孫玉霜,耿顯亞

(安徽理工大學(xué) 數(shù)學(xué)與大數(shù)據(jù)學(xué)院,安徽 淮南 232001)

本文所考慮的都是有限簡(jiǎn)單圖,設(shè)G=(V(G),E(G))是一個(gè)圖,V(G)表示圖的頂點(diǎn)集,E(G)表示圖的邊集,對(duì)于一個(gè)頂點(diǎn)v∈V,圖G-u是由V-{u}給定的,對(duì)于一條邊e,圖G-e是由一個(gè)圖G刪去一條邊e給定的[1-8],另外N(v)={u|uv∈E}表示在G中與v有邊相連的點(diǎn),NG[v]={v}∪N(v)表示與v有邊連接的點(diǎn)且點(diǎn)v包括在內(nèi)[3,11,15].

在后續(xù)計(jì)算中,我們會(huì)用到如下公式:

1) Gutman and Polansky[9],如果uv是G中的一條邊,則

m(G)=m(G-uv)+m(G-{u,v})

(1)

2) Gutman and Polansky[9],如果v是G中的一個(gè)頂點(diǎn),則

i(G)=i(G-v)+i(G-NG[v])

(2)

3) Gutman and Polansky[9],如果G是由G1,G2,…,Gk組成,則

(3)

4)m(p2)=2,m(p3)=3,m(p4)=5,m(p5)=8,m(p6)=13,m(C7)=29.

5)i(p1)=2,i(p2)=3,i(p3)=5,i(p4)=8,i(p5)=13,i(P6)=21.

其中:Pn表示n個(gè)頂點(diǎn)的路,Cn表示n個(gè)頂點(diǎn)的圈.

Gn可以視為具有n個(gè)七邊形的鏈,也可以表示為末端有一個(gè)七邊形的,如圖1所示.

圖1 有個(gè)七邊形的鏈Figure 1 A chain with a hexagon

圖2 七邊形鏈的三種排列方式Figure 2 Three arrangements of heptagonal chain

一個(gè)有n個(gè)七邊形的隨機(jī)鏈Gn(p1,p2,p3)在之后的每一步k=(3,4,…,n)中都會(huì)有三種隨機(jī)排列:

其中:p1,p2,p3是常數(shù),與k無(wú)關(guān).特別的,當(dāng)p1=1,p2=1,p3=1時(shí),圖Gn分別可記為鄰鏈Q(jìng)n,元鏈Mn,對(duì)鏈Ln.

1 在隨機(jī)七邊形鏈中Hosoya指數(shù)的期望值探究

圖3 末端連接的三類(lèi)連接方式Figure 3 Three types of end connection

m(Gn)=m(Gn-e)+m(Gn-{u,v})=m(C7)m(Gn-1)+m(P6)m(An-2)=29m(Gn-1)+13m(An-2)

m(An-2)=m(P6)m(Gn-2)+m(P5)m(An-3)=13m(Gn-2)+8m(An-3)

m(An-2)=m(P6)m(Gn-2)+m(P5)m(Bn-3)=13m(Gn-2)+8m(Bn-3)

m(An-2)=m(P6)m(Gn-2)+m(P5)m(Cn-3)=13m(Gn-2)+8m(Cn-3)

m(Gn)=m(C7)m(Gn-1)+m(P6)m(Bn-2)=

29m(Gn-1)+13m(Bn-2)

m(Bn-2)=m(p6)m(Gn-2)+m(p1)m(p4)m(An-3)=

13m(Gn-2)+5m(An-3)

m(Bn-2)=m(p6)m(Gn-2)+m(p1)m(p4)m(Bn-3)=

13m(Gn-2)+5m(Bn-3)

m(Bn-2)=m(p6)m(Gn-2)+m(p1)m(p4)m(Cn-3)=

13m(Gn-2)+5m(Cn-3)

m(Gn)=m(C7)m(Gn-1)+m(p6)m(Cn-2)=

29m(Gn-1)+13m(Cn-2)

m(Cn-2)=m(p6)m(Gn-2)+m(p2)m(p3)m(An-3)=

13m(Gn-2)+6m(An-3)

m(Cn-2)=m(p6)m(Gn-2)+m(p2)m(p3)m(Bn-3)=

13m(Gn-2)+6m(Bn-3)

m(Cn-2)=m(p6)m(Gn-2)+m(p2)m(p3)m(Cn-3)=

13m(Gn-2)+6m(Cn-3)

根據(jù)式(1)～(3)且p1+p2+p3=1可以得到期望:

E(m(Gn))=29E(m(Gn-1))+13p1E(m(An-2))+

13p2E(m(Bn-2))+13p3E(m(Cn-2))=

29E(m(Gn-1))+169E(m(Gn-2))+

顯然有

8p1E(m(An-4))+8p2E(m(Bn-4))+

8p3E(m(Cn-4))

5p1E(m(An-4))+5p2E(m(Bn-4))+

5p3E(m(Cn-4))

6p1E(m(An-4))+6p2E(m(Bn-4))+

6p3E(m(Cn-4))

根據(jù)上式,可分別得到

8p1[E(m(Gn-1))-29E(m(Gn-2))-

169E(m(Gn-3))]

5p2[E(m(Gn-1))-29E(m(Gn-2))-

169E(M(Gn-3))]

6p3[E(m(Gn-1))-29E(m(Gn-2))-

169E(m(Gn-3))]

綜上所述且根據(jù)p1+p2+p3=1即得關(guān)于Hosoya指數(shù)期望值的遞推公式:

E(m(Gn))=(29+8p1+5p2+6p3)E(m(Gn-1))+

(169-232p1-145p2-174p3)E(m(Gn-2))=

(2p1-p2+35)E(m(Gn-1))-

(58p1-29p2-5)E(m(Gn-2))

另外,期望值存在兩個(gè)極限:

E(m(G1)) =m(C7)=29E(m(G2))=1 010

利用上述遞推關(guān)系和邊界條件,我們可以得到:

定理2.1 在隨機(jī)七邊形鏈中Hosoya指數(shù)的期望值

E(m(Gn))=

分別令p1=1,p2=1,p3=1我們可以從定理2.1中得到On,Mn,Ln的Hosoya指數(shù)的推論.

推論2.2

2 在隨機(jī)七邊形鏈中Merrifield-Simmons指數(shù)的期望值探究

i(Gn)=i(Gn-v)+i(Gn-NG[v])=

i(P6)i(Gn-1)+i(P4)i(An-2)=

21i(Gn-1)+8i(An-2)

i(An-2)=i(P5)i(Gn-2)+i(P4)i(An-3)=

13i(Gn-2)+8i(An-3)

i(An-2)=i(P5)i(Gn-2)+i(P4)i(Bn-3)=

13i(Gn-2)+8i(Bn-3)

i(An-2)=i(P5)i(Gn-2)+i(P4)i(Cn-3)=

13i(Gn-2)+8i(Cn-3)

i(Gn)=i(P6)i(Gn-1)+i(P4)i(Bn-2)=

21i(Gn-1)+8i(Bn-2)

i(Bn-2)=i(p1)i(p4)i(Gn-2)+i(p3)i(An-3)=

16i(Gn-2)+5i(An-3)

i(Bn-2)=i(p1)i(p4)i(Gn-2)+i(p3)i(Bn-3)=

16i(Gn-2)+5i(Bn-3)

i(Bn-2)=i(p1)i(p4)i(Gn-2)+i(p3)i(Cn-3)=

16i(Gn-2)+5i(Cn-3)

i(Gn)=i(P6)i(Gn-1)+i(P4)i(Cn-2)=

21i(Gn-1)+8i(Cn-2)

i(Cn-2)=i(P2)i(P3)i(Gn-2)+

i(P1)i(P2)i(An-3)=15i(Gn-2)+6i(An-3)

i(Cn-2)=i(P2)i(P3)i(Gn-2)+

i(P1)i(P2)i(Bn-3)=15i(Gn-2)+6i(Bn-3)

i(Cn-2)=i(P2)i(P3)i(Gn-2)+

i(P1)i(P2)i(Cn-3)=15i(Gn-2)+6i(Vn-3)

根據(jù)式(1)～(3)且p1+p2+p3=1可以得到期望:

顯然有

E(i(Gn))=21E(i(Gn-1))+(104p1+128p2+

48p2p3)E(i(Bn-3))+(64p1p3+40p2p3+

8p2E(i(Bn-4))+8p3E(i(Cn-4))

5p2E(i(Bn-4))+5p3E(i(Cn-4))

6p2E(i(Bn-4))+6p3E(i(Cn-4))

根據(jù)上式,可分別得到

8p1[E(i(Gn-1))-21E(i(Gn-2))-

(104p1+128p2+120p3)E(i(Gn-3))]

5p2[E(i(Gn-1))-21E(i(Gn-2))-

(104p1+128p2+120p3)E(i(Gn-3))]

6p3[E(i(Gn-1))-21E(i(Gn-2))-

(104p1+128p2+120p3)E(i(Gn-3))]

綜上所述且根據(jù)p3=1-p1-p2可以得到關(guān)于Merrifield-Simmons指數(shù)期望值的遞推公式:

E(i(Gn))=(21+8p1+5p2+6p3)E(i(Gn-1))+

(-64p1+23p2-6p3)E(i(Gn-2))=

(2p1-p2+27)E(i(Gn-1))-

(58p1-29p2+6)E(i(Gn-2))

E(i(G1))=i(C4)=7E(i(G2))=777

利用上述遞推關(guān)系和邊界條件,可以得到:

定理3.1 在隨機(jī)七邊形鏈中Merrifield-Simmons指數(shù)的期望值

E(i(Gn))=

分別令p1=1,p2=1,p3=1,我們可以從定理3.1中得到On,Mn,Ln的Merrifield-Simmons指數(shù)的推論:

推論3.2

3 結(jié) 語(yǔ)

本文得到了含有n個(gè)七邊形的隨機(jī)鏈的Hosoya指數(shù)Merrifield-Simmons指數(shù)的期望值的具體推導(dǎo)解析公式,并分別討論了m(Gn)和i(Gn)的期望值,它們的組成和結(jié)構(gòu)也正在向圖論的相關(guān)研究方向發(fā)展.