Survey
* Your assessment is very important for improving the workof artificial intelligence, which forms the content of this project
* Your assessment is very important for improving the workof artificial intelligence, which forms the content of this project
Graph Theory Conference in honor of Egawa’s 60th birthday, September 10 to 14, 2013 On the Resistance–Distance Spectral Radius A. Dilek Maden ∗ A. Sinan Cevik † In this study, we present our results for the maximum eigenvalue λ1 = λ1 (G) (i. e., the spectral radius) of the resistance–distance matrix R = (Rij ). We also provide some lower and upper bounds for λ1 (G) for molecular graphs and a few Nordhaus–Gaddum–type results . THEOREM 1. Let G be a connected graph with n ≥ 2 vertices. Then √ ∑n √ n 2 ∑ R Rj i i=1 ≤ λ1 (G) ≤ max Rij 1≤j≤n n Ri i=1 (1) where Ri is the sum of i-th row of the matrix R . Moreover equality holds in (1) if and only if R1 = R2 = · · · = Rn . COROLLARY 2. Let G be a connected graph with n ≥ 2 vertices. Then λ1 (G) ≥ with where Kf = ∑ 2 Kf n (2) Rij denote the Kirchhoff index. Equality holds if and only if i<j R1 = R2 = · · · = Rn or G ∼ = Kn . Let G be the class of connected graphs whose resistance–distance matrices have exactly one positive eigenvalue. Denote by S = S(G) the trace of R2 . THEOREM 3. Let G ∈ G with n ≥ 2 vertices. Then √ n−1 S(G) λ1 (G) ≤ n with equality holding if and only if G ∼ = Kn . (3) THEOREM 4. Let G ∈ G with n ≥ 2 . Then √ S(G) . λ1 (G) ≥ 2 Equality holds in (4) if and only if G ∼ = K2 . (4) ∗ Department of Mathematics, Science Faculty, Selcuk University, Campus, Konya, TURKEY. E-mail:[email protected] † E-mail:[email protected] THEOREM 5. Let G be a connected graph on n > 2 vertices, m edges, maximum degree ∆ , second maximum degree ∆2 and minimum degree δ. Then [ ]} { 1 1 (∆2 − δ)2 2 λ1 (G) ≥ 2 + (n − 2) + (5) ∆ + 1 2m − ∆ − 1 ∆2 δ and { } 1 n n(n − 3)2 λ1 (G) ≥ 2 + + . (6) n δ 2m − ∆ − δ − 1 THEOREM 6. Let G be a connected graph on n > 2 vertices and m edges, maximum degree ∆ , second maximum degree ∆2 and minimum degree δ such that its complement G is also connected. Then { [ 1 1 1 2 λ1 (G) + λ1 (G) ≥ 2 + + (n − 2) + ∆+1 n−δ 2m − ∆ − 1 ] ( )( ) 1 1 (∆2 − δ)2 + + + n(n − 2) − 2m + 1 + δ 2m − ∆ − 1 ∆2 δ )( )} ( (∆ − δ2 )2 1 (7) + n(n − 2) − 2m + 1 + δ (n − 1 − δ2 )(n − 1 − ∆) with equality holding if and only if G ∼ = Kn . THEOREM 7. Let G be a connected graph on n > 2 vertices and m edges, maximum degree ∆ and minimum degree δ such that its complement G is also connected. such that its complement G is also connected. Then {( ) 1 4 1 λ1 (G) + λ1 (G) ≥ + 2 + + n δ (n − 1 − ∆) [ ]} 1 1 2 + (n − 3) + . 2m − ∆ − δ − 1 n(n − 3) − 2m + 1 + ∆ + δ THEOREM 8. Let G ∈ G with n > 2 vertices, and let G be connected. Then, √ ] √ n − 1 [√ λ1 (G) + λ1 (G) ≤ S(G) + n3 + S(G) n ∼ with equality holding if and only if G = Kn . References [ 1 ] R. B. Bapat, Resistance distance in graphs, Math. Student 68 (1999), 87-98. [ 2 ] K. C. Das, A. D. (Güngör) Maden, A. S. Çevik, On the Kirchoff index and the resistence distance energy of a graph, MATCH Commun. Math. Comput. Chem. 67(2) (2012), 541-556. [ 3 ] D. J. Klein, M. Randić, Resistance distance, J. Math. Chem. 12 (1993), 81-95. [ 4 ] E. A. Nordhaus, J. W. Gaddum, On complementary graphs, Am. Math. Montly 63 (1956), 175-177. 2