﻿ 三圈图的最小离心距离指标
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 北京化工大学学报(自然科学版)  2018, Vol. 45 Issue (1): 109-114  DOI: 10.13543/j.bhxbzr.2018.01.018 0

### 引用本文

FEI JunQi, TU JianHua. Tricyclic graphs with a minimum eccentric distance sum[J]. Journal of Beijing University of Chemical Technology (Natural Science), 2018, 45(1): 109-114. DOI: 10.13543/j.bhxbzr.2018.01.018.

### 文章历史

Tricyclic graphs with a minimum eccentric distance sum
FEI JunQi , TU JianHua
Faculty of Science, Beijing University of Chemical Technology, Beijing 100029, China
Abstract: The eccentric distance sum (EDS) has been widely used in the fields of chemistry and biology. In order to study its mathematical properties further, a new method has been used in this paper. Firstly, three transformations of edge-grafting on the eccentricity distance sum of a connected graph were studied. Then by using these transformations and induction, we obtained the minimum eccentric distance sum among all tricyclic graphs with order n, and also characterized the structure of the corresponding tricyclic graphs. The results demonstrate that our method is simple and effective, and offers a new way of studying the eccentric distance sum of general graphs.
Key words: graph invariant    eccentric distance sum    tricyclic graphs

1 特定移边变换对ξd(G)的影响

 ${\xi ^d}\left( G \right) = \sum\limits_{\left\{ {u,v} \right\} \subset V\left( G \right)} {\left( {{\varepsilon _G}\left( u \right) + {\varepsilon _G}\left( v \right)} \right){d_G}\left( {u,v} \right)}$

Li等[12]在研究离心阻尼距离指标(ERDS)的数学性质时刻画了3种移边变换，具体内容如下。

 图 1 3种移边变换 Fig.1 Three edge-grafting transformations
 $G' = G - \left\{ {v{v_1},v{v_2}, \cdots ,v{v_p}} \right\} + \left\{ {u{v_1},u{v_2}, \cdots ,u{v_p}} \right\}$

 $G' = G - \left\{ {v{v_1},v{v_2}, \cdots ,v{v_q}} \right\} + \left\{ {u{v_1},u{v_2}, \cdots ,u{v_q}} \right\}$
 $G'' = G - \left\{ {u{u_1},u{u_2}, \cdots ,u{u_p}} \right\} + \left\{ {v{u_1},v{u_2}, \cdots ,v{u_p}} \right\}$

 $G - e = {G_1} \cup {G_2},u \in {V_{{G_1}}},v \in {V_{{G_2}}}$

 $\left\{ \begin{array}{l} V\left( G \right) = V\left( {G'} \right),\varepsilon \left( v \right) = \varepsilon '\left( v \right)\\ \forall x \in V\left( H \right),\varepsilon \left( x \right) \ge \varepsilon '\left( x \right)\\ \forall y \in W,\varepsilon \left( y \right) = \varepsilon '\left( y \right) + 1 \end{array} \right.$ (1)
 $\left\{ \begin{array}{l} \forall x,y \in V\left( H \right),d\left( {x,y} \right) = d'\left( {x,y} \right)\\ \forall x,y \in W,d\left( {x,y} \right) = d'\left( {x,y} \right) \end{array} \right.$ (2)

 $\begin{array}{l} {\xi _1}: = \left( {\sum\limits_{\left\{ {x,y} \right\} \subseteq {V_H}} + \sum\limits_{\left\{ {x,y} \right\} \subseteq W} {} } \right)\left[ {\left( {\varepsilon \left( x \right) + \varepsilon \left( y \right)} \right)} \right.\\ \left. {d\left( {x,y} \right) - \left( {\varepsilon '\left( x \right) + \varepsilon '\left( y \right)} \right)d'\left( {x,y} \right)} \right] + \left[ {\sum\limits_{x \in {V_H}\backslash \left\{ u \right\},y \in W} + } \right.\\ \left. {\sum\limits_{x \in {V_H},y = v} {} } \right)\left[ {\left( {\varepsilon \left( x \right) + \varepsilon \left( y \right)} \right)d\left( {x,y} \right) - \left( {\varepsilon '\left( x \right) + } \right.} \right.\\ \left. {\left. {\varepsilon '\left( y \right)} \right)d'\left( {x,y} \right)} \right] > 0 \end{array}$
 $\begin{array}{l} {\xi _2}: = \sum\limits_{x \in W,y = u} {\left( {\varepsilon \left( x \right) + \varepsilon \left( y \right)} \right)d\left( {x,y} \right)} - \\ \sum\limits_{x \in W,y = u} {\left( {\varepsilon '\left( x \right) + \varepsilon '\left( y \right)} \right)d'\left( {x,y} \right)} + \sum\limits_{x \in W,y = v} {\left( {\varepsilon \left( x \right) + } \right.} \\ \left. {\varepsilon \left( y \right)} \right)d\left( {x,y} \right) - \sum\limits_{x \in W,y = v} {\left( {\varepsilon '\left( x \right) + \varepsilon '\left( y \right)} \right)d'\left( {x,y} \right)} = \\ \left| W \right|\left[ {2\left( {\varepsilon \left( u \right) + \varepsilon \left( {{v_1}} \right)} \right) - \left( {\varepsilon '\left( u \right) + \varepsilon '\left( {{v_1}} \right)} \right)} \right] + \left| W \right|\\ \left[ {\left( {\varepsilon \left( v \right) + \varepsilon \left( {{v_1}} \right)} \right) - 2\left( {\varepsilon '\left( v \right) + \varepsilon '\left( {{v_1}} \right)} \right)} \right] \ge 3\left| W \right|\left( \varepsilon \right.\\ \left. {\left( {{v_1}} \right) + \varepsilon '\left( {{v_1}} \right)} \right) + \left| W \right|\left( {2\varepsilon \left( u \right) - 2\varepsilon '\left( v \right) + \varepsilon \left( v \right) - } \right.\\ \left. {\varepsilon '\left( u \right)} \right) \ge 3\left| W \right| + \left| W \right|\left( {\varepsilon \left( u \right) - \varepsilon '\left( v \right)} \right) = \left| W \right|\left( {\varepsilon \left( u \right) - } \right.\\ \left. {\varepsilon \left( v \right) + 3} \right) \end{array}$

 $\varepsilon \left( u \right) - \varepsilon \left( v \right) + 3 > 0$

 $\varepsilon \left( u \right) - \varepsilon \left( v \right) + 3 > 0$

 ${\xi ^d}\left( G \right) - {\xi ^d}\left( {G'} \right) = {\xi _1} + {\xi _2} > 0$

2 三圈图的最小离心距离指标

1) T1(n, r, s, t)={G}，其中rst分别表示图$\hat G$中的3个圈CrCsCtrst，且CrCsCt彼此之间在图$\hat G$中至多有1个公共顶点；

2) T2(n, x, y, z, w)={G}，其中x, y, z分别表示$\hat G$中连通顶点pq的3条内部不交的路PxPyPzw表示$\hat G$中连通顶点gh的路Pwwxyz，且$\left\{ {g, h} \right\} \subset \bigcup\limits_{i = x, y, z} {V\left( {{P_i}} \right)}$

3) T3(n, x, y, z, w)={G}，其中x, y, z分别表示$\hat G$中连通顶点pq的3条内部不交的路PxPyPz(xyz)，w表示$\hat G$中悬挂于顶点g的圈Cw(w≥3)，且$g \in \bigcup\limits_{i = x, y, z} {V\left( {{P_i}} \right)}, V\left( {{C_w}-g} \right) \cap \bigcup\limits_{i = x, y, z} {V\left( {{P_i}} \right)} = \emptyset$

 图 2 n=5, 6, 7时的某些特殊三圈图 Fig.2 Some special tricyclic graphs with order n=5, 6, 7
 $G \in \left\{ {{T^{1\left( 1 \right)}},{T^{1\left( 2 \right)}}} \right\}$

 $G \in \left\{ {{T^{2\left( 1 \right)}},{T^{2\left( 2 \right)}},{T^{2\left( 3 \right)}}} \right\}$

 $G \in \left\{ {{T^{3\left( 1 \right)}},{T^{3\left( 2 \right)}}} \right\}$

 $\left\{ \begin{array}{l} {\xi ^d}\left( {{T^{1\left( 1 \right)}}} \right) = 126,{\xi ^d}\left( {{T^{1\left( 2 \right)}}} \right) = 211\\ {\xi ^d}\left( {{T^{2\left( 1 \right)}}} \right) = 48,{\xi ^d}\left( {{T^{2\left( 2 \right)}}} \right) = 48,{\xi ^d}\left( {{T^{2\left( 3 \right)}}} \right) = 48\\ {\xi ^d}\left( {{T^{3\left( 1 \right)}}} \right) = 83,{\xi ^d}\left( {{T^{3\left( 2 \right)}}} \right) = 124 \end{array} \right.$

 图 3 具有最小ξd(G)值的三圈图 Fig.3 The tricyclic graphs with minimum ξd(G)

 $\left\{ \begin{array}{l} \varepsilon \left( w \right) \ge 2,\varepsilon \left( {w'} \right) \ge 1,d\left( {w,w'} \right) = 1\\ \varepsilon \left( a \right) \ge 2,d\left( {a,w} \right) \ge 2,\forall a \in V\left( G \right)\backslash \left\{ {w,w'} \right\} \end{array} \right.$ (3)

G′=GwG′仍是三圈图，由分析可知

 $\left\{ \begin{array}{l} \varepsilon \left( a \right) \ge \varepsilon '\left( a \right),\varepsilon \left( b \right) \ge \varepsilon '\left( b \right),\forall a,b \in V\left( {G'} \right)\\ d\left( {a,b} \right) \ge d'\left( {a,b} \right),\forall a,b \in V\left( {G'} \right)\\ {\xi ^d}\left( {G'} \right) \ge 4{\left( {n - 1} \right)^2} - 9\left( {n - 1} \right) - 7 = \\ \;\;\;\;\;\;\;\;\;\;4{n^2} - 17n + 6 \end{array} \right.$ (4)

 $\begin{array}{l} {\xi ^d}\left( G \right) - {\xi ^d}\left( {G'} \right) = \sum\limits_{a,b \in V\left( {G'} \right)} {\left[ {\left( {\varepsilon \left( a \right) + \varepsilon \left( b \right)} \right)} \right.} \\ \left. {d\left( {a,b} \right) - \left( {\varepsilon '\left( a \right) + \varepsilon '\left( b \right)} \right)d'\left( {a,b} \right)} \right] + \left( {\varepsilon \left( {w'} \right) + } \right.\\ \left. {\varepsilon \left( w \right)} \right)d\left( {w',w} \right) + \sum\limits_{a \in V\left( {G'} \right)\backslash \left\{ {w'} \right\}} {\left( {\varepsilon \left( a \right) + \varepsilon \left( w \right)} \right)d\left( {a,} \right.} \\ \left. w \right) \ge 0 + \left( {1 + 2} \right) \times 1 + \left( {n - 2} \right) \times \left( {2 + 2} \right) \times 2 = \\ 8n - 13 \end{array}$

 $\left\{ \begin{array}{l} \varepsilon \left( a \right) \ge 2,\forall a \in V\left( G \right)\\ d\left( {w,w'} \right) = 1,d\left( {w,w''} \right) = 1,d\left( {w',w''} \right) = 2\\ d\left( {a,w} \right) \ge 2,\forall a \in V\left( G \right)\backslash \left\{ {w,w',w''} \right\} \end{array} \right.$ (5)

G′=Gw+ww″，G′仍是三圈图，由分析可得

 $\left\{ \begin{array}{l} \varepsilon \left( a \right) \ge \varepsilon '\left( a \right),\varepsilon \left( b \right) \ge \varepsilon '\left( b \right),\forall a,b \in V\left( {G'} \right)\\ d\left( {a,b} \right) \ge d'\left( {a,b} \right),\forall a,b \in V\left( {G'} \right)\\ {\xi ^d}\left( {G'} \right) \ge 4{\left( {n - 1} \right)^2} - 9\left( {n - 1} \right) - 7 = \\ \;\;\;\;\;\;\;\;\;\;4{n^2} - 17n + 6 \end{array} \right.$ (6)

 $\begin{array}{l} {\xi ^d}\left( G \right) - {\xi ^d}\left( {G'} \right) \ge 0 + \left( {\varepsilon \left( {w'} \right) + \varepsilon \left( {w''} \right)} \right)d\left( {w',} \right.\\ \left. {w''} \right)\left( {\varepsilon' \left( {w'} \right) + \varepsilon' \left( {w''} \right)} \right)d'\left( {w',w''} \right) + \sum\limits_{a \in \left\{ {w',w''} \right\}} {\left( \varepsilon \right.} \\ \left. {\left( a \right) + \varepsilon \left( w \right)} \right)d\left( {a,w} \right) + \sum\limits_{a \in V\left( {G'} \right)\backslash \left\{ {w',w''} \right\}} {\left( {\varepsilon \left( a \right) + \varepsilon \left( w \right)} \right)} \\ d\left( {a,w} \right) \ge \left( {2 + 2} \right) + 2 \times \left( {2 + 2} \right) + 2 \times \left( {2 + 2} \right) \times \left( {n - } \right.\\ \left. 3 \right) = 8n - 12 \end{array}$

 ${\xi ^d}\left( G \right) \ge {\xi ^d}\left( {G'} \right) + 8n - 12 > 4{n^2} - 9n - 7$

 $\left\{ \begin{array}{l} \varepsilon \left( u \right) \ge 2,\varepsilon \left( {u'} \right) \ge 1,d\left( {u,u'} \right) = 1\\ \varepsilon \left( a \right) \ge 2,d\left( {a,u} \right) \ge 2,\forall a \in V\left( G \right)\backslash \left\{ {u,u'} \right\} \end{array} \right.$ (7)

G′=GuG′仍是三圈图。由分析可知

 $\left\{ \begin{array}{l} \varepsilon \left( a \right) \ge \varepsilon '\left( a \right),\varepsilon \left( b \right) \ge \varepsilon '\left( b \right),\forall a,b \in V\left( {G'} \right)\\ d\left( {a,b} \right) \ge d'\left( {a,b} \right),\forall a,b \in V\left( {G'} \right)\\ {\xi ^d}\left( {G'} \right) \ge 4{\left( {n - 1} \right)^2} - 9\left( {n - 1} \right) - 7 = \\ \;\;\;\;\;\;\;\;\;\;4{n^2} - 17n + 6 \end{array} \right.$ (8)

 $\begin{array}{l} {\xi ^d}\left( G \right) - {\xi ^d}\left( {G'} \right) = \sum\limits_{a,b \in V\left( {G'} \right)} {\left[ {\left( {\varepsilon \left( a \right) + \varepsilon \left( b \right)} \right)} \right.} \\ \left. {d\left( {a,b} \right) - \left( {\varepsilon '\left( a \right) + \varepsilon '\left( b \right)} \right)d'\left( {a,b} \right)} \right] + \left( {\varepsilon \left( {u'} \right) + } \right.\\ \left. {\varepsilon \left( u \right)} \right)d\left( {u,u'} \right) + \sum\limits_{a \in V\left( {G'} \right)\backslash \left\{ {u'} \right\}} {\left( {\varepsilon \left( a \right) + \varepsilon \left( u \right)} \right)d\left( {a,u} \right)} \ge \\ 0 + \left( {1 + 2} \right) \times 1 + \left( {n - 2} \right) \times \left( {2 + 2} \right) \times 2 = 8n - 13 \end{array}$

 $\left\{ \begin{array}{l} \varepsilon \left( a \right) \ge 2,\forall a \in V\left( G \right)\\ d\left( {u,u'} \right) = 1,d\left( {u,u''} \right) = 1,d\left( {u',u''} \right) = 2\\ d\left( {a,u} \right) \ge 2,\forall a \in V\left( G \right)\backslash \left\{ {u,u',u''} \right\} \end{array} \right.$ (9)

G′=Gu+uu″，G′仍是三圈图，由分析可知

 $\left\{ \begin{array}{l} \varepsilon \left( a \right) \ge \varepsilon '\left( a \right),\varepsilon \left( b \right) \ge \varepsilon '\left( b \right),\forall a,b \in V\left( {G'} \right)\\ d\left( {a,b} \right) \ge d'\left( {a,b} \right),\forall a,b \in V\left( {G'} \right)\\ {\xi ^d}\left( {G'} \right) \ge 4{n^2} - 17n + 6 \end{array} \right.$ (10)

 $\begin{array}{l} {\xi ^d}\left( G \right) - {\xi ^d}\left( {G'} \right) \ge 0 + \left( {\varepsilon \left( {u'} \right),\varepsilon \left( {u''} \right)} \right)d\left( {u',} \right.\\ \left. {u''} \right) - \left( {\varepsilon '\left( {u'} \right),\varepsilon '\left( {u''} \right)} \right)d'\left( {u',u''} \right) + \sum\limits_{a \in \left\{ {u',u''} \right\}} {\left( {\varepsilon \left( a \right) + } \right.} \\ \left. {\varepsilon \left( u \right)} \right)d\left( {a,u} \right) + \sum\limits_{a \in V\left( {G'} \right)\backslash \left\{ {u',u''} \right\}} {\left( {\varepsilon \left( a \right) + \varepsilon \left( u \right)} \right)d\left( {a,} \right.} \\ \left. u \right) \ge \left( {2 + 2} \right) + 2 \times \left( {2 + 2} \right) + 2 \times \left( {2 + 2} \right) \times \left( {n - 3} \right) = \\ 8n - 12 \end{array}$

 ${\xi ^d}\left( G \right) \ge {\xi ^d}\left( {G'} \right) + 8n - 12 > 4{n^2} - 9n - 7$

 $\left\{ \begin{array}{l} \varepsilon \left( u \right) \ge 2,\varepsilon \left( {u'} \right) \ge 1,d\left( {u,u'} \right) = 1\\ \varepsilon \left( a \right) \ge 2,d\left( {a,u} \right) \ge 2,\forall a \in V\left( G \right)\backslash \left\{ {u,u'} \right\} \end{array} \right.$ (11)

G′=GuG′仍是三圈图。由分析可知

 $\left\{ \begin{array}{l} \varepsilon \left( a \right) \ge \varepsilon '\left( a \right),\varepsilon \left( b \right) \ge \varepsilon '\left( b \right),\forall a,b \in V\left( {G'} \right)\\ d\left( {a,b} \right) \ge d'\left( {a,b} \right),\forall a,b \in V\left( {G'} \right)\\ {\xi ^d}\left( {G'} \right) \ge 4{\left( {n - 1} \right)^2} - 9\left( {n - 1} \right) - 7 = \\ \;\;\;\;\;\;\;\;\;\;4{n^2} - 17n + 6 \end{array} \right.$ (12)

 $\begin{array}{l} {\xi ^d}\left( G \right) - {\xi ^d}\left( {G'} \right) = \sum\limits_{a,b \in V\left( {G'} \right)} {\left[ {\left( {\varepsilon \left( a \right) + \varepsilon \left( b \right)} \right)} \right.} \\ \left. {d\left( {a,b} \right) - \left( {\varepsilon '\left( a \right) + \varepsilon '\left( b \right)} \right)d'\left( {a,b} \right)} \right] + \left( {\varepsilon \left( {u'} \right) + } \right.\\ \left. {\varepsilon \left( u \right)} \right)d\left( {u,u'} \right) + \sum\limits_{a \in V\left( {G'} \right)\backslash \left\{ {u'} \right\}} {\left( {\varepsilon \left( a \right) + \varepsilon \left( u \right)} \right)d\left( {a,u} \right)} \ge \\ 0 + \left( {1 + 2} \right) \times 1 + \left( {n - 2} \right) \times \left( {2 + 2} \right) \times 2 = 8n - 13 \end{array}$

 ${\xi ^d}\left( G \right) > {\xi ^d}\left( {{G_2}} \right) = 126$

 $\left\{ \begin{array}{l} \varepsilon \left( a \right) \ge 2,\forall a \in V\left( G \right)\\ d\left( {u,u'} \right) = 1,d\left( {u,u''} \right) = 1,d\left( {u',u''} \right) = 2\\ d\left( {a,u} \right) \ge 2,\forall a \in V\left( G \right)\backslash \left\{ {u,u',u''} \right\} \end{array} \right.$ (13)

G′=Gu+uu″，G′仍是三圈图。由分析可知

 $\left\{ \begin{array}{l} \varepsilon \left( a \right) \ge \varepsilon '\left( a \right),\varepsilon \left( b \right) \ge \varepsilon '\left( b \right),\forall a,b \in V\left( {G'} \right)\\ d\left( {a,b} \right) \ge d'\left( {a,b} \right),\forall a,b \in V\left( {G'} \right)\\ {\xi ^d}\left( {G'} \right) \ge 4{n^2} - 17n + 6 \end{array} \right.$ (14)

 $\begin{array}{l} {\xi ^d}\left( G \right) - {\xi ^d}\left( {G'} \right) \ge 0 + \left( {\varepsilon \left( {u'} \right),\varepsilon \left( {u''} \right)} \right)d\left( {u',} \right.\\ \left. {u''} \right) - \left( {\varepsilon '\left( {u'} \right),\varepsilon '\left( {u''} \right)} \right)d'\left( {u',u''} \right) + \sum\limits_{a \in \left\{ {u',u''} \right\}} {\left( {\varepsilon \left( a \right) + } \right.} \\ \left. {\varepsilon \left( u \right)} \right)d\left( {a,u} \right) + \sum\limits_{a \in V\left( {G'} \right)\backslash \left\{ {u',u''} \right\}} {\left( {\varepsilon \left( a \right) + \varepsilon \left( u \right)} \right)d\left( {a,} \right.} \\ \left. u \right) \ge \left( {2 + 2} \right) + 2 \times \left( {2 + 2} \right) + 2 \times \left( {2 + 2} \right) \times \left( {n - 3} \right) = \\ 8n - 12 \end{array}$

ξd(G)≥ξd(G′)+8n－12>4n2－9n－7成立。

 ${G^ * } \notin \left\{ {{G_1},{G_2},{G_3},{G_4},{G_5}} \right\}$

 ${\xi ^d}\left( {{G^ * }} \right) > 4{n^2} - 9n - 7$

3 结束语

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