Analyzing pathway energy and time complexity in flower families through vertex-disjoint paths

Proof. Let us examine the vertex-disjoint pathway matrix $A_p(F_n)$ associated with a flower graph. The elements of this matrix can be expressed in a structured manner:

$$A_p(F_n)=\{\begin{array}{ll}{\omega }_{ij}=0,& \text{sum of all null entries on the maindiagonal where }i=j,\\ {\mathrm{\Lambda }}_{ij}=4,& \text{max disjoint paths between innerblock,}\\ & if2\le i\le n+1,2\le j\le n+1,\\ {⋎}_{ij}=3,& \text{min disjoint paths in the inner blockwhere }i\ne j,i,j\le n+1,\\ {\mathrm{\top }}_{ij}=2,& \text{disjoint paths between inner and outerblocks}\\ & \text{if}i\ge n+2,1\le j\le 2n+1,1\le i\le n+1,j\ge n+2\end{array}$$ The following equation defines the characteristic polynomial of this $2n+1\times 2n+1$ square matrix: $$a_0{\mu }^{2n+1}+a_1{\mu }^{2n}+a_2{\mu }^{2n-1}+\dots +a_n=0,$$ where the characteristic values are denoted as ${\mu }_1,{\mu }_2,\dots ,{\mu }_{n-1},{\mu }_n,{\mu }_{n+1},\dots ,{\mu }_{2n+1}$. The squared characteristic values added together can be expressed as follows: $$\begin{array}{rl}\sum _{i=1}^{2n+1}{\mu }_i^2& =\left[\sum _{i=j}^{2n+1}\left({\omega }_{ij}\right)\right]+[\sum _{i<j}^{n+1}\left({⋎}_{ij}\right)+\sum _{i>j}^{n+1}\left({⋎}_{ij}\right)]+[\sum _{i\ge 2}^{n+1}\sum _{j\ge 2}^{n+1}({\mathrm{\Lambda }}_{ij})]\\ & +[\left(\sum _{i=1}^n\sum _{j=n+1}^{2n+1}\left({\mathrm{\top }}_{ij}\right)\right)+\left(\sum _{i=n+1}^{2n+1}\sum _{j=1}^{2n+1}\left({\mathrm{\top }}_{ij}\right)\right)+\left(\sum _{i=n+1}^{2n+1}\sum _{j=n+1}^{2n+1}\left({\mathrm{\top }}_{ij}\right)\right)]\\ & =4\mathrm{\Lambda }+3⋎+2\mathrm{\top }.\end{array}$$ Here $\mathrm{\Lambda },\mathrm{\top },\text{ and }⋎$ denotes the sum of all the elements 4, 3 and 2 vertex-disjoint path matrix respectively. 

Proof. Consider the flower graph $F_n$ with an vertex-disjoint pathway matrix of order $l\times l$, where $l=2n+1$ represents the vertices. The characteristic values ${\mu }_1,{\mu }_2,\dots ,{\mu }_l$ are obtained by solving the characteristic polynomial $|A_p(F_n)–\mu I|=0$. The pathway energy $E_p(F_n)$ is the sum of the absolute values of the characteristic values of the vertex-disjoint path matrix. The following condition describes this. $${}^2={\left(\sum _{i=1}^l|{\mu }_i|\right)}^2.$$

Moreover, this can also be written as the product of the absolute characteristic value summations:

Case 1: Proof for the pathway energy lower bounds of flower graph

From the Chebyshev’s Inequality 3, we derive that, $${}^2\ge \sum _{i=1}^l|{\mu }_i{|}^2+\frac{2}{l(l-1)}\left(\sum _{i=1}^l|{\mu }_i|\right)\left(\sum _{i=1}^l|{\mu }_i|\right).$$

From the Weighted AM–GM inequality 4, we conclude, $$\begin{array}{rl}{\left[E_p\left(F_n\right)\right]}^2& \ge \sum _{i=1}^l|{\mu }_i{|}^2+\frac{2}{l(l-1)}\left(\sum _{i=1}^l{\mu }_i{w}_i\right)\left(\sum _{i=1}^l{\mu }_i{w}_i\right)\text{where}\sum _{i=1}^l{w}_i=1,\\ & \ge \sum _{i=1}^l|{\mu }_i{|}^2+\frac{2}{l(l-1)}\prod _{i=1}^l\left({\mu }_i^{w_i}\right)\prod _{i=1}^l\left({\mu }_i^{w_i}\right),\\ & \ge 4\mathrm{\Lambda }+3⋎+2\mathrm{\top }+\frac{2}{l(l-1)}\prod _{i=1}^l{\left({\mu }_i^{w_i}\right)}^2.\text{…………. [by Theorem 4.1]}.\end{array}$$

Thus, the lower bounds of the pathway energy for flower graph is, $$E_p(F_n)\ge \sqrt{4\mathrm{\Lambda }+3⋎+2\mathrm{\top }+\frac{2}{l(l-1)}\prod _{i=1}^l{\left({\mu }_i^{w_i}\right)}^2}.$$

Case 2: Proof for the pathway energy improved lower bounds.

From the AM-GM inequalities in Eq. (2), we derive that $$\frac{1}{l(l-1)}\sum _{\begin{array}{c}i\ne j\end{array}}|{\mu }_i||{\mu }_j|⩾{\left(\prod _{\begin{array}{c}i\ne j\end{array}}|{\mu }_i||{\mu }_j|\right)}^{\frac{1}{l(l-1)}},$$

$$\begin{array}{rl}{\left[E_p\left(F_n\right)\right]}^2& \ge \sum _{i=1}^l{\left|{\mu }_i\right|}^2+l(l-1){\left(\prod _{i\ne j}\left|{\mu }_i\right|\left|{\mu }_j\right|\right)}^{\frac{1}{l(l-1)}}\\ & =\sum _{i=1}^l{\left|{\mu }_i\right|}^2+l(l-1){\left|\prod _{i=1}^l\left({\mu }_i\right)\right|}^{\frac{2}{l}}\\ & =4\mathrm{\Lambda }+3⋎+2\mathrm{\top }+l(l-1)D^{\frac{2}{l}}\text{by Theorem 4.1}.\end{array}$$

Thus, the improved lower bounds of the pathway energy for flower graph is, $$E_p(F_n)\ge \sqrt{4\mathrm{\Lambda }+3⋎+2\mathrm{\top }+l(l-1)D^{\frac{2}{l}}}.$$ 

Proof. The methodology employed to ascertain the characteristic values echoes that utilized in Theorem 4.2. Let ${\mu }_1\ge {\mu }_2\ge {\mu }_3\ge \cdots \ge {\mu }_{n-1}\ge {\mu }_n\ge \cdots \ge {\mu }_l$ denote the characteristic values of the $l\times l$ vertex-disjoint path matrix of $F_n$.

Case 1: Proof for the pathway energy upper bound of flower graph.

From the Cauchy-Schwarz inequality in the Eq. (1) we derive that, $${\left(\sum _{i=1}^l{\alpha }_i{\beta }_i\right)}^2\le \left(\sum _{i=1}^l{\alpha }_i^2\right)\left(\sum _{i=1}^l{\beta }_i^2\right).$$

To establish the upper bound, consider the expression $[E_p(F_n){]}^2\le {\left(\sum _{i=1}^l{\alpha }_i{\beta }_i\right)}^2$, where the pathway energy is bounded by the right-hand side. Substituting ${\alpha }_i=1$ and ${\beta }_i=|{\mu }_i|$ into the inequality yields the desired result.

$$\begin{array}{rl}{}^2& \le {\left(\sum _{i=1}^{l}1.|{\mu }_i|\right)}^2\le {\left(\sum _{i=1}^{l}1\right)}^2.\left(\sum _{i=1}^l|{\mu }_i{|}^2\right)\text{……………… by 1)},\\ & =l(4\mathrm{\Lambda }+3⋎+2\mathrm{\top })\text{…………………. by Theorem 4.1},\\ [E_p(F_n)]& \le \sqrt{l(4\mathrm{\Lambda }+3⋎+2\mathrm{\top })}.\end{array}$$

Thus, the upper bound of the pathway energy of the flower graph concluded by, $$E_p(F_n)\le \sqrt{l(4\mathrm{\Lambda }+3⋎+2\mathrm{\top })}.$$

Case 2: Proof for the pathway energy improved upper bound.

The Weighted Cauchy Schwarz inequality in (5) allows us to deduce that,

$${\left(\sum _{i=1}^l{a}_i{b}_i{w}_i\right)}^2\le \left(\sum _{i=1}^l{a}_i^2{w}_i\right)\left(\sum _{i=1}^l{b}_i^2{w}_i\right).$$

For proving the improved upper bound, consider $[E_p(F_n){]}^2\le {(\sum _{i=1}^l{a}_i.b_i.w_i)}^2$, such that the pathway energy of the graph is less than RHS of the inequality, and substituting $a_i=\frac{1}{(l{)}^{\frac{3}{2}}}|{\mu }_i|$ and $b_i=1$ $w_i=\sqrt{5}$, in the above inequality,

$$\begin{array}{rl}{}^2& \le (\sum _{i=1}^l\frac{{\mu }_i^2}{(l{)}^3}.\sqrt{5})\left(\sum _{i=1}^{l}1.\sqrt{5}\right),\\ & \le \frac{\sqrt{5}l}{l^3}\left(\sum _{i=1}^l{\mu }_i^2\right).l^2\sqrt{5},\\ & =5(4\mathrm{\Lambda }+3⋎+2\mathrm{\top })\text{……………………. by Theorem 4.1},\\ [E_p(F_n)]& \le \sqrt{5(4\mathrm{\Lambda }+3⋎+2\mathrm{\top })}.\end{array}$$

Thus, the improved upper bound of the pathway energy for flower graph is, $$[E_p(F_n)]\le \sqrt{5(4\mathrm{\Lambda }+3⋎+2\mathrm{\top })}.$$ 

Proof. The structure of the vertex-disjoint path matrix $A_p(S{F}_n)$ for the sunflower graph is depicted as follows,

$$A_p(S{F}_n)=\{\begin{array}{ll}{ϝ}_{ij}=0,& \text{Total count of zeros positioned along themain diagonal}i=j,\\ {\delta }_{ij}=4,& \text{Largest set of disjoint paths inside thecentral block, where }\\ & |i–j|=1\text{or}|i–j|=n-1,1\le i,j\le n+1,\\ {\varkappa }_{ij}=3,& \text{the inner block's minimum disjointpath if}i\ne j,i,j\le n+1,\\ {\varpi }_{ij}=2,& \text{vertex-disjoint paths spanning the innerand outer regions}\\ & \text{if}i\ge n+2,1\le j\le 2n+1,1\le i\le n+1,j\ge n+2\end{array}$$ The characteristic equation of this $2n+1\times 2n+1$ square matrix is given by: $$a_0{\mu }^{2n+1}+a_1{\mu }^{2n}+a_2{\mu }^{2n-1}+\dots +a_n=0,$$ where the characteristic values are denoted as ${\mu }_1,{\mu }_2,\dots ,{\mu }_{n-1},{\mu }_n,{\mu }_{n+1},\dots ,{\mu }_{2n+1}$. The sum of the squares of these characteristic values is defined as: $$\begin{array}{rl}\sum _{i=1}^{2n+1}{\mu }_i^2& =\left[\sum _{i=j}^{2n+1}\left({ϝ}_{ij}\right)\right]+[\sum _{i<j}^{n+1}\left({\varkappa }_{ij}\right)+\sum _{i>j}^{n+1}\left({\varkappa }_{ij}\right)]+[\sum _{\begin{array}{c}1\le i\le n\\ |i–j|=1\end{array}}\sum _{\begin{array}{c}1\le j\le n\\ |i–j|=n-1\end{array}}({\delta }_{ij})]\\ & +[\left(\sum _{i=1}^n\sum _{j=n+1}^{2n+1}\left({\varpi }_{ij}\right)\right)+\left(\sum _{i=n+1}^{2n+1}\sum _{j=1}^{2n+1}\left({\varpi }_{ij}\right)\right)+\left(\sum _{i=n+1}^{2n+1}\sum _{j=n+1}^{2n+1}\left({\varpi }_{ij}\right)\right)]\\ & =4\delta +3\varkappa +2\varpi .\end{array}$$

In this instance, $\delta ,\varkappa ,\text{ and }\phi$ denote the total of all elements 4, 3, and 2 respectively, present in the vertex-disjoint path matrices. 

Proof. The proof is similar to Theorem 4.2 and is left to the interest of the reader. 

Proof. The proof is similar to Theorem 4.4 and is left to the interest of the reader. 

Proof. The arrangement of elements in the vertex-disjoint path matrix $A_p(CS{F}_n)$ of a closed sunflower graph is shown below:

$$A_p(S{F}_n)=\{\begin{array}{ll}{\lambda }_{ij}=0,& \text{Sum of zeros on the matrix's maindiagonal where }i=j,\\ {\kappa }_{ij}=5,& \text{Optimal disjoint path count within theinner block}\\ & \text{where}i\ne j,i,j\le n+1,\\ {\chi }_{ij}=4,& \text{maximum possible disjoint paths between outerblock},\\ & \text{if}i\ge n+2,1\le j\le 2n+1,1\le i\le n+1,j\ge n+2.\end{array}$$

The characteristic equation associated with this $2n+1\times 2n+1$ square matrix is: $$a_0{\mu }^{2n+1}+a_1{\mu }^{2n}+a_2{\mu }^{2n-1}+\dots +a_n=0,$$ where the characteristic values are denoted as ${\mu }_1,{\mu }_2,\dots ,{\mu }_{n-1},{\mu }_n,\dots ,{\mu }_{2n+1}$.
The squared sum of these characteristic values is expressed as, $$\begin{array}{rl}\sum _{i=1}^{2n+1}{\mu }_i^2=& \left[\sum _{i=j}^{2n+1}\left({\lambda }_{ij}\right)\right]+[\sum _{i<j}^{n+1}\left({\kappa }_{ij}\right)+\sum _{i>j}^{n+1}\left({\kappa }_{ij}\right)]\\ & +[\left(\sum _{i=1}^n\sum _{j>n}^{2n+1}\left({\chi }_{ij}\right)\right)+\left(\sum _{i>n}^{2n+1}\sum _{j=1}^{2n+1}\left({\chi }_{ij}\right)\right)+\left(\sum _{i>n+1}^{2n+1}\sum _{j>n+1}^{2n+1}\left({\chi }_{ij}\right)\right)]\\ =& 0\left(\alpha \right)+5\left(\kappa \right)+4\left(\chi \right)\\ =& 5\kappa +4\chi .\end{array}$$In this context, $\kappa \text{ and }\chi$ represent the sums of all the elements 5 and 4 in the path matrices, respectively. 

Proof. The vertex-disjoint path matrix of the closed sunflower network $CS{F}_n$ has order $l\times l$, where the vertices are represented by $l=2n+1$. Solving the polynomial equation $|A_p(CS{F}_n)–\mu I|=0$ yields the characteristic values ${\mu }_1,{\mu }_2,\dots ,{\mu }_l$. Following the method in Theorem 4.2 and using terminology that has already been explained, the pathway energy $E_p(CS{F}_n)$ is defined as the total of the absolute values of these characteristic values.

Case 1: Proof for the pathway energy lower bound of $\mathbf{C}\mathbf{S}{\mathbf{F}}_{\mathbf{n}}$

From the Chebyshev’s Inequality (3), we derive that, $${}^2\ge \sum _{i=1}^l|{\mu }_i{|}^2+\frac{2}{l(l-1)}\left(\sum _{i=1}^l|{\mu }_i|\right)\left(\sum _{i=1}^l|{\mu }_i|\right).$$

From the Weighted AM–GM inequality (4), we conclude, $$\begin{array}{rl}{\left[E_p\left(CS{F}_n\right)\right]}^2& \ge \sum _{i=1}^l|{\mu }_i{|}^2+\frac{2}{l(l-1)}\left(\sum _{i=1}^l{\mu }_i{w}_i\right)\left(\sum _{i=1}^l{\mu }_i{w}_i\right)\text{where}\sum _{i=1}^l{w}_i=1,\\ & \ge \sum _{i=1}^l|{\mu }_i{|}^2+\frac{2}{l(l-1)}\prod _{i=1}^l\left({\mu }_i^{w_i}\right)\prod _{i=1}^l\left({\mu }_i^{w_i}\right)\\ & \ge 5\kappa +4\chi +\frac{2}{l(l-1)}\prod _{i=1}^l{\left({\mu }_i^{w_i}\right)}^2\text{……………. by (Theorem6.1)}.\end{array}$$

Thus, the lower bound is $$E_p(CS{F}_n)\ge \sqrt{5\kappa +4\chi +\frac{2}{l(l-1)}\prod _{i=1}^l{\left({\mu }_i^{w_i}\right)}^2}.$$

Case 2: Proof for the pathway energy improved lower bound.

From the AM-GM inequalities in Eq. (2), we derive that

$$\begin{array}{rl}{\left[E_p\left(CS{F}_n\right)\right]}^2& =\sum _{i=1}^l{\left|{\mu }_i\right|}^2+l(l-1){\left|\prod _{i=1}^l\left({\mu }_i\right)\right|}^{\frac{2}{l}}=5\kappa +4\chi +l(l-1)D^{\frac{2}{l}}\text{[by Theorem6.1]}.\end{array}$$

Thus, the improved lower bound of the pathway energy for closed sunflower graph is, $$E_p(CS{F}_n)\ge \sqrt{5\kappa +4\chi +l(l-1)D^{\frac{2}{l}}}.$$ 

Proof. The vertex-disjoint path configuration of the $CS{F}_n$ graph is elaborated in detail in the path adjacency matrix (6). It has $2n+1$ roots, implying that it contains $2n+1$ characteristic values.

The characteristic equation of (6) is $$\begin{array}{}\text{(7)}& (-\mu {)}^{2n+1}+\mathrm{tr}(-\mu {)}^{2n}+\cdots +det(A_p(CS{F}_n))=0.\end{array}$$

Upon solving the Eq. (7), the $2n+1$ characteristic values are determined to be,

$${\mu }_1={\mu }_2={\mu }_3=\cdots ={\mu }_{n-1}=5,{\mu }_n={\mu }_{n+1}={\mu }_{n+2}=\cdots ={\mu }_{2n-1}=4,$$ $${\mu }_{2n}=(n–10)(0.47)+0.47,\text{ and }{\mu }_{2n+1}=(n–10)(8.53)+83.53.$$

The Spectrum is expressed in the form of, $$\mathtt{\text{Spec}}({\mathbb{A}}_p({CSF}_n))=\left[\begin{array}{cccc}-5& -4& (n-10)(0.47)+(0.47)& (n-10)(8.53)+83.53\\ n& n-1& 1& 1\end{array}\right].$$

The exact closed sunflower graph pathway energy is given by, $$\begin{array}{rl}{E}_p[(CS{F}_n)]& =\sum _{i=1}^{2n+1}|{\mu }_i|=\mid -5\mid (n)+\mid -4\mid (n-1)\\ & +\mid (n-10)(0.47)–0.47\mid +\mid (n-10)(8.53)+85.53\mid \\ & =9n-4+\mid (n-10)(0.47)+(0.47)\mid +\mid (n-10)(8.53)+85.53\mid .\end{array}$$

The value $\mid -5\mid$ occurs $(n)$ times, $\mid -4\mid$ repeated $(n-1)$ times, $[(n-10)(0.47)]–0.47$, and $[(n-10)(8.53)]+85.53$ arrives once.

Therefore the pathway energy exact bound for closed sunflower graph is denoted by $E_p(CS{F}_n)=9n-4+\mid (n-10)(0.47)+(0.47)\mid +\mid (n-10)(8.53)+85.53\mid .$ 

Proof. In this segment to establish the upper bounds, follow the approach of characteristic values determination in Theorem 4.2. For the purpose of the proof, it is a essential to take into account the following consideration,

Case 1: Proof for the pathway energy upper bound of $\mathbf{C}\mathbf{S}{\mathbf{F}}_{\mathbf{n}}$.

To establish the upper bound, apply the Cauchy-Schwarz inequality in (1).

Consider $[E_p(CS{F}_n){]}^2\le {\left(\sum _{i=1}^l{\alpha }_i{\beta }_i\right)}^2$, where the pathway energy is bounded above. Substituting ${\alpha }_i=1$ and ${\beta }_i=|{\mu }_i|$ into this inequality yields the desired result.

$$\begin{array}{rl}{}^2& \le {\left(\sum _{i=1}^{l}1.|{\mu }_i|\right)}^2\le {\left(\sum _{i=1}^{l}1\right)}^2.\left(\sum _{i=1}^l|{\mu }_i{|}^2\right)\text{……………….. [by(1)]}\\ & =l(5\kappa +4\chi )\text{…………………… [by Theorem 6.1]}\\ [E_p(CS{F}_n)]& \le \sqrt{l(5\kappa +4\chi )}.\end{array}$$

Thus, the upper bound of the pathway energy of the closed sunflower graph concluded by, $E_p(CS{F}_n)\le \sqrt{l(5\kappa +4\chi )}.$

Case 2: Proof for the pathway energy improved upper bound.

Using the Weighted Cauchy-Schwarz inequality from Eq. (5), we deduce the improved upper bound: $[E_p(CS{F}_n){]}^2\le {\left(\sum _{i=1}^l{a}_i{b}_i{w}_i\right)}^2$ Substituting $a_i=\frac{1}{l^{3/2}}|{\mu }_i|$, $b_i=1$, and $w_i=\sqrt{5}$ in RHS gives the result.

$$\begin{array}{rl}{[E_p(CS{F}_n)]}^2& \le (\sum _{i=1}^l\frac{{\mu }_i^2}{(l{)}^3}.\sqrt{5})\left(\sum _{i=1}^{l}1.\sqrt{5}\right)\\ & =5(5\kappa +4\chi )\text{ …………………………. [by Theorem 6.1]}\\ [E_p(CS{F}_n)]& \le \sqrt{5(5\kappa +4\chi )}.\end{array}$$

The improved upper bound of the path energy for the closed sunflower graph is: $$[E_p(CS{F}_n)]\le \sqrt{5(5\kappa +4\chi )}.$$ 

Proof. The configuration of the vertex-disjoint path adjacency matrix $A_p(B{l}_n)$ for a blossom graph is outlined as follows,$$\begin{array}{}\text{(8)}& A_p(B{l}_n)=\{\begin{array}{ll}{\alpha }_{ij}=0,& \text{whenever}i=j,\\ {\beta }_{ij}=5,& \text{whenever.}i\ne j,i,j\le 2n+1.\end{array}\end{array}$$ here $\begin{array}{r}{\varrho }_1=\sum _{i=j}^{2n+1}({\alpha }_{ij}),{\varrho }_2=\sum _{i<j}\left({\beta }_{ij}\right)+\sum _{i>j}\left({\beta }_{ij}\right).\end{array}$

For this $2n+1\times 2n+1$ matrix, the characteristic equation is, $$a_0{\mu }^{2n+1}+a_1{\mu }^{2n}+a_2{\mu }^{2n-1}+\cdots +a_n=0.$$ All the characteristic values are represented by ${\mu }_1,{\mu }_2,\dots ,{\mu }_{n-1},{\mu }_n,{\mu }_{n+1},\dots ,{\mu }_{2n+1}.$

The total of the squared characteristic values can be outlined as, $$\begin{array}{rl}\sum _{i=1}^{2n+1}{\mu }_i^2& =\sum _{i=j}^{2n+1}(a_{ij})+\sum _{i<j}(b_{ij})+\sum _{i>j}(b_{ij})\\ & =\sum _{i<j}(b_{ij}{)}^2+\sum _{i>j}(b_{ij}{)}^2\\ \sum _{i=1}^{2n+1}{\mu }_i^2& =5[{\varrho }_1+{\varrho }_2]\end{array}$$ where ${\varrho }_1$ and ${\varrho }_2$ are specific terms related to the matrix components. 

Proof. In general, the blossom graph $B{l}_n$ has a vertex-disjoint adjacency matrix of order $l\times l$, with the number of vertices denoted by $l=2n+1$. The characteristic values ${\mu }_1,{\mu }_2,\dots ,{\mu }_l$ can be determined by solving $|A_p(B{l}_n)–\mu I|=0$. As similar to the method earlier in Theorem 4.2, the pathway energy $E_p(B{l}_n)$, is the total of the absolute values of these characteristic values.

Case 1: Proof for the pathway energy lower bound of blossom graph

From the Chebyshev’s Inequality (3), we derive that, $${}^2\ge \sum _{i=1}^l|{\mu }_i{|}^2+\frac{2}{l(l-1)}\left(\sum _{i=1}^l|{\mu }_i|\right)\left(\sum _{i=1}^l|{\mu }_i|\right).$$

We derive the following from the Weighted AM–GM inequality (4): $$\begin{array}{rlr}{\left[E_p\left(B{l}_n\right)\right]}^2& \ge \sum _{i=1}^l|{\mu }_i{|}^2+\frac{2}{l(l-1)}\left(\sum _{i=1}^l{\mu }_i{w}_i\right)\left(\sum _{i=1}^l{\mu }_i{w}_i\right)\text{where}& \sum _{i=1}^l{w}_i=1,\\ & \ge \sum _{i=1}^l|{\mu }_i{|}^2+\frac{2}{l(l-1)}\prod _{i=1}^l\left({\mu }_i^{w_i}\right)\prod _{i=1}^l\left({\mu }_i^{w_i}\right)\\ & \ge 5({\varrho }_1+{\varrho }_2)+\frac{2}{l(l-1)}\prod _{i=1}^l{\left({\mu }_i^{w_i}\right)}^2\text{ by Theorem7.1}.\end{array}$$

Thus, the lower bound of the pathway energy for blossom graph is,

Case (ii): Proof for the pathway energy improved lower bound.

We obtain that from the AM-GM inequality in Eq. (2). $$\begin{array}{rl}{\left[E_p\left(B{l}_n\right)\right]}^2& =\sum _{i=1}^l{\left|{\mu }_i\right|}^2+l(l-1){\left|\prod _{i=1}^l\left({\mu }_i\right)\right|}^{\frac{2}{l}}\\ & =5({\varrho }_1+{\varrho }_2)+l(l-1)D^{\frac{2}{l}}\text{by Theorem 7.1}.\end{array}$$

Thus, the improved lower bound of the pathway energy for blossom graph is, $$E_p(B{l}_n)\ge \sqrt{5({\varrho }_1+{\varrho }_2)+l(l-1)D^{\frac{2}{l}}}.$$