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ИРКУТСКИЙ ГОСУДАРСТВЕННЫЙ УНИВЕРСИТЕТ ПУТЕЙ СООБЩЕНИЯ
As a teacher, through this year teaching and research practice, he has realized that learner autonomy and teacher development have much in common. Having shared with responsibilities with students, both the students and the teacher benefited from the process. The objective of learning in the second language classroom has changed from language form to language use and development (Janice, 1987). As qualified teachers, "we should be able to motivate the students both on a daily basis by designing good tasks" (Wang, 1999) and in a long run by helping them set up meaningful and achievable long-term goals. For these students who have special majors, teachers should especially have the duty to study the learners characteristics and features, adopt proper methodology and attitude toward the teaching.
So when implementing the formative assessment, teachers should be very sensitive to the characteristics of the students, as their different cognitive, affective, social and physical experiences are not so mature. It will be damaging and unrealistic to demand all the students to
perform at perfect conditions and compare them with one another. Therefore it is necessary for the students to understand their individual nature, set up their own learning goals in relation to curriculum objectives so that they develop their learning strategies accordingly. Directed properly, they will respond positively to a systematic formative assessment.
REFERENCES
1. Research on Teaching Assessment System for Foreign Teachers in Colleges, Journal of East China Jiaotong University, Vol.24 No.6, Dec., 2007.
2. Research on Credit-Transfer System -Important Factor of Education Globalization, Jiangxi Provincial Education Research Project, Dec. 2005.
3. Zhou Liping Associate Professor and Deputy Director, International School /International Office East China Jiaotong University, specializing in English teaching method, international education.
Baogen Xu, Li deng
УДК519.218.2
ON RECIPROCAL SIGNED DOMINATION IN GRAPHS
1. Introduction. We use Bondy and Murty [1] for terminology and notation not defined here and consider simple graphs only.
Let G = (V, E) be a graph, ifv e V(G), then NG (v) denotes the open neighbourhood of v in G, and Ng [v]= Ng (v) Y{v} for the close one. For simplicity, sometimes, NG (v) and NG [v] are denoted by N(v) and N[v], respectively. If S c V(G) then G[S] denotes the subgraph of G
induced by S.. S( G) and A(G) denote the minimum and maximum degree of G, respectively.
In recent years, some kinds of domination in graphs have been investigated [2~7]. T.W.Haynes, etc.[2~3] surveyed the major research accomplishments on domination theory.
Let G = (V, E) be a graph, a signed domination function (SDF) of G is a function f: V(G) ^{-1,+ 1} satisfying £f(v) > 1 for all
v eN [u ]
vertices u e V(G), and the signed domination number of G is defined as y s (G) = min{ £ f(v)f is an SDF of G}. v eV (g )
In this paper, we initiate the study of a new graph parameter by changing > to «= in the definition of signed domination function. Now we introduce a kind of new domination parameter as follows:
Definition 1.1. Let G = (V, E) be a graph, a function f: V ^ {-1, +1} is said to be a reciprocal signed domination function (RSDF) of G if £ f (v) < 1holds for every vertex u e V. The
v eN [u ]
reciprocal signed domination number of G is defined as yr (G) = max { £f(v) } f is an RSDF of G}. ' v eV (G)
If f be a RSDF of a graph G, andS c V(G), for convenience, we write f( S) = £ f(v).
СИСТЕМНЫЙ АНАЛИЗ И МЕЖДИСЦИПЛИНАРНЫЕ ПОДХОДЫ В ИССЛЕДОВАНИЯХ
By the above definition, we have the following
Lemma 1.2. Let G be a graph of order n, then
(1) y re (G) = n if and only if G = ~K~n;
(2) For any two disjoint graphs G1 andG2, y ra (G iYG 2) = y ra (G1)+y ra (G 2)^
(3) yra(G) - n(mod2).
In this paper we obtain mainly some upper bounds of yr (G) for general graphs, and determine the exact values of yr (G) for some special classes of graphs G such as the complete graphs Kn paths Pn cycles Cn and complete bipartite graphs Km,netc.. In addition, we pose some open problems and conjectures.
2. Some bounds of reciprocal signed domination number
In this section, we give some upper bounds of y rs (G )for general graphs G.
Theorem 2.1. For any graph G without isolate vertex, ifjV (G )| = n, then
yrs (G) < n + 2-2 Vl + n |.
And this bound is sharp.
Proof: Let f be such a RSDF of G that f(V(G)) =yr (G). Define
' A = {v e V(G)f(v) =1}, B = {v e V(G)|f(v) = -1} , | A| = s and |B| = t. Obviously, s +1 = n and yr (G) = s -1 = n -2t. Since S(G)>1, each vertex of A is joined to at least one vertex of B, it implies
{{uv e E(G)|u e A,v e B }|> s. Thus, there exists a
s
vertex v e B so that |N(v)IA| > [ — ], by Definition
s
1.1 v is joined to at least [ — ]-2 vertices of B,
that is , t =|B|>[ - ]-1 = [ — ]-1 >—-1, 1 1 L t J L t J t
t2 + 2t -n > 0, it implies t > -1W1 + n, note that t is
an integer, sot >-1+| V1 + n |, we have
y^ (G) = n-2t < n + 2-2 -Jl+n | (2.1) Next we construct a graph G such that yrs(G) = n + 2 -2 VT+n |.
Let t = -1 +| V1+n |, s = n +1 - V1 + n |, clearly,
s +1 = n. Note that n > 2 and hence t > 1. Since t2 + 2t -n > 0, that is, t(t +1) > n -1 = s, thus, we construct a graph G as follows: Let G be the graph obtained from Kt by adding exactly s pendant-edges such that each vertex of Kt adds at most t +1 pendant-edges. Clearly, |V(G )| = s +1 = n.We define a function f: V (G) ^{-1,+ 1} as follows:
f (v) =
Г-1 when v e V(Kt);
+ 1 when v € V(Kt).
It is easy to see that f is a RSDF of G. Thus yr (G) > f((V(G)) = n - 2t, combining with (2.1) , we have yrs(G) = n + 2 -2 V1+n |, this proof is complete.
Theorem 2.2. For any graph G without isolate vertex, if |V(G )| = n and |E(G )| = m, then
y rs (G ) <
5
be such a RSDF of G that
Define A = {v e V(G)f(v) =1} -1} , I A| = s and
Proof: Let f
f( V (G)) =y rs (G). B = {v e V(G)f(v) = -1}
|B| = t,obviously, s +1 = n and yr (G) = n -2t. And let E(A,B) = {uv e E(G)|u e A,v e bB }.
As each vertex in A must be joined to at least one vertex in B, it implies
|E( A, B)> s = n -1. (2.2)
For each vertex v e B, |N(v)IB|>|N(v)IA|-2, so we have
s < |E(A,B) = ^|N(v)IA| < ^(|N(v)IB| + 2) =
veB veB
= 2|E(G[B])|+ 2t, thus we have
|E(G[B])| > s-2t = n-3t. (2.3)
Combining with 2.2 and (2.3), we have
m >|E(A,B)+|£(G[B])|>
3n - 5t
2
So,
t>
3n - 2m
5
and
hence,
4m — n
уrs(G) = n — 2t <-. The proof is complete.
Theorem 2.3. For any graph G of order n (n > 2), ifE(G)|= m then
. A-5 + 2
yrs(G) < A s o n. A+S + 2
Where A and S denote the maximum and minimum degree of G, resp..
Proof: Let f be such a RSDF of G, that f(V(G)) =yr (G). Define
A ={v e V(G)|f(v) = 1}, B = {v e V(G)|f(v) = -1}, | A| = s and |B| = t,obviously, yr (G) = s-1. So, we have
s+t=n and
s=
n + У rs (G)
and
t = n-yJG). (2.4)
22 We know from the definition 1.1 that f(N[u]) < 1holds for every vertexu e V(G), thus £f(N[u]) < n, that is,
ueV (G)
ИРКУТСКИЙ ГОСУДАРСТВЕННЫЙ УНИВЕРСИТЕТ ПУТЕЙ СООБЩЕНИЯ
£(d(u) +1)f(u) < n. f(v 1) + f(v2) < 0. Analogously,
ueV(G) f(vn-1) + f (vn) < 0. Thus, we have
Note thatu£G((u) = y"(G* thus y, P ) = f(V(Pn )) < £f(N[v3,+1]) < k -1 =
1—1 1—1 ' =1
£d(u)f(u) + £d(u)f(u) < n-yra (G), it implies n
ueA ueB = (3k +1)-2(k +1)=n-2[ - ].
8(n/jaIg'aJ'ai-:/:^ < &-}!'4^))we haand Case 2.3. When,, = 3k + 3(k >1); Note that
A-5 + 2 f(v 1)+f(v 2) < 0, we have hence y re (G) < n. i
A+S + 2 yrs(Pn ) = f(V(Pn ))<£f(N[vs+1])< k =
We have completed the proof of Theorem 2.3. i=1 .
The following statements are immediate from =(3k + 2) 2(k +1) = n 2[ 3 ]
Theorem 2.3. Combining with 3.1, we have completed the
Corollary 2.4. For any k-regular graph G of proof of Theorem 3.1.
order n, then ys(G) < n . Similar to the proof of Theorem 3.1, we can
k +1 obtain easily the following 3. The Special graphs. In this section, we give Corollary 3.2. If n > 3, then the exact values of yrs (G )for some special l graphs, C 2p n
such as the complete graphs Kn paths Pn cycles Cn yrs ( n) = n [ 3 ].
and complete bipartite graphs Km, netc.. It is easy Theorem 3.3. If m> 2 and n > 2 are two
1 -(-1)n (-1)m +(-1)n - 2
to check that ys (Kn ) =--—holds for all integers, then yrs (Km,n) = -—---—--.
integersn> 1. Proof let f be such a RSDF of Kmn, that
Theorem 3.1; If n > 2 is an integer, then f(V( Kmn )) =y ^ (Km,n ). WriteV (Km,n ) = V1YV2,
y rs (Pn) = n - 2[ n ]. where |V11= m and |V21= n. We define 3 Vt + ={ue Vi |f(u) = +1},
Proof: LelVP) = {v"v 2'^.} and V, = {u ef) = -1} (,=1,2).
E(Pn ) = {v,v,+1|0 < i < n-1},define a function + +
f: v.p ) ^ {-1,+1} as follows: Case 1 If V1 * ^ and V2 * v;for any vertex
Case1.1, When n =1(bmod3); let f(vn ) = -1and u e V+ ,by Definition 1.1, we have f(N[u]) < 1, and
wheni< n-1, hence, |V2+|-|V2-|< 0, note that |V2+|+|V2-|= n, f-1, when i = 2(mod3) (^n -1
(v,t)=|+1 otherwise; which imply |V2+|-|V2-|< —2—. ^^g^^
Case1.2 Whenn = 0,2(mod3); For each integer . . . , (-1)m -1
„ , , |V,+ |-|V. |<-—--. So, we have
,(1 <, < n), let I 1 I I 1 I 2
f-1 when i = 2(mod3),
{V' ) = {+1 OtherwiSe. y rs ( Km,n ) = V+ | -|Vr| +|V2+ | -|V2-| < (-1)m +2-1)n -2 .
It is easy to check that f is a RSDF of Pn, and + + _
hence , we have Case 2 If V * v and V2 =9; For any u e V2 ,
y (P )> f(V(P )=n-2[ n ] . (3.1) then f(N[u])<1, |V1+1—|V1-1< 2 note that
n 3 |v1+|+|V1-| = m and 2n > 5-(-1)n, so
On the other hand, let f be such a RSDF of Pn 1 111
that f(V(p- )) =Yr. (p- ). |V1+1 - |V1-1 < 2-1 -(-1)m , we have
Case 2.1. When n = 3k( k > 1); I 1 I I 1 I 2
4-1 n y (K ) = iv+mvmv-! <
y r. (Pn ) =f(V(Pn )) = £ f( N [v 3,+2]) < k = n - 2[ n ]. y r.(Km,n) |V | |V | V 2 |<
i=0 3 1 -(-1)m (-1)m +(-1)n - 2 Case 2.2. When n = 3k +1( k > 1); Note that < 2--2-- n <-2-.
f(N[v 1]) = f(v 1)+f(v2) < 1, it implies
СИСТЕМНЫЙ АНАЛИЗ И МЕЖДИСЦИПЛИНАРНЫЕ ПОДХОДЫ В ИССЛЕДОВАНИЯХ
Case 3. If V1+ = ф and V2+ ф ф ; It is similar to
Case 2.
Case 4. If V1+ = ф and V2+ = ф ;
(-1)m +(-1)n - 2
Y . ( Кш,п ) HM-jV^-m - n <-
2
Sum up the above Case 1~4, we have (-1)m +(-1)n - 2
Y, (Kmin ) <
2
(3.2)
On the other hand, we can partition easily V1 = A1YA2 and V2 = B1YB2, sothat1 >|A11-|A21>0 and1 >|B1|-|B21>0. Define f:V(Km,n )^{-1,+ 1} as follows:
f-1 when v e A1YB1;
f (v) = J 1 11
[+1 when v e A 2 YB 2.
It is easy to see that f is a RSDF of Kmn, so, r,(K„ )> f(V ( Km,. ) = '-1)m ^ - 2. ,
Combining with 3.2,, we have completed the proof of Theorem 3.3.
If true, the lower bound is the best possible. For example, Let G be a graph whose every component is K33. By Lemma 1.2 and Theorem
, 11 i 3.3, we have yrs(G) = -—|V(G)|.
Problem 3. Is it true that y rs (T) > 0 holds for all trees T? If true, characterize the trees T satisfying y rs (T) = 0.
REFERENCES
1. J.A.Bondy,V.S.R.Murty, Graph Theory with Applications[M], Elsevier, Amsterdam, 1976
2. T.W.Haynes,S. T.Hedetniemi, and P.J.Slater, Domination in graphs[M], New York,1998.
3. T.W.Haynes, S. T.Hedetniemi, and P.J.Slater, Fundamentals of Domination in graphs[M], New York,1998.
4. Z.Zhang, Baogen.Xu, Y.Li, L.Liu, A note on the lower bounds of signed domination number of agraph[J], Discrete Math. 195(1999),295-298.
4. Some open problems and conjectures.
In this section, we pose some open problems and conjectures. Although some upper bounds of yrs(G) are obtained in this paper, it seems more difficult to give a good lower bound for the reciprocal signed domination number of G, we have the following
Problem 1. Determine the exact value of g(n) = min{y rs (G )G is a graph of order n} for every
positive integer n.
Conjecture 2. For any graph G of order n,
theny rs (G) >-n.
5. Baogen.Xu, E.J.Cockayne, T.W.Haynes, S.T.Hedetniemi, S.Zhou, Extremal graphs for inequalities involving domination parameters, Discrete Math. 216(2000) 1-10
6. Baogen.Xu, On minus domination and signed domination in graphs, J. Math. Res. Exposition, 4(2003), 586-590
7. E.J.Cockayne, C.M.Mynhart, On a generalization of signed domination functions of graphs, Ars. Combin. 43 (1996) 235-245.
Fan Yong, Zhang Xuan
AN EMPIRICAL STUDY ON CONCENTRATION RATIO IN CHINA'S SECURITIES TRADERS
Among the framework of industrial structure comprehensive analysis of it will reflect the
theory, concentration ratio is regarded as a main distribution of inner-industry business and
criterion to measure the market structure. Since competition in the market. The most common
the change of concentration ratio will directly criterion is concentration ratio (CRn, see Formula
reflect the competition status in market, a A ), an index refers to the proportion of
УДК 681.3
