Dilation index on graphs
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If we define $k$-dilation index for an infinite graph $G$ as follow
$$mathcalD_k(G)=supBiggdfracB(v,kr)mid vin V,rin NBigg.$$
And given $2le k<k'$ where $k,k'$ are natural numbers. How ca we prove the following:
$$mathcalD_k'(G)le mathcalD_k(G)^[fraclog k' log k]+1.$$
I have tried many ways but I could note. I will be appreciated if someone gives me at least a hint.
graph-theory
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up vote
-1
down vote
favorite
If we define $k$-dilation index for an infinite graph $G$ as follow
$$mathcalD_k(G)=supBiggdfracB(v,kr)mid vin V,rin NBigg.$$
And given $2le k<k'$ where $k,k'$ are natural numbers. How ca we prove the following:
$$mathcalD_k'(G)le mathcalD_k(G)^[fraclog k' log k]+1.$$
I have tried many ways but I could note. I will be appreciated if someone gives me at least a hint.
graph-theory
add a comment |Â
up vote
-1
down vote
favorite
up vote
-1
down vote
favorite
If we define $k$-dilation index for an infinite graph $G$ as follow
$$mathcalD_k(G)=supBiggdfracB(v,kr)mid vin V,rin NBigg.$$
And given $2le k<k'$ where $k,k'$ are natural numbers. How ca we prove the following:
$$mathcalD_k'(G)le mathcalD_k(G)^[fraclog k' log k]+1.$$
I have tried many ways but I could note. I will be appreciated if someone gives me at least a hint.
graph-theory
If we define $k$-dilation index for an infinite graph $G$ as follow
$$mathcalD_k(G)=supBiggdfracB(v,kr)mid vin V,rin NBigg.$$
And given $2le k<k'$ where $k,k'$ are natural numbers. How ca we prove the following:
$$mathcalD_k'(G)le mathcalD_k(G)^[fraclog k' log k]+1.$$
I have tried many ways but I could note. I will be appreciated if someone gives me at least a hint.
graph-theory
asked Aug 23 at 8:33
majduleen zeyadeh
305
305
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add a comment |Â
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