Full Text
L. M. Vitaver
ON A VERTEX-EDGE FUNCTION OF GRAPHS
(Presented by Academician S. L. Sobolev on 23 I 1962)
A systematic study of recurrent functions of graphs was begun by A. A. Zykov (¹–⁴). We shall consider one of such functions, belonging to the class of vertex-edge functions (³), and in doing so shall use the definitions and notation of (¹–⁴).
- Let \(L\) be an arbitrary nonempty graph whose edges are ordered in some way; let \(ab\) be its first edge. Let \(L_\alpha\) be the graph obtained from \(L\) by deleting the edge \(ab\) without deleting vertices; let \(L_\mu\) be the graph obtained from \(L\) by deleting the edge \(ab\) and replacing the vertices \(a\) and \(b\) by a single vertex \(\bar a\), adjacent to those of the remaining vertices of \(L\) which were adjacent to one and only one of the vertices \(a,b\) (the order of the edges in the graphs \(L_\alpha, L_\mu\) is induced by the order of the edges in the graph \(L\)).
Fig. 1
Further, let \(K\) be a ring with generators \(\alpha,\mu,1\), and let \(\Phi(L)\) be a function of graphs taking values in \(K\) and satisfying the conditions
\[ \Phi(L)=\alpha\Phi(L_\alpha)+\mu\Phi(L_\mu)+1; \tag{1} \]
\[ \Phi(E_n)=0,\qquad n=0,1,2,\ldots, \tag{2} \]
where \(E_n\) is the empty \(n\)-vertex graph.
In order that the value of the function \(\Phi(L)\) not depend on the manner of ordering the edges of the graph \(L\), it is necessary and sufficient that the conditions
\[ (\alpha\mu-\mu\alpha)(\alpha+\mu)^m\mu^n(\alpha+\mu+1)=0; \tag{3_1} \]
\[ (\alpha^2-1)\mu^{\,n+1}(\alpha+\mu+1)=0, \tag{3_2} \]
where \(m,n=0,1,2,\ldots\), be fulfilled.
The method of proof of this assertion is borrowed from (⁴). Necessity is proved by comparing the values of \(\Phi\) for the corresponding graph of Fig. 1 for two ways of ordering its edges. In deriving \((3_2)\) we use the relation
\[ (\alpha^m\mu^n-\mu^n\alpha^m)(\alpha+\mu+1)=0, \tag{3_3} \]
which is an algebraic consequence of \((3_1)\).
To prove sufficiency, we first show that, with the help of (3), the general expression \(\Phi(L)\) is brought to the form
\[ \Phi(L)=\{f_L(\alpha)+\alpha\varphi_L(\mu)+\psi_L(\mu)\}(\alpha+\mu+1)+\frac{1-(-1)^{d_2(L)}}{2}; \tag{4} \]
where
\[ f_L(\alpha)= \begin{cases} \alpha^{d_2(L)-2}+\alpha^{d_2(L)-4}+\cdots+\alpha^2+1, & \text{if } d_2(L)\equiv 0 \pmod 2,\\ \alpha^{d_2(L)-2}+\alpha^{d_2(L)-4}+\cdots+\alpha^3+\alpha, & \text{if } d_2(L)\equiv 1 \pmod 2, \end{cases} \tag{5} \]
and \(\varphi_L(\mu)\), \(\psi_L(\mu)\) are polynomials with integer nonnegative coefficients, moreover
\[ \varphi_L(-\mu)=(-1)^{d_2(L)+1}\varphi_L(\mu);\qquad \psi_L(-\mu)=(-1)^{d_2(L)}\psi_L(\mu). \tag{6} \]
After this it remains to consider three cases of the mutual arrangement of the pair of edges \(ac\) and \(bc\) (or \(bd\)) in an arbitrary graph (Fig. 2); applying induction on the number of edges of the graph and using once again relations (3), we complete the proof of sufficiency.
Fig. 2
- The expression (4) for the function \(\Phi(L)\) is uniquely recovered from the general expression of the function \(\Psi(L)\), satisfying the conditions
\[ \Psi(L)=\Psi(L_\alpha)+\mu\Psi(L_\mu)+1\,*; \tag{1'} \]
\[ \Psi(E_n)=0,\qquad n=0,1,2,\ldots \tag{2'} \]
Obviously, \(\Psi(L)\) is a polynomial with integer nonnegative coefficients:
\(\Psi(L)=\sum_{k\ge 0} a_k(L)\mu^k\), moreover
\[ a_k(L)=a_k(L_\alpha)+a_{k-1}(L_\mu),\qquad k=1,2,\ldots; \tag{7} \]
\[ a_0(L)=a_0(L_\alpha)+1. \tag{8} \]
We shall now show that
\[ a_k(L)=\sum_{i\ge 0}(-2)^i P_{i+k+1,i}(L), \tag{9} \]
where \(P_{ji}(L)\) is the number of edge subgraphs \(M\) of the graph \(L\) for which \(d_2(M)=j\) and \(l(M)=i\).
It is easy to see that \(P_{ji}(L)=P_{ji}(L_\alpha)+\overline{P}_{ji}(L)\), where \(\overline{P}_{ji}(L)\) is the number of those edge subgraphs (with number of edges \(j\) and cyclomatic number \(i\)) of the graph \(L\) which contain the edge \(ab\).
Let \(\rho\) be the number of triangles of the graph \(L\) containing the edge \(ab\); the vertices of these triangles distinct from \(a\) and \(b\) will be called the \(c\)-vertices of the graph \(L\).
Let \(P_{ji}^{r,\varepsilon,s}(L_\mu)\) be the number of those edge subgraphs (with number of edges \(j\) and cyclomatic number \(i\)) of the graph \(L_\mu\) whose composition includes \(r\) \(c\)-vertices of the graph \(L\), of which \(\varepsilon\) belong to one connected component with the vertex \(\overline{a}\) in the graph \(L_\mu\), while the remaining \(r-\varepsilon\) are distributed among \(s\) connected components of the graph \(L_\mu\).
In any edge subgraph \(M\) of the graph \(L\) we distinguish classes of vertices belonging to the separate connected components of \(M\) itself or of the graph obtained from \(M\) as a result of the operation \(L\to L_\mu\). Among the \(c\)-vertices of each of these classes we distinguish vertices adjacent to both \(a\) and \(b\) simultaneously, adjacent to only one of the vertices \(a,b\), and, finally, not adjacent—
\[ \text{* It is clear that for } \alpha=1 \text{ the conditions (3) are fulfilled.} \]
connected neither with \(a\) nor with \(b\). The notation needed below for the numbers of \(c\)-vertices of one or another type is given in Fig. 3.
Fig. 3
Then
\[ \bar P_{ji}(L)= \sum_{r=0}^{\rho}\sum_{\omega=0}^{r}\sum_{\omega_1=0}^{\omega} C_{\omega}^{\omega_1}2^{\omega_1} \sum_{\varepsilon=0}^{r-\omega}\sum_{\varepsilon_1=0}^{\varepsilon} C_{\varepsilon}^{\varepsilon_1}2^{\varepsilon_1} \sum_{\varepsilon_2=0}^{\varepsilon-\varepsilon_1} C_{\varepsilon-\varepsilon_1}^{\varepsilon_2} \times \]
\[ \times \sum_{s\ge 0}\sum_{t=0}^{s} \left\{ \sum_{\substack{\sum_{n=1}^{t} q_n\le r-\omega-\varepsilon}} \prod_{n=1}^{t} \left[ \sum_{q'_n=1}^{q_n} C_{q_n}^{q'_n} \sum_{q_{n1}=0}^{q'_n} C_{q'_n}^{q_{n1}}2^{q_{n1}} \right] \right\} P_{j_1,i_1}^{\,r-\omega,\varepsilon,s}(L_\mu), \]
where the expression in braces is regarded as equal to one for \(t=0\);
\[ j_1=j-1-(2\omega-\omega_1)-(\varepsilon_1+2\varepsilon_2) -\sum_{n=1}^{t}(2q'_n-q_{n1}); \]
\[ i_1=i-(\omega-\omega_1)-(\varepsilon_1+2\varepsilon_2) -\sum_{n=1}^{t}(2q'_n-q_{n1})+t. \]
Hence*
\[ \sum_{i\ge 0}(-2)^i P_{i+k+1,i}(L) - \sum_{i\ge 0}(-2)^i P_{i+k+1,i}(L_\alpha) = \sum_{i\ge 0}(-2)^i \bar P_{i+k+1,i}(L) = \]
\[ = \sum_{r=0}^{\rho}\sum_{\omega=0}^{r}\sum_{\omega_1=0}^{\omega} C_{\omega}^{\omega_1}2^{\omega_1} \sum_{\varepsilon=0}^{r-\omega}\sum_{\varepsilon_1=0}^{\varepsilon} C_{\varepsilon}^{\varepsilon_1}2^{\varepsilon_1} \sum_{\varepsilon_2=0}^{\varepsilon-\varepsilon_1} C_{\varepsilon-\varepsilon_1}^{\varepsilon_2} \times \]
\[ \times \sum_{s\ge 0}\sum_{t=0}^{s} \left\{ \sum_{\substack{\sum_{n=1}^{t} q_n\le r-\omega-\varepsilon}} \prod_{n=1}^{t} \left[ \sum_{q'_n=1}^{q_n} C_{q_n}^{q'_n} \sum_{q_{n1}=0}^{q'_n} C_{q'_n}^{q_{n1}}2^{q_{n1}} \right] \right\} \times \]
\[ \times \sum_{i_1\ge 0}(-2)^i P_{\,i_1+k-\omega-t,\;i_1}^{\,r-\omega,\varepsilon,s}(L_\mu) = \]
\[ \text{\* In replacing the summation index } i_1 \text{ by } i \text{ we used the fact that } P_{ji}=0 \text{ for } i<0. \]
\[ = \sum_{r=0}^{\rho} \sum_{\omega=0}^{r} \left[ \sum_{\omega_1=0}^{\omega} C_{\omega}^{\omega_1} 2^{\omega_1}(-2)^{\omega-\omega_1} \right] \sum_{\varepsilon=0}^{r-\omega} \left[ \sum_{\varepsilon_1=0}^{\varepsilon} C_{\varepsilon}^{\varepsilon_1} 2^{\varepsilon_1}(-2)^{\varepsilon_1} \sum_{\varepsilon_2=0}^{\varepsilon-\varepsilon_1} C_{\varepsilon-\varepsilon_1}^{\varepsilon_2}(-2)^{2\varepsilon_2} \right]\times \]
\[ \times \sum_{s\geq 0}\sum_{t=0}^{s} \left\{ \sum_{\substack{\sum_{n=1}^{t} q_n \leq r-\omega-\varepsilon}} \prod_{n=1}^{t} \left[ \sum_{q'_n=1}^{q_n} C_{q_n}^{q'_n}(-2)^{q'_n} \left( \sum_{q_{n1}=0}^{q'_n} C_{q'_n}^{q_{n1}}2^{q_{n1}}(-2)^{q'_n-q_{n1}} \right) \right] \right\}\times \]
\[ \times \sum_{i_1\geq 0}(-2)^{i_1-t} P_{i_1+k-\omega-t,i_1}^{r-\omega,\varepsilon,s}(L_\mu) = \sum_{r=0}^{\rho}\sum_{\varepsilon=0}^{r}\sum_{s\geq 0}\sum_{i_1\geq 0} (-2)^{i_1} P_{i_1+k,i_1}^{r,\varepsilon,s}(L_\mu) = \]
\[ = \sum_{i_1\geq 0}(-2)^{i_1} \sum_{r,\varepsilon,s} P_{i_1+k,i_1}^{r,\varepsilon,s}(L_\mu) = \sum_{i_1\geq 0}(-2)^i P_{i_1+k,i_1}(L_\mu). \]
Thus
\[ \sum_{i\geq 0}(-2)^i P_{i+k+1,i}(L) = \sum_{i\geq 0}'(-2)^i P_{i+k+1,i}(L_\alpha) + \sum_{i\geq 0}(-2)^i P_{i+k,i}(L_\mu), \tag{7'} \]
\[ k=1,2,\ldots \]
Moreover,
\[ P_{10}(L)=P_{10}(L_\alpha)+1, \tag{8'} \]
for \(P_{10}(L)=d_2(L)\). Comparing (7), (8) and (7′), (8′), and taking into account the uniqueness of the function \(\Psi(L)\), we obtain (9).
Thus,
\[ \Psi(L)=\sum_{i,k}(-2)^i P_{i+k+1,i}(L)\mu^k. \]
- It is easy to show that the quantities \(d_2(L)\) and \(l(L)\) are connected by the relation
\[ l(L)\leq d_2(L)+\tfrac12\left[1-\sqrt{d_2(L)+1}\right], \]
where the equality sign is attained if \(L\) is a graph almost isomorphic to a complete graph. It follows that edge subgraphs \(M\) of the graph \(L\), for which \(d_2(M)=j\) and \(l(M)=i\), can exist only when
\[ j\leq d_2(L);\qquad i\leq \min\{l(L);\ j+\tfrac12(1-\sqrt{8j+1})\}. \]
Taking this into account, we finally obtain
\[ \psi(L)= \sum_{k=0}^{d_2(L)-l(L)-1} \sum_{i=0}^{C_{k+1}^{2}-1} (-2)^i P_{i+k+1,i}(L)\mu^k. \]
The present work was carried out under the supervision of A. A. Zykov, to whom the author expresses gratitude for posing the problem and for a number of useful discussions.
Novosibirsk Institute
of Water Transport Engineers
Received
20 II 1962
REFERENCES CITED
¹ A. A. Zykov, Izv. Sibirsk. otd. AN SSSR, No. 5 (1959). ² A. A. Zykov, Izv. Sibirsk. otd. AN SSSR, No. 9 (1960). ³ A. A. Zykov, Izv. Sibirsk. otd. AN SSSR, No. 12 (1960). ⁴ A. A. Zykov, DAN, 139, No. 4 (1961).