In [10], by means of the concept of effective resistances, the global cyclicity index has been proposed:
$$ C(G)= \sum_{(i,j) \in E} \frac{1}{R_{ij}} - m. $$
(11)
Yang in [18] continued the study of this new cyclicity measure for connected graphs. Following Bianchi et al. [16, 25] and computing the extremal values of the Schur-convex function \(f(R_{ij})=\sum_{(i,j)\in E}\frac{1}{R_{ij}}\) on the set \(S= \{ R_{ij}\in \mathbb{R}^{m}:\sum_{(i,j)\in E}R_{ij}=n-1,\frac {2}{n}\leq R_{ij}\leq1 \} \), he obtained the following bounds for \(C(G)\):
$$ \frac{m(m-n+1)}{n-1} \le C(G) \le\frac{n(m-n+1)}{2}, $$
(12)
where \((m-n+1)\) is the well-known cyclomatic number of a graph (see Theorems 3.13, 3.15 and Corollary 3.14 in [18]).
The aim of this section is to extend bounds (12) to the case of general networks where a resistance \(r_{ij}\), \(k \le r_{ij}\le K\), is associated to any edge. We show next how the weighted majorization technique proposed in Section 2 can be a fruitful tool to bound the weighted global cyclicity index, throughout the p-Schur-convex functions.
First of all, let us define the weighted global cyclicity index as a natural extension of the global cyclicity index (11) which can be recovered when \(r_{ij}=1\) for all \((i,j) \in E\):
$$CW(G)= \sum_{(i,j) \in E} \biggl( \frac{1}{R_{ij}} - \frac{1}{r_{ij}} \biggr). $$
If we define the variables \(x_{ij}=\frac{R_{ij}}{\sqrt{r_{ij}}}\) and the weights \(p_{ij}= \frac{1}{\sqrt{r_{ij}}}\), the weighted global cyclicity index can be written as a function of \(x_{ij}\) and \(p_{ij}\) as follows:
$$CW(G)= f(x_{ij},p_{ij}) = \sum_{(i,j) \in E} \biggl( \frac{p_{ij}}{ x_{ij}} - p_{ij}^{2} \biggr). $$
The choice of the variables and of the weights ensures that the function f is p-Schur-convex. Indeed, applying Theorem 2 we get
$$(x_{ij}-x_{i'j'}) \biggl(\frac{1}{p_{ij}} \frac{\partial f}{\partial x_{ij}} - \frac{1}{p_{i'j'}} \frac{\partial f}{\partial x_{i'j'}} \biggr) = \frac{(x_{ij}-x_{i'j'})^{2} (x_{ij}+x_{i'j'})}{x_{ij}^{2} x_{i'j'}^{2}}\ge0 $$
for all \((i,j), (i',j') \in E\).
Remark 8
Note that other possible choices of the variables and of the weights are not fruitful:
-
(1)
if we use as variables \(x_{ij}=R_{ij}\) and as weights \(p_{ij}= \frac{1}{r_{ij}}\) the function \(CW(G)\) is not p-Schur-convex;
-
(2)
if we use as variables \(x_{ij}=\frac{R_{ij}}{r_{ij}}\) and as weights \(p_{ij}=1\) the function \(CW(G)\) is not Schur-convex.
We can now state our main result.
Theorem 9
Let
\(G=(V,E)\)
a connected network with
n
vertices and
m
edges. Let
\(r_{ij}\), \(k \le r_{ij} \le K\), be the resistances associated to any edge
\((i,j) \in E\)
and let
$$C=\sum_{(i,j) \in E} \frac{1}{r_{ij}},\qquad C'= \sum_{(i,j) \in E} \frac{1}{\sqrt{r_{ij}}},\qquad C'' = \max \biggl\{ \frac{nK}{2k}, \frac{\sqrt{k}}{\sqrt{K}} + \frac{n \sqrt{K}}{2k\sqrt{k}} - \frac{1}{K} \biggr\} . $$
Then
$$ \frac{(C')^{2}}{n-1} - C \le CW(G) \le C'' + \biggl( \frac{n \sqrt{K}}{ 2k} + \frac{\sqrt{k}}{K} \biggr) C' - \frac{n}{2k} - \frac{n^{2}-n}{2 \sqrt{k} \sqrt{K}} - C. $$
(13)
Proof
Let \(e_{1}, e_{2}, \ldots,e_{m}\) be the edges of G. For simplicity denote the resistances and the effective resistances between the end vertices of the edge \(e_{i}\), as \(r_{i}\), and \(R_{i}\), respectively and let \(p_{i}= \frac{1}{\sqrt{r_{i}}} \). Moreover, let us assume, without loss of generality, that the variables \(x_{i}= \frac{R_{i}}{\sqrt{r_{i}}}\) are arranged in nonincreasing order: \(x_{1} \ge x_{2} \ge\cdots\ge x_{m}\).
Recalling that by Foster’s first formula
$$\sum_{ e_{i} \in E} p_{i} \cdot x_{i} =(n-1), $$
and that
$$\frac{2k}{n \sqrt{K}} \le x_{i} \le\frac{K}{\sqrt{k}}, \quad\mbox{for all } 1 \le i \le m, $$
let us now consider the set
$$S= \Biggl\{ \mathbf{x} \in\mathbb{R}^{m}: \sum _{i=1}^{m} x_{i} p_{i} = (n-1), \frac{K}{\sqrt{k}} \ge x_{1} \ge x_{2} \ge\cdots\ge x_{m} \ge\frac{2k}{n \sqrt{K}} \Biggr\} . $$
The function \(CW(G)=f(x_{i},p_{i})\) is p-Schur-convex and thus its lower and upper bounds on S are attained at the minimum and maximum element of S with respect to the p-majorization order, respectively.
From Remark 5 we know that the minimal element of S is \(\mathbf{x}_{\mathbf{\ast p}}(S)= [(\frac{n-1}{\sum_{i=1}^{m} p_{i}} )]^{m}\). Thus the lower bound is
$$f(\mathbf{x}_{\mathbf{\ast p}}, \mathbf{p})= \sum_{i=1}^{m} p_{i} \cdot \frac {\sum_{i=1}^{m} p_{i}}{n-1} - C = \frac{(C')^{2}}{n-1} -C. $$
The maximal element \(\mathbf{x}^{\mathbf{\ast p}}\) of the set S can be computed with Corollary 4, yielding
$$\mathbf{x}^{\mathbf{\ast p}} = \biggl[\underbrace{\frac{K}{\sqrt{k}}, \ldots, \frac{K}{\sqrt{k}}}_{l\mbox{-}\mathrm{times}}, \theta, \underbrace{\frac{2k}{n \sqrt{K}}, \ldots,\frac{2k}{n \sqrt{K}} }_{(m-l-1)\mbox{-}\mathrm{times}} \biggr], $$
where l is the first integer such that
$$ \frac{K}{\sqrt{k}} \sum_{i=1}^{l} p_{i} + \frac{2k}{n \sqrt{K}} \sum_{i=l+1}^{m} p_{i} \le(n-1) < \frac{K}{\sqrt{k}} \sum_{i=1}^{l+1} p_{i} + \frac{2k}{n \sqrt{K}} \sum_{i=l+2}^{m} p_{i} $$
(14)
and \(\theta= p_{l+1} \cdot (n-1 -\frac{K}{\sqrt{k}} \sum_{i=1}^{l} p_{i} - \frac{2k}{n \sqrt{K}} \sum_{i=l+2}^{m} p_{i} )\). Let \(D=\sum_{i=1}^{l} p_{i} \) and
$$H=\frac{n-1- \frac{2k}{n \sqrt{K}} C'}{\frac{K}{\sqrt{k}} - \frac {2k}{n \sqrt{K}}}. $$
From (14) easy computations show that
$$0 \le(H-D) < p_{l+1}. $$
Moreover,
$$f\bigl(\mathbf{x}^{\mathbf{\ast p}},\mathbf{p}\bigr)= \biggl( \frac{n \sqrt{K}}{{2k}} - \frac{\sqrt{k}}{K} \biggr) (H-D) + \frac{1}{ ( \frac{K}{\sqrt {k}} - \frac{{2k}}{n \sqrt{K}} )(H-D) + \frac{2k}{n \sqrt{K}} \cdot p_{l+1}} + T, $$
where
$$T= \biggl( \frac{n \sqrt{K}}{2k}+ \frac{\sqrt{k}}{K} \biggr) C' - \frac{n \sqrt{K}}{2k} \cdot p_{l+1} - \frac{n^{2}-n}{2 \sqrt{k} \sqrt {K}} -C. $$
Let \(y=H-D\) and consider the function
$$h(y)= \biggl( \frac{n \sqrt{K}}{{2k}} - \frac{\sqrt{k}}{K} \biggr) y + \frac{1}{ ( \frac{K}{\sqrt{k}} - \frac{{2k}}{n \sqrt{K}} )y + \frac{2k}{n \sqrt{K}} \cdot p_{l+1}} + T. $$
The first derivative is
$$h^{\prime}(y)= - \frac{\frac{K}{\sqrt{k}} - \frac{{2k}}{n \sqrt {K}}}{ [ ( \frac{K}{\sqrt{k}} - \frac{{2k}}{n \sqrt{K}} )y + \frac{2k}{n \sqrt{K}} \cdot p_{l+1} ]^{2}} + \frac{n \sqrt{K}}{{2k}} - \frac{\sqrt{k}}{K} $$
and the only nonnegative stationary point is
$$\widehat{y} = \frac{ \sqrt{ \frac{2 \sqrt{kK}}{n} } - \frac{2k}{n \sqrt{K}} \cdot p_{l+1}}{\frac{K}{\sqrt{k}} - \frac{{2k}}{n \sqrt{K}}}. $$
Assuming, without loss of generality that \(k \le1 \le K\), we can also be assured that \(\widehat{y} < p_{l+1}\). The stationary point \(\widehat {y}\) turns out to be a minimum. Thus the maximum value of the function h is attained at the extremum of the interval \([0, p_{l+1}]\). We have
$$\begin{aligned}& h(0)= \frac{1}{\frac{2k}{n \sqrt{K}} \cdot p_{l+1}} +T,\\& h ( p_{l+1} )= \frac{1}{\frac{K}{\sqrt{k}} \cdot p_{l+1} } + \biggl(\frac{n \sqrt{K}}{2k} - \frac{\sqrt{k}}{K} \biggr) \cdot p_{l+1} + T. \end{aligned}$$
We can get rid of \(p_{l+1}\) by using the bounds on the resistances. We obtain
$$\begin{aligned}& h(0) \le\frac{nK}{2k} + T,\\& h ( p_{l+1} ) \le\frac{\sqrt{k}}{\sqrt{K}} + \frac{n \sqrt{K}}{2k\sqrt{k}} - \frac{1}{K} + T. \end{aligned}$$
The assertion easily follows from the bound
$$T \le \biggl( \frac{n \sqrt{K}}{2k} + \frac{\sqrt{k}}{K} \biggr) C' - \frac{n}{2k} - \frac{n^{2}-n}{2 \sqrt{k} \sqrt{K}} - C. $$
□
Noting that \(\frac{m}{\sqrt{K}} \le C' \le\frac{m}{\sqrt{k}} \) and \(\frac{m}{K} \le C \le\frac{m}{k} \), we get the following corollary.
Corollary 10
Let
\(G=(V,E)\)
a connected network with
n
vertices and
m
edges. Let
\(r_{ij}\), \(k \le r_{ij} \le K\), be the resistances associated to any edge. Then
$$ \frac{m^{2}}{K(n-1)} - \frac{m}{k} \le CW(G) \le C'' + \frac{nm\sqrt {K}}{2k\sqrt{k}} - \frac{n}{2k} - \frac{n^{2}-n}{2 \sqrt{k} \sqrt{K}}, $$
(15)
where
\(C'' = \max \{\frac{nK}{2k}, \frac{\sqrt{k}}{\sqrt{K}} + \frac{n \sqrt{K}}{2k\sqrt{k}} - \frac{1}{K} \}\).
If in inequality (15) we set \(k=K=1\), that is, \(r_{ij}= 1\) for all \((i,j) \in E\), we get
$$\frac{m(m-n+1)}{n-1} \le CW(G) \le\frac{n(m-n+1)}{2} $$
i.e. the bounds provided by Yang in [18], Theorem 3.13 and Corollary 3.14 for the global cyclicity index.
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