---
# System prepended metadata

title: 研究所 離散

---

# 研究所 離散
![image](https://hackmd.io/_uploads/rJIXx0_Bkx.png)
![image](https://hackmd.io/_uploads/B1--QFzNyx.png)
## ==可數集不可數集==
![image](https://hackmd.io/_uploads/r1EFbw3dyl.png)
![image](https://hackmd.io/_uploads/B1--QFzNyx.png)
![image](https://hackmd.io/_uploads/Sks7Gvhdye.png)
![image](https://hackmd.io/_uploads/HkNEzw3uJx.png)
![image](https://hackmd.io/_uploads/Bkfy7vnd1l.png)
![image](https://hackmd.io/_uploads/rkr77Pnuyx.png)
![image](https://hackmd.io/_uploads/HySuXP2_kg.png)
![image](https://hackmd.io/_uploads/HkEs7Ph_yg.png)

## ==歸納法用來證明C(m,n)==

![image](https://hackmd.io/_uploads/B1P0jVULke.png)
![image](https://hackmd.io/_uploads/B1--QFzNyx.png)


![image](https://hackmd.io/_uploads/Bys1nEI8yx.png)
![image](https://hackmd.io/_uploads/B1--QFzNyx.png)


![image](https://hackmd.io/_uploads/S1QghNI8ke.png)
![image](https://hackmd.io/_uploads/B1--QFzNyx.png)
## ==close commutative associative==

![image](https://hackmd.io/_uploads/rJS19NFPkg.png)
![image](https://hackmd.io/_uploads/B1--QFzNyx.png)

![image](https://hackmd.io/_uploads/BkwAFEtDyx.png)
![image](https://hackmd.io/_uploads/B1--QFzNyx.png)
## ==單位元素==

![image](https://hackmd.io/_uploads/BJVXqNYPye.png)
![image](https://hackmd.io/_uploads/B1--QFzNyx.png)

## ==生成樹形個數==

![image](https://hackmd.io/_uploads/B1vmzPFByx.png)
![image](https://hackmd.io/_uploads/SJtNMvYS1x.png)
![image](https://hackmd.io/_uploads/B1--QFzNyx.png)

![image](https://hackmd.io/_uploads/Bk_DMPKHJl.png)
![image](https://hackmd.io/_uploads/B1--QFzNyx.png)

## ==內部節點外部節點計算==
:::info
1. n=n0+n1+n2+...
1. n=B+1
2. B=1 * n1+2 * n2 +3 * n3+...

:::
![image](https://hackmd.io/_uploads/Sy117Kf4Jx.png)

![image](https://hackmd.io/_uploads/S1bibV8L1l.png)
![image](https://hackmd.io/_uploads/Sy117Kf4Jx.png)

![image](https://hackmd.io/_uploads/SyN2-NLL1g.png)
![image](https://hackmd.io/_uploads/Sy117Kf4Jx.png)

## ==power set==

![image](https://hackmd.io/_uploads/B1_MsHHEke.png)
![image](https://hackmd.io/_uploads/Sy117Kf4Jx.png)

## ==cross==
![image](https://hackmd.io/_uploads/SkbM2HB4kl.png)

![image](https://hackmd.io/_uploads/Sy117Kf4Jx.png)

## ==組合==

![image](https://hackmd.io/_uploads/r12jo_zVkx.png)
![image](https://hackmd.io/_uploads/Sy117Kf4Jx.png)


![image](https://hackmd.io/_uploads/ByuJpdM41e.png)
![image](https://hackmd.io/_uploads/HkzkmtMNyl.png)

![image](https://hackmd.io/_uploads/rJbpAOz4Jg.png)![image](https://hackmd.io/_uploads/SyVJ7Yz4ye.png)

![image](https://hackmd.io/_uploads/Syy6-tMNke.png)

![image](https://hackmd.io/_uploads/rJ1-mtG41l.png)
![image](https://hackmd.io/_uploads/rJMpzFGEkx.png)

![image](https://hackmd.io/_uploads/B1--QFzNyx.png)
![image](https://hackmd.io/_uploads/r1-g_FMEke.png)

![image](https://hackmd.io/_uploads/B1--QFzNyx.png)
![image](https://hackmd.io/_uploads/rknkF5GNJl.png)

![image](https://hackmd.io/_uploads/By02K5G41l.png)
![image](https://hackmd.io/_uploads/B1--QFzNyx.png)

![image](https://hackmd.io/_uploads/rJHEi9GNJg.png)
![image](https://hackmd.io/_uploads/B1--QFzNyx.png)


![image](https://hackmd.io/_uploads/SkZkh9G4yl.png)
![image](https://hackmd.io/_uploads/B1--QFzNyx.png)

![image](https://hackmd.io/_uploads/HJpcnqMVkg.png)
![image](https://hackmd.io/_uploads/ryJ225f41l.png)

![image](https://hackmd.io/_uploads/B1--QFzNyx.png)
## ==遞迴 格式1 用帶入法==

![image](https://hackmd.io/_uploads/rka2VAmEJl.png)
![image](https://hackmd.io/_uploads/Sy70E074yl.png)
![image](https://hackmd.io/_uploads/B1--QFzNyx.png)


## ==遞迴 格式2 只有an項 求eigen value==
:::info
超好用 
:::
![image](https://hackmd.io/_uploads/HkTiHA74Jx.png)
![image](https://hackmd.io/_uploads/B1--QFzNyx.png)




![image](https://hackmd.io/_uploads/S1xaPC74Jl.png)
![image](https://hackmd.io/_uploads/ryH6ORXE1x.png)
![image](https://hackmd.io/_uploads/B1--QFzNyx.png)

:::danger
eigen value有重根時令法
:::
![image](https://hackmd.io/_uploads/BJVnORm41l.png)
![image](https://hackmd.io/_uploads/B1--QFzNyx.png)

:::danger
eigen value有共惡根時令法
:::
![image](https://hackmd.io/_uploads/S1U55RX4yx.png)
![image](https://hackmd.io/_uploads/B1--QFzNyx.png)

## ==遞迴 格式3 有an項和n項 特解+其次解==

![image](https://hackmd.io/_uploads/SkaCo0QVyl.png)

![image](https://hackmd.io/_uploads/rkwb30mV1l.png)
![image](https://hackmd.io/_uploads/r1LG3RX4yl.png)
![image](https://hackmd.io/_uploads/B1--QFzNyx.png)
![image](https://hackmd.io/_uploads/Syu73CmN1l.png)
![image](https://hackmd.io/_uploads/H1mNhAQ4kl.png)

![image](https://hackmd.io/_uploads/B1--QFzNyx.png)
## ==遞迴 有an項和指數項 格式4==

![image](https://hackmd.io/_uploads/SJ486CmE1x.png)
![image](https://hackmd.io/_uploads/Bys8pA7Vkl.png)
![image](https://hackmd.io/_uploads/ByEva07E1e.png)
![image](https://hackmd.io/_uploads/B1--QFzNyx.png)

## ==遞迴 生成函數法==

![image](https://hackmd.io/_uploads/BJk1YS2NJl.png)
![image](https://hackmd.io/_uploads/B1--QFzNyx.png)
![image](https://hackmd.io/_uploads/ByPZYB3EJe.png)
![image](https://hackmd.io/_uploads/HkUMFH2EJg.png)
![image](https://hackmd.io/_uploads/B1--QFzNyx.png)

![image](https://hackmd.io/_uploads/BJhnFB3VJg.png)
![image](https://hackmd.io/_uploads/B1--QFzNyx.png)

![image](https://hackmd.io/_uploads/rJnptr2Vyl.png)
![image](https://hackmd.io/_uploads/B1--QFzNyx.png)



## ==函數個數==
![image](https://hackmd.io/_uploads/H17izVUUkl.png)
![image](https://hackmd.io/_uploads/B1--QFzNyx.png)


![image](https://hackmd.io/_uploads/HyER-P3V1x.png)
![image](https://hackmd.io/_uploads/B1--QFzNyx.png)

![image](https://hackmd.io/_uploads/BJ-2bw2Nyx.png)
![image](https://hackmd.io/_uploads/B1--QFzNyx.png)

## ==stirling數==

![image](https://hackmd.io/_uploads/BJ6YmD24kl.png)
![image](https://hackmd.io/_uploads/B1--QFzNyx.png)
![image](https://hackmd.io/_uploads/ryn2mP2Vkl.png)
![image](https://hackmd.io/_uploads/B1--QFzNyx.png)
![image](https://hackmd.io/_uploads/SkDTmvnEJe.png)
![image](https://hackmd.io/_uploads/B1--QFzNyx.png)
# ==數論==
## ==inverse==
![image](https://hackmd.io/_uploads/SklLyIYPyl.png)
![image](https://hackmd.io/_uploads/B1--QFzNyx.png)

![image](https://hackmd.io/_uploads/BJ_xJItPJx.png)
![image](https://hackmd.io/_uploads/B1--QFzNyx.png)
## ==費馬小定理==
![image](https://hackmd.io/_uploads/rJeDxLKD1e.png)
![image](https://hackmd.io/_uploads/B1--QFzNyx.png)

## ==各種餘數定理==
![image](https://hackmd.io/_uploads/ByfzXBFw1g.png)
![image](https://hackmd.io/_uploads/B1--QFzNyx.png)

![image](https://hackmd.io/_uploads/SJUezBFPkl.png)
![image](https://hackmd.io/_uploads/B1--QFzNyx.png)

![image](https://hackmd.io/_uploads/BJD1MHFPyg.png)
![image](https://hackmd.io/_uploads/B1--QFzNyx.png)

## ==gcd(a,b)=c ，c能寫成a和b的線性組合==
![image](https://hackmd.io/_uploads/rJfvCrFvkg.png)
![image](https://hackmd.io/_uploads/B1--QFzNyx.png)

![image](https://hackmd.io/_uploads/r1GT8BYvJg.png)
![image](https://hackmd.io/_uploads/B1--QFzNyx.png)

![image](https://hackmd.io/_uploads/HJinQrKDkl.png)
![image](https://hackmd.io/_uploads/B1--QFzNyx.png)

![image](https://hackmd.io/_uploads/SyLwmrYv1e.png)
![image](https://hackmd.io/_uploads/B1--QFzNyx.png)

![image](https://hackmd.io/_uploads/SyBjESYD1x.png)
![image](https://hackmd.io/_uploads/B1--QFzNyx.png)

![image](https://hackmd.io/_uploads/B1tpVrYvkx.png)
![image](https://hackmd.io/_uploads/B1--QFzNyx.png)

## ==各種證明==
![image](https://hackmd.io/_uploads/SkcJxrFvkl.png)
![image](https://hackmd.io/_uploads/B1--QFzNyx.png)

![image](https://hackmd.io/_uploads/rkAvMrYv1l.png)
![image](https://hackmd.io/_uploads/B1--QFzNyx.png)

![image](https://hackmd.io/_uploads/S1JyXrtDye.png)
![image](https://hackmd.io/_uploads/B1--QFzNyx.png)

## ==質數個數有無限個==

![image](https://hackmd.io/_uploads/SyBSJrFv1g.png)
![image](https://hackmd.io/_uploads/B1--QFzNyx.png)



# ==關係==
## ==關係個數==
![image](https://hackmd.io/_uploads/SkyVSK24Je.png)
![image](https://hackmd.io/_uploads/B1--QFzNyx.png)

![image](https://hackmd.io/_uploads/HJDkbv2NJg.png)
![image](https://hackmd.io/_uploads/B1--QFzNyx.png)

![image](https://hackmd.io/_uploads/ryZxZDhE1x.png)
![image](https://hackmd.io/_uploads/B1--QFzNyx.png)

## ==等價關係個數==
![image](https://hackmd.io/_uploads/HJiHieHSkl.png)
![image](https://hackmd.io/_uploads/B1--QFzNyx.png)

:::danger

* P0 1
* P1 1
* P2 2 
* P3 5
* p4 15
* p5 52

:::
![image](https://hackmd.io/_uploads/B1--QFzNyx.png)

## ==paritcal order set 、 total order set==

![image](https://hackmd.io/_uploads/S1-JSNLIJx.png)
![image](https://hackmd.io/_uploads/B1--QFzNyx.png)

![image](https://hackmd.io/_uploads/r1W-HNILye.png)
![image](https://hackmd.io/_uploads/B1--QFzNyx.png)

![image](https://hackmd.io/_uploads/S1IIvgHSke.png)
![image](https://hackmd.io/_uploads/B1--QFzNyx.png)

![image](https://hackmd.io/_uploads/H1cgOlBr1l.png)
![image](https://hackmd.io/_uploads/B1--QFzNyx.png)

![image](https://hackmd.io/_uploads/r12CseBBkl.png)
![image](https://hackmd.io/_uploads/B1--QFzNyx.png)

![image](https://hackmd.io/_uploads/HyZzAxBrJe.png)
![image](https://hackmd.io/_uploads/B1--QFzNyx.png)

## ==maximal 、 minimal 、 greatest 、 least==
:::danger
* maximal 極大
* minimal 極小
* greatest 最大(maximul)
* least 最小(minimul)
:::
![image](https://hackmd.io/_uploads/B1--QFzNyx.png)

![image](https://hackmd.io/_uploads/S1P5xZrSJe.png)
![image](https://hackmd.io/_uploads/B1--QFzNyx.png)


## ==lattice==

![image](https://hackmd.io/_uploads/BykWXbBryg.png)
![image](https://hackmd.io/_uploads/B1--QFzNyx.png)

![image](https://hackmd.io/_uploads/ByUI7WHHJg.png)
![image](https://hackmd.io/_uploads/B1--QFzNyx.png)

## ==bounded lattice==

![image](https://hackmd.io/_uploads/SJrj4bSH1e.png)
![image](https://hackmd.io/_uploads/B1--QFzNyx.png)

# ==圖論==

## ==simple graph 、multiple graph==

![image](https://hackmd.io/_uploads/B11cjbHrJe.png)
![image](https://hackmd.io/_uploads/B1--QFzNyx.png)

![image](https://hackmd.io/_uploads/HyhAsZHSyl.png)
![image](https://hackmd.io/_uploads/B1--QFzNyx.png)


## ==sum of deg(V) = 2E==
![image](https://hackmd.io/_uploads/B14OKfSHkl.png)
![image](https://hackmd.io/_uploads/B1--QFzNyx.png)
![image](https://hackmd.io/_uploads/SyxFFGrBJg.png)
![image](https://hackmd.io/_uploads/B1--QFzNyx.png)


![image](https://hackmd.io/_uploads/SyrXMHYr1l.png)
![image](https://hackmd.io/_uploads/B1--QFzNyx.png)

![image](https://hackmd.io/_uploads/SJFy4HYryg.png)
![image](https://hackmd.io/_uploads/B1--QFzNyx.png)
## ==regular graph==
![image](https://hackmd.io/_uploads/ry8t4SFrJx.png)
![image](https://hackmd.io/_uploads/B1--QFzNyx.png)

## ==subgraph==

![image](https://hackmd.io/_uploads/ry-aErtHJx.png)
![image](https://hackmd.io/_uploads/B1--QFzNyx.png)

![image](https://hackmd.io/_uploads/B1cAEHFBke.png)
![image](https://hackmd.io/_uploads/B1--QFzNyx.png)

![image](https://hackmd.io/_uploads/B1ByBStH1g.png)
![image](https://hackmd.io/_uploads/B1--QFzNyx.png)

## ==incident matrix==
![image](https://hackmd.io/_uploads/ByDEvrYH1g.png)
![image](https://hackmd.io/_uploads/B1--QFzNyx.png)

![image](https://hackmd.io/_uploads/SkHMrBtHJe.png)
![image](https://hackmd.io/_uploads/B1--QFzNyx.png)

![image](https://hackmd.io/_uploads/BymuHBtSyl.png)
![image](https://hackmd.io/_uploads/B1--QFzNyx.png)


## ==path trail circuit circle==

![image](https://hackmd.io/_uploads/rJbiHSFBJx.png)
![image](https://hackmd.io/_uploads/B1--QFzNyx.png)

## ==相鄰矩陣與walk個數的關係==
:::danger
1. 相鄰矩陣 Mii 只有在有 loop 時才等於1
2. walk 是不管有沒有經過重複點或邊
:::
![image](https://hackmd.io/_uploads/B1--QFzNyx.png)

![image](https://hackmd.io/_uploads/Sy_SUBtHJx.png)
![image](https://hackmd.io/_uploads/B1--QFzNyx.png)

![image](https://hackmd.io/_uploads/ryV_LrKBke.png)
![image](https://hackmd.io/_uploads/B1--QFzNyx.png)

![image](https://hackmd.io/_uploads/rJl9LSYrJx.png)
![image](https://hackmd.io/_uploads/B1--QFzNyx.png)

![image](https://hackmd.io/_uploads/rJkCPrtHJx.png)
![image](https://hackmd.io/_uploads/B1--QFzNyx.png)

![image](https://hackmd.io/_uploads/ByDk_BtS1x.png)
![image](https://hackmd.io/_uploads/SkTfuHYSJx.png)
![image](https://hackmd.io/_uploads/B1--QFzNyx.png)


## ==connect==

![image](https://hackmd.io/_uploads/SJyHuBFr1x.png)
![image](https://hackmd.io/_uploads/B1--QFzNyx.png)

![image](https://hackmd.io/_uploads/Syk8_rFryg.png)
![image](https://hackmd.io/_uploads/B1--QFzNyx.png)

![image](https://hackmd.io/_uploads/HkwI_rFH1l.png)
![image](https://hackmd.io/_uploads/B1--QFzNyx.png)

## ==cut point 、cut set、 bridge==
![image](https://hackmd.io/_uploads/BkNOuSKHyx.png)
![image](https://hackmd.io/_uploads/B1--QFzNyx.png)

## ==若有點的degree>=2 => 有循環==

![image](https://hackmd.io/_uploads/BJLsOrFSkl.png)
![image](https://hackmd.io/_uploads/B1--QFzNyx.png)



## ==isomorphic==

![image](https://hackmd.io/_uploads/SkbShHKHye.png)
![image](https://hackmd.io/_uploads/rJ8v2SFSye.png)

![image](https://hackmd.io/_uploads/B1--QFzNyx.png)

![image](https://hackmd.io/_uploads/BJnooSKryx.png)
![image](https://hackmd.io/_uploads/B1--QFzNyx.png)

![image](https://hackmd.io/_uploads/S1T3srtryl.png)
![image](https://hackmd.io/_uploads/B1--QFzNyx.png)

![image](https://hackmd.io/_uploads/SJD6oSYrkg.png)
![image](https://hackmd.io/_uploads/B1--QFzNyx.png)

![image](https://hackmd.io/_uploads/rkZ0sBKB1g.png)
![image](https://hackmd.io/_uploads/B1--QFzNyx.png)

![image](https://hackmd.io/_uploads/HJif3HKSkg.png)
![image](https://hackmd.io/_uploads/B1--QFzNyx.png)

![image](https://hackmd.io/_uploads/ByYm2rYSJl.png)
![image](https://hackmd.io/_uploads/B1--QFzNyx.png)

## ==self complement==
![image](https://hackmd.io/_uploads/r1xY2HFr1e.png)
![image](https://hackmd.io/_uploads/B1--QFzNyx.png)






# ==常見的圖==
## ==Pn==
![image](https://hackmd.io/_uploads/BJjeVLFS1x.png)
![image](https://hackmd.io/_uploads/B1--QFzNyx.png)
## ==Gmn==
![image](https://hackmd.io/_uploads/HkJfNIYS1x.png)
![image](https://hackmd.io/_uploads/B1--QFzNyx.png)

## ==complete graph => Kn==

![image](https://hackmd.io/_uploads/H1WgYrYSyx.png)
![image](https://hackmd.io/_uploads/B1--QFzNyx.png)
 
![image](https://hackmd.io/_uploads/HJapFrKHJx.png)
![image](https://hackmd.io/_uploads/B1--QFzNyx.png)

## ==用Kn判斷是否為聯通圖==
![image](https://hackmd.io/_uploads/ByIXcrtSJl.png)
![image](https://hackmd.io/_uploads/B1--QFzNyx.png)

![image](https://hackmd.io/_uploads/rk8V9BtHkl.png)
![image](https://hackmd.io/_uploads/B1--QFzNyx.png)

![image](https://hackmd.io/_uploads/Syj8qSYrkx.png)
![image](https://hackmd.io/_uploads/B1--QFzNyx.png)

## ==hypercube => Qn==

![image](https://hackmd.io/_uploads/S1qs9SYSyg.png)
![image](https://hackmd.io/_uploads/B1--QFzNyx.png)

![image](https://hackmd.io/_uploads/Bksp9BYB1g.png)
![image](https://hackmd.io/_uploads/B1--QFzNyx.png)

![image](https://hackmd.io/_uploads/r16AqSFrke.png)
![image](https://hackmd.io/_uploads/B1--QFzNyx.png)

## ==Gray code==

![image](https://hackmd.io/_uploads/r1QeoStHkl.png)
![image](https://hackmd.io/_uploads/B1--QFzNyx.png)



## ==cycal graph  => Cn==

![image](https://hackmd.io/_uploads/ryT7iHYrkl.png)
![image](https://hackmd.io/_uploads/B1--QFzNyx.png)

![image](https://hackmd.io/_uploads/H1mEoHKByl.png)
![image](https://hackmd.io/_uploads/B1--QFzNyx.png)

## ==whell graph  => Wn==

![image](https://hackmd.io/_uploads/S1oBjBKBJe.png)
![image](https://hackmd.io/_uploads/B1--QFzNyx.png)

![image](https://hackmd.io/_uploads/HytwsSKS1l.png)
![image](https://hackmd.io/_uploads/B1--QFzNyx.png)




## ==bipratite graph==
:::danger
* 用 0 1分兩邊
* 具最大邊數 => Kmn
* 不含有奇數長之cycle 
:::
![image](https://hackmd.io/_uploads/B1WNHUtrJx.png)
![image](https://hackmd.io/_uploads/B1--QFzNyx.png)

![image](https://hackmd.io/_uploads/rk0StLKH1e.png)
![image](https://hackmd.io/_uploads/B1--QFzNyx.png)

![image](https://hackmd.io/_uploads/rkb8HLKS1e.png)
![image](https://hackmd.io/_uploads/B1--QFzNyx.png)

![image](https://hackmd.io/_uploads/B10aSLYryg.png)
![image](https://hackmd.io/_uploads/B1--QFzNyx.png)


![image](https://hackmd.io/_uploads/BJWpKIYr1l.png)
![image](https://hackmd.io/_uploads/BJFBwIFBkl.png)



# ==圖形判斷==
## ==用matrix算有多少spanning tree==
![image](https://hackmd.io/_uploads/HJEDcRuv1g.png)
![image](https://hackmd.io/_uploads/BJFBwIFBkl.png)
![image](https://hackmd.io/_uploads/SJ8t9Cdvkg.png)
![image](https://hackmd.io/_uploads/BJFBwIFBkl.png)

## ==Planner==
:::danger
K5，K33或以上都不是Planner
:::
![image](https://hackmd.io/_uploads/BkmBsQLI1g.png)
![image](https://hackmd.io/_uploads/B1--QFzNyx.png)

![image](https://hackmd.io/_uploads/HJT-IUKByx.png)
![image](https://hackmd.io/_uploads/B1--QFzNyx.png)


![image](https://hackmd.io/_uploads/Sy4c8UYBkg.png)
![image](https://hackmd.io/_uploads/B1--QFzNyx.png)

![image](https://hackmd.io/_uploads/HJM3ULtryg.png)
![image](https://hackmd.io/_uploads/B1--QFzNyx.png)

![image](https://hackmd.io/_uploads/S1C0LUYB1g.png)
![image](https://hackmd.io/_uploads/B1--QFzNyx.png)

![image](https://hackmd.io/_uploads/S1cGDUYSye.png)
![image](https://hackmd.io/_uploads/B1--QFzNyx.png)

![image](https://hackmd.io/_uploads/HkLNwLKHJx.png)
![image](https://hackmd.io/_uploads/B1--QFzNyx.png)

![image](https://hackmd.io/_uploads/BJFBwIFBkl.png)
![image](https://hackmd.io/_uploads/B1--QFzNyx.png)

## ==proper coloring==
![image](https://hackmd.io/_uploads/H1uL_LtHJl.png)
![image](https://hackmd.io/_uploads/B1--QFzNyx.png)

![image](https://hackmd.io/_uploads/ryFsO8trkl.png)
![image](https://hackmd.io/_uploads/B1--QFzNyx.png)

![image](https://hackmd.io/_uploads/SkXKOLYr1x.png)
![image](https://hackmd.io/_uploads/B1--QFzNyx.png)

![image](https://hackmd.io/_uploads/ryX2dIKSye.png)
![image](https://hackmd.io/_uploads/B1--QFzNyx.png)

![image](https://hackmd.io/_uploads/H18pYItBke.png)
![image](https://hackmd.io/_uploads/B1--QFzNyx.png)

## ==chromatic polynomial==
:::danger
最小著色數就是最小的 人 使得 chromatic polynomial >0
:::
![image](https://hackmd.io/_uploads/S1lB5UKrJx.png)
![image](https://hackmd.io/_uploads/B1--QFzNyx.png)

![image](https://hackmd.io/_uploads/Sybqi8YH1x.png)
![image](https://hackmd.io/_uploads/B1--QFzNyx.png)

![image](https://hackmd.io/_uploads/SJoW38Yrke.png)
![image](https://hackmd.io/_uploads/B1--QFzNyx.png)

![image](https://hackmd.io/_uploads/ryF-T8FSyx.png)
![image](https://hackmd.io/_uploads/B1--QFzNyx.png)

![image](https://hackmd.io/_uploads/HJrla8KS1l.png)
![image](https://hackmd.io/_uploads/B1--QFzNyx.png)


## ==用Kn判斷是否為聯通圖==
![image](https://hackmd.io/_uploads/BkwmqHYr1e.png)
![image](https://hackmd.io/_uploads/B1--QFzNyx.png)

![image](https://hackmd.io/_uploads/rkdNqSYHye.png)
![image](https://hackmd.io/_uploads/B1--QFzNyx.png)

![image](https://hackmd.io/_uploads/BkvIcBFHkl.png)
![image](https://hackmd.io/_uploads/B1--QFzNyx.png)


## ==euler circuit、hamilton cycle==


![image](https://hackmd.io/_uploads/r1h7CBKBkl.png)
![image](https://hackmd.io/_uploads/B1--QFzNyx.png)


## ==判斷euler circuit==

![image](https://hackmd.io/_uploads/SyDUCHYrkx.png)
![image](https://hackmd.io/_uploads/B1--QFzNyx.png)

![image](https://hackmd.io/_uploads/SJRF0rKS1e.png)
![image](https://hackmd.io/_uploads/B1--QFzNyx.png)
## ==判斷hamilton cycle==


![image](https://hackmd.io/_uploads/HkhT-ItBJl.png)
![image](https://hackmd.io/_uploads/B1--QFzNyx.png)

![image](https://hackmd.io/_uploads/S1D1fLFrJl.png)
![image](https://hackmd.io/_uploads/B1--QFzNyx.png)

![image](https://hackmd.io/_uploads/ry5lMIKHJg.png)
![image](https://hackmd.io/_uploads/B1--QFzNyx.png)

![image](https://hackmd.io/_uploads/SkkMzItSJl.png)
![image](https://hackmd.io/_uploads/B1--QFzNyx.png)

![image](https://hackmd.io/_uploads/ryFzM8Yrye.png)
![image](https://hackmd.io/_uploads/B1--QFzNyx.png)

## ==hamilton cycle整理==

![image](https://hackmd.io/_uploads/ByQYMUtHkx.png)
![image](https://hackmd.io/_uploads/H1eqG8KByg.png)
![image](https://hackmd.io/_uploads/B1--QFzNyx.png)

![image](https://hackmd.io/_uploads/r15DX8tHJx.png)
![image](https://hackmd.io/_uploads/B1--QFzNyx.png)

![image](https://hackmd.io/_uploads/HyROmItSke.png)
![image](https://hackmd.io/_uploads/Sy75XIFS1l.png)
![image](https://hackmd.io/_uploads/B1--QFzNyx.png)

![image](https://hackmd.io/_uploads/rkQimUYr1g.png)
![image](https://hackmd.io/_uploads/B1--QFzNyx.png)


# ==tree==
:::info
定義跟DS不一樣
離散的
:::
![image](https://hackmd.io/_uploads/SJhxTtQvyg.png)
![image](https://hackmd.io/_uploads/B1--QFzNyx.png)

:::info
定義跟DS不一樣
DS的
:::
![image](https://hackmd.io/_uploads/Syz86tXwJg.png)
![image](https://hackmd.io/_uploads/B1--QFzNyx.png)

![image](https://hackmd.io/_uploads/rk43JDFSke.png)
![image](https://hackmd.io/_uploads/B1--QFzNyx.png)


![image](https://hackmd.io/_uploads/SJ1t-wKHJg.png)
![image](https://hackmd.io/_uploads/B1--QFzNyx.png)

![image](https://hackmd.io/_uploads/rkuobDtr1g.png)
![image](https://hackmd.io/_uploads/B1--QFzNyx.png)

## ==深度與externel的關系==
![image](https://hackmd.io/_uploads/By8lbDtrye.png)
![image](https://hackmd.io/_uploads/ByI-bwYSkl.png)
![image](https://hackmd.io/_uploads/B1--QFzNyx.png)

