|B|=\sum_{a=3}^{\left\lfloor\frac{n}{3}\right\rfloor}\left(\left\lfloor\frac{n-1}{2}\right\rfloor-\left\lfloor\frac{n-a}{2}\right\rfloor\right)+\sum_{a=\left\lfloor\frac{n}{3}\right\rfloor+1}^{\left\lfloor\frac{n-1}{2}\right\rfloor}\left(\left\lfloor\frac{n-1}{2}\right\rfloor-(a-1)\right)$

We can then simplify this expression using Mathematica [11]. ∎

######  Lemma 4.5 (The size of $C$).

With $C$ is as in the statement of Theorem 4.1, and $n\geq 5$, let $n=6k+i$
where $k\geq 0$ and $0\leq i\leq 5$. Then

(11) $|C|=\begin{cases}3k^{2}-3&i=0\\\ 3k^{2}+k-3&i=1\\\
3k^{2}+2k-2&i=\mbox{2}\\\ 3k^{2}+3k-2&i=3\\\ 3k^{2}+4k-2&i=4\\\
3k^{2}+5k&i=5.\end{cases}$

###### Proof.

Recall that $C=\left\\{(a,a,d,d)\in\mathbb{N}^{4}\ |\ 3\leq a<d\leq n-a,\
n\leq 2a+d-1\right\\}$. Let $a$ and $d$ be integers such that

$a\geq 3\quad\mbox{and}\quad\max\\{a,n-2a+1\\}<d\leq n-a.$

So for a fixed $a\geq 3$ we consider two cases.

  1. (1)

When $a\leq n-2a+1$, or equivalently when
$a\leq\left\lfloor\frac{n+1}{3}\right\rfloor$ we must have ${n-2a+1\leq d\leq
n-a}$. So, the number of possibilities for $d$ is $a$.

  2. (2)

When $a>n-2a+1$, or equivalently when
$a>\left\lfloor\frac{n+1}{3}\right\rfloor$, we have $a<d\leq n-a.$ So, the
possible $d$ in this case is $n-2a$.

$|C|=\sum_{a=3}^{\left\lfloor\frac{n+1}{3}\right\rfloor}a+\sum_{a=\left\lfloor\frac{n+1}{3}\right\rfloor+1}^{\left\lfloor\frac{n-1}{2}\right\rfloor}n-2a$

Mathematica [11] will do the rest of the calculations resulting. ∎

######  Lemma 4.6 (The size of $D$).

If $D$ is as in the statement of Theorem 4.1 and $n=5,$ then $|D|=0.$ When
$n>5$, let $n=6k+i$ where $k>0$ and $0\leq i\leq 5$. Then

$\displaystyle|D|$
$\displaystyle=\begin{cases}\frac{1}{8}(18k^{3}-45k^{2}+34k-8)&i=0\quad\mbox{and}\quad
k\ \text{even}\\\ \frac{1}{8}(18k^{3}-45k^{2}+34k-7)&i=0\quad\mbox{and}\quad
k\ \text{odd}\\\ \frac{1}{8}(18k^{3}-27k^{2}-14k+24)&i=1\quad\mbox{and}\quad
k\ \text{even}\\\ \frac{1}{8}(18k^{3}-27k^{2}-14k+23)&i=1\quad\mbox{and}\quad
k\ \text{odd}\\\ \frac{1}{8}(18k^{3}-27k^{2}+10k)&i=2\quad\mbox{and}\quad k\
\text{even}\\\ \frac{1}{8}(18k^{3}-27k^{2}+10k-1)&i=2\quad\mbox{and}\quad k\
\text{odd}\\\ \frac{1}{8}(18k^{3}-9k^{2}-26k+16)&i=3\quad\mbox{and}\quad k\
\text{even}\\\ \frac{1}{8}(18k^{3}-9k^{2}-26k+17)&i=3\quad\mbox{and}\quad k\
\text{odd}\\\ \frac{1}{8}(18k^{3}-9k^{2}-2k)&i=4\quad\mbox{and}\quad k\
\text{even}\\\ \frac{1}{8}(18k^{3}-9k^{2}-2k+1)&i=4\quad\mbox{and}\quad k\
\text{odd}\\\ \frac{1}{8}(18k^{3}+9k^{2}-26k+8)&i=5\quad\mbox{and}\quad k\
\text{even}\\\ \frac{1}{8}(18k^{3}+9k^{2}-26k+7)&i=5\quad\mbox{and}\quad k\
\text{odd}\\\ \end{cases}$

###### Proof.

Recall that $D=\\{(a,r,d,d)\in\mathbb{N}^{4}\ |\ 3\leq a<r<d<n-r,\ n+2\leq
a+r+d\\}$. Now $r<d<n-r$ implies that $r\leq d-1\leq n-r-2$, and so we must
have $r\leq\frac{n-2}{2}=\frac{n}{2}-1$. So we fix integers $a$ and $r$ such
that

$3\leq a\leq\left\lfloor\frac{n}{2}\right\rfloor-2\quad\mbox{and}\quad a+1\leq
r\leq\left\lfloor\frac{n}{2}\right\rfloor-1.$

Now using both the first and second inequalities in $D$, we can deduce the
following conditions on $d$

$\max\\{r+1,n-a-r+2\\}\leq d<n-r,$

which leads us to consider two scenarios.

  1. (1)

When $r+1\leq n-a-r+2$, or in other words, ${r<\frac{(n-a)}{2}}+1$, then $d$
satisfies the inequality $n-r-a+2\leq d<n-r$. Therefore, we have $(a-2)$ such
integers $d$.

  2. (2)

When $r+1>n-a-r+2$, or equivalently ${r\geq\frac{(n-a)}{2}+1}$, we can find
the number of $d$’s by the inequality $r<d<n-r$. Thus, the number of such $d$
is $(n-2r-1)$.

So the total number of $d$’s for a fixed $a$ and $r$ is

$\displaystyle\sum_{r=a+1}^{\left\lfloor\frac{n-a}{2}\right\rfloor}(a-2)+\sum_{r=\left\lfloor\frac{n-a}{2}\right\rfloor+1}^{\left\lfloor\frac{n}{2}\right\rfloor-1}(n-2r-1)$
$\displaystyle=$
$\displaystyle(a-2)\left(\left\lfloor\frac{n-a}{2}\right\rfloor-a\right)+\left(\left\lfloor\frac{n}{2}\right\rfloor-1-\left\lfloor\frac{n-a}{2}\right\rfloor\right)\left(n-1-\left(\left\lfloor\frac{n}{2}\right\rfloor+\left\lfloor\frac{n-a}{2}\right\rfloor\right)\right).$

Now summing over $a$ we have

$|D|=\sum_{a=3}^{\left\lfloor\frac{n}{2}\right\rfloor-2}\left((a-2)\left(\left\lfloor\frac{n-a}{2}\right\rfloor-a\right)+\left(\left\lfloor\frac{n}{2}\right\rfloor-1-\left\lfloor\frac{n-a}{2}\right\rfloor\right)\left(n-1-\left\lfloor\frac{n}{2}\right\rfloor-\left\lfloor\frac{n-a}{2}\right\rfloor\right)\right).$

The rest of the computation is carried out by Mathematica [11]. ∎

######  Theorem 4.7.

With
$\operatorname{\operatorname{CW}_{\operatorname{depth},\operatorname{reg},\dim,\operatorname{degh}}}(n)$
is as in the statement of Theorem 4.1,
$|\operatorname{\operatorname{CW}_{\operatorname{depth},\operatorname{reg},\dim,\operatorname{degh}}}(n)|=2$
when $n=5$. When $n>5$, let $n=6k+i$ where $k>0$ and $0\leq i\leq 5$. Then

$\displaystyle|\operatorname{\operatorname{CW}_{\operatorname{depth},\operatorname{reg},\dim,\operatorname{degh}}}(n)|$
$\displaystyle=\begin{cases}\begin{cases}\frac{1}{8}(18k^{3}-9k^{2}+22k-16)&k\
\text{is even}\\\ \frac{1}{8}(18k^{3}-9k^{2}+22k-15)&k\ \text{is
odd}\end{cases}&i=0\\\ \begin{cases}\frac{1}{8}(18k^{3}+9k^{2}-2k+16)&k\
\text{is even}\\\ \frac{1}{8}(18k^{3}+9k^{2}-2k+15)&k\ \text{is
odd}\end{cases}&i=1\\\ \begin{cases}\frac{1}{8}(18k^{3}+9k^{2}+22k)&k\
\text{is even}\\\ \frac{1}{8}(18k^{3}+9k^{2}+22k-1)&k\ \text{is
odd}\end{cases}&i=2\\\ \begin{cases}\frac{1}{8}(18k^{3}+27k^{2}+10k+16)&k\
\text{is even}\\\ \frac{1}{8}(18k^{3}+27k^{2}+10k+17)&k\ \text{is
odd}\end{cases}&i=3\\\ \begin{cases}\frac{1}{8}(18k^{3}+27k^{2}+34k)&k\
\text{is even}\\\ \frac{1}{8}(18k^{3}+27k^{2}+34k+1)&k\ \text{is
odd}\end{cases}&i=4\\\ \begin{cases}\frac{1}{8}(18k^{3}+27k^{2}+34k+32)&k\
\text{is even}\\\ \frac{1}{8}(18k^{3}+27k^{2}+34k+31)&k\ \text{is
odd}\end{cases}&i=5\\\ \end{cases}$

###### Proof.

Using Lemmas 4.2,4.3,4.4,4.5,and 4.6. ∎

It is natural to wonder how many Cameron–Walker graphs achieve a given tuple
of integers on their sequence of homological invariants. This question was
raised by Adam Van Tuyl [10], and we are able to partially answer it below.

######  Proposition 4.8.

Let $2\leq a\leq b\leq n-2$, and let $G$ be a Cameron–Walker graph.

  1. (1)

$\operatorname{depth}(G)=\dim(G)=(b,b)$ if and only if $G$ is a Cohen Macaulay
Cameron–Walker graph with exactly $b$ vertices in its bipartite part.

  2. (2)

Suppose $(\operatorname{depth}(G),\dim(G))=(2,b)$.

    1. (a)

$\dim(G)=n-2$ if and only if $m=2$ and $t_{j}=0$ for all $1\leq j\leq p$.

    2. (b)

$\dim(G)=n-3$ if and only if $m=p=1$ and $t_{1}=1$.

    3. (c)

${\dim(G)=\frac{n-1}{2}}$ and $n$ is odd if and only if $m=p=1$, and
$s_{1}=1$, and $t_{1}\geq 2$.

###### Proof.

  1. (1)

$(\Rightarrow)$ is clear by the Cohen–Macaulay property.

$(\Leftarrow)$ By [4, Theorem 1.3], if $G$ is a Cohen–Macaulay Cameron–Walker
graph then $s_{i}=t_{j}=1$ for each $i=1,2,\ldots,m$ and $j=1,2,\ldots,p$.

So, $(\operatorname{depth}(G),\dim(G))=(b,b)\Leftrightarrow$ $G$ is of the
form in the Figure 4 where $G$ is a Cohen Macaulay Cameron–Walker graph with
exactly $b$ vertices in its bipartite part.

$\bullet$$a$$\bullet$$b$$\bullet$$c$$\bullet$$d$$\bullet$$e$$\bullet$$f$$\bullet$$g$$\bullet$$j$$\bullet$$l$$\bullet$$m$$\bullet$$n$$\bullet$$p$$\bullet$$q$$\bullet$$r$$\bullet$$s$$\bullet$$t$
Figure 4. A Cohen Macaulay Cameron–Walker Graph as in Proposition 4.8.

  2. (2)

In this case, $\operatorname{depth}(G)=2$ and $\dim(G)=b$. By [5, Lemma 3.11]
we have three types of Cameron–Walker graphs with the property
$\operatorname{depth}(G)=2$.

    1. (a)

$(\Rightarrow)$ is clear by [5, Proposition 3.13].

$(\Leftarrow)$ Suppose $m=2$ and $t_{j}=0$ for all $1\leq j\leq p$.

By [5, Lemma 3.12], ${\dim(G)=|V(G)|-2.}$ Then we have $b=n-2$. i.e. if
${\dim(G)=n-2}$ then Cameron–Walker graph is of the form $m=2,\ t_{j}=0$ for
all $j$.

    2. (b)

$(\Rightarrow)$ is clear by [5, Proposition 3.13].

$(\Leftarrow)$ Suppose $m=p=1$, $t_{1}=1$. Then by [5, Lemma 3.12],
${\dim(G)=|V(G)|-3.}$ i.e. $b=n-3$. Therefore, if dim $G=n-3$ then
Cameron–Walker graph is of the form $m=p=1,\ t_{1}=1$.

    3. (c)

$(\Rightarrow)$ is clear by [5, Proposition 3.13].

$(\Leftarrow)$ Suppose $m=p=1$, $s_{1}=1$, $t_{1}\geq 2$.

Then by [5, Lemma 3.12], ${\dim(G)=\frac{|V(G)|-1}{2}.}$ i.e.
${b=\frac{n-1}{2}}$.

Thus, dim ${(G)=\frac{n-1}{2}}$ and $n$ is odd.

∎

##  5\. Acknowledgment

We would like to express our very great appreciation to Dr. Adam Van Tuyl for
his valuable suggestions to improve this research work.

## References

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  * [5] T. Hibi, H. Kanno, K. Kimura, K. Matsuda and A. Van Tuyl, Homological invariants of Cameron-Walker graphs, Transactions of the American Mathematical Society 374 (2021), 6559-6582.
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