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r = arg in Java Access qr codes in Java r = arg

r = arg use jboss qr code 2d barcode maker toproduce qr codes for java QR Codes r R(v0 ,vd ). et (vi , vi+1 ) . 1 (G vi ) y (13.12). In the following, we g ive some simple lemmas on the lower bound and upper bound of the metric used in the keep-connect algorithm. Lemma 13.5.

1 [The lower bound of the MTEKC(y) metric] For each route, the MTEKC(y) employing the Fiedler-value metric has the following property:. Connectivity-aware network lifetime optimization et (vi , vi+1 )W (vi ) y n 2 1 n 1 mini dvi (G) (n 2)n 2(n 1)m y d 1 y d 1 et (vi , vi+1 ). et (vi , vi+1 ),. (13.13). where dvi is the degre QRCode for Java e of node vi in the graph, n is the number of vertices in the graph, and m is the number of edges in the graph. P ROOF. To prove these inequalities, we require the upper bound of the Fiedler value as follows (stated in Lemma 13.

3.5). Consider G(V, E) and let dvi (G) be the degree of node vi in graph G.

Then, 0 < 1 (G) n min dvi (G). n 1 i (13.14).

Now, consider the grap h G vi obtained from graph G by removing node vi and all edges connected to node vi . Obviously, 1 (G vi ) n 1 n 1 min dvi (G vi ) min dvi (G), n 2 i n 2 i (13.15).

since the minimum degr Java qrcode ee of graph G vi is smaller than or equal to the minimum degree of graph G. Now, using the keep-connect algorithm with the Fiedler value, we have. d 1 d 1 et (vi , vi+1 ) W (vi ) =. y i=0 i=0 et (vi , vi+1 ) 1 (G vi ) y y d 1 Since n mini dvi n 2 (n 1)mini dvi (G). (13.16) et (vi , vi+1 )..

dvi = 2m, we have 1 mi ni dvi (G). n 2m (13.17). Then, we obtain the se QRCode for Java cond inequality n 2 (n 1)mini dvi (G). y d 1 et (vi , vi+1 ) . (n 2)n 2(n 1)m et (vi , vi+1 ). (13.18).

Lemma 13.5.2 [The uppe r bound of the MTEKC(y) metric] For each route, the MTEKC(y) employing the Fiedler-value metric has the following property:.

13.5 The upper bound on the energy consumption et (vi , vi+1 ) W (vi ) y 1 2( (G) 1)[1 co QR-Code for Java s( /(n 1))]. y d 1 et (vi , vi+1 ), (13.19). where (G) is the edge -cut or edge connectivity of the graph. The edge-cut/edge connectivity is de ned as the minimal number of edges whose removal would result in a disconnected graph. n is the number of vertices in the graph.

P ROOF. As in Lemma 13.5.

1, we use the lower bound of the Fiedler value in Lemma 13.3.6.

Consider G(V, E) and let (G) be the edge-cut of the graph. Then, 1 (G) 2 (G)[1 cos( /n)]. Now, consider the graph G vi obtained from graph G by removing node vi and all edges connected to node vi .

From the upper bound of the Fiedler value, we have 1 (G vi ) 2 (G vi )[1 cos( /(n 1))]. Since (G vi ) (G) 1, 1 (G vi ) 2[ (G) 1][1 cos( /(n 1))]. By following a proof similar to that in Lemma 13.

5.1, we can obtain. d 1 d 1 (13.20). (13.21). et (vi , vi+1 ) W (vi ) y = i=0 i=0 et (vi , vi+1 ) 1 (G vi ) y y d 1 1 2[ (G) 1][1 cos( /(n 1))]. et (vi , vi+1 ).. Lemma 13.5.3 [Complete graph] For a complete graph, the MTEKC(y) route employing the Fiedler-value metric is the same as the MTE route.

P ROOF. From the de nitions of the MTE route and MTEKC(y) with Fiedler-value route, we have the following inequalities. d(r ) 1 d(r ) 1. et (vi , vi+1 ) . i=1 d(r ) 1 i=1 i=1 et (vi , vi+1 ),. d(r ) 1 i=1 (13.22). et (vi , vi+1 ) 1 (G vi ) y et (vi , vi+1 ) . 1 (G vi ) y (13.23). We note that removing one node from a complete graph with n nodes results in another complete graph with n 1 nodes. Therefore, we have 1 (G vi ) = n 1. On simplifying (13.

23), we have. Connectivity-aware network lifetime optimization d(r ) 1. d(r ) 1. et (vi , vi+1 ) . i=1 i=1 et (vi , vi+1 ).. (13.24). On combining this with (13.22), we have d(r ) 1 d(r ) 1. et (vi , vi+1 ) =. i=1 i=1 et (vi , vi+1 ).. (13.25). Since it is impossible to have two different routes with the same total transmit energy in random network deployment for xed source node v0 and destination node vd , we conclude that r = r . Now we are ready to develop the upper bound on the energy consumed in the MTEKC(y) algorithm with the Fiedler value. The following theorem gives the upper bound on the consumed energy.

Theorem 13.5.1 The energy consumed for MTEKC(y) using the Fiedler value satis es the following upper bound:.

d(r ) 1. et (vi , vi+1 ). (n 1)m n(n 2)( ( QR Code 2d barcode for Java G) 1)[1 cos( /(n 1))]. y d(r ) 1. et (vi , vi+1 ).. (13.26).
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