Extremal Graphs for Randić Energy

Transcription

1 MATCH Communications in Mathematical and in Computer Chemistry MATCH Commun. Math. Comput. Chem. 77 (2017) ISSN Extremal Graphs for Randić Energy Kinkar Ch. Das, Shaowei Sun Department of Mathematics, Sungkyunkwan University, Suwon , Republic of Korea (Received December 11, 2015) Abstract The Randić matrix R(G) = (r ij ) n n of a graph G whose vertex v i has degree d i is defined by r ij = 1/ d i d j if the vertices v i and v j are adjacent and r ij = 0 otherwise. The Randić energy RE is the sum of absolute values of the eigenvalues of R(G). In MATCH Commun. Math. Comput. Chem. 74 (2015) , Maden obtained several bounds on Randić energy and characterized the extremal graphs. We found some errors in the characterization of extremal graphs. Some of these are now corrected, whereas some are stated as conjectures. 1 Introduction Let G = (V, E) be a simple connected graph with vertex set V = {v 1, v 2,..., v n } and edge set E = E(G) ( E(G) = m). Also let d i be the degree of vertex v i for i = 1, 2,..., n. The maximum degree and the minimum degree are denoted by = (G) and δ = δ(g), respectively. If the vertices v i and v j are adjacent, we denote that by v i v j E(G). Let A(G) and D(G) be the adjacency matrix and the diagonal matrix of vertex degrees of G, respectively. The Laplacian matrix of a graph G is denoted by L(G) and is defined by L(G) = D(G) A(G). The normalized Laplacian matrix L(G) of G is defined as D 1/2 (G)L(G)D 1/2 (G). Let ρ 1 ρ 2 ρ n 1 ρ n = 0 denote the eigenvalues of L(G). Then we have n i=1 ρ i = n. Another popular matrix in spectral graph theory, Randić matrix, is defined by R(G) = I n L(G) with eigenvalues θ 1, θ 2,..., θ n and label them in non-increasing order. From the definitions of the normalized Laplacian matrix

2 -78- and the Randić matrix, one can easily see that ρ i = 1 θ n i+1, i = 1, 2,..., n. The Randić energy [8] of G is defined as Note that E L (G) = RE = RE(G) = n ρ i 1 = i=1 n θ i. i=1 n θ i = RE(G). Several lower and upper bounds on Randić energy are mentioned, see [2, 6 9]. The general Randić index when α = 1 is R 1 (G) = i=1 u v E(G) 1 d u d v. The complete product G 1 G2 of graphs G 1 and G 2 is the graph obtained from G 1 G2 by joining every vertex of G 1 with every vertex of G 2. by K n and K p, q As usual, we denote (p + q = n) the complete graph and complete bipartite graph of order n, respectively. In [10], Maden obtained several bounds on the Randić energy for graphs and bipartite graphs. Moreover, she characterized the extremal graphs. But we found some errors on the characterization of extremal graphs. Some of the errors are corrected by us whereas some we state in the form of conjectures. 2 Preliminaries In this section, we list some previously known results that will be needed in the next section. Lemma 1. [5] Let G be a graph of order n without isolated vertices. Then ρ 1 (G) = ρ 2 (G) = = ρ n 2 (G) = ρ n 1 (G) if and only if G = K n. In [1], Cavers found the spectrum of G = K 1 r Ks with n = 1 + r s: s + 1,, s + 1, 1 s s }{{} s,, 1, 0. (1) }{{ s} s r r+1 r 1 Let ˆd u = 1 d v v: u v E(G) be the normalized valency of u, and let w: w N u N v 1 d w be the normalized number of common neighbors of two distinct vertices u and v. We denote this

3 -79- normalized number of common neighbors by ˆλ uv if u and v are adjacent, and by ˆµ uv if they are not. The next two results are obtained by Dam and Omidi [11]: Lemma 2. [11] Let G be a connected graph with m edges. Then G has three normalized Laplacian eigenvalues 0, θ 1, θ 2 if and only if the following three properties hold. (i) ˆd u = t d 2 u (θ 1 1)(θ 2 1)d u for all vertices u, (ii) ˆλ uv = t d u d v + 2 θ 1 θ 2 for adjacent vertices u and v, (iii) ˆµ uv = t d u d v for non-adjacent vertices u and v, where t = Lemma 3. θ1 θ2. 2 m [11] Let G be a graph obtained by adjoining a new vertex to all vertices of a regular graph G. Then G has three distinct normalized Laplacian eigenvalues if and only if G is a disjoint union of (at least two) cliques of the same size t. In this case, the non-trivial normalized Laplacian eigenvalues are 1 t and t. Lemma 4. [3] Let G be a bipartite graph of order n. For each ρ i, the value 2 ρ i is also an eigenvalue of G. 3 Characterization of graphs extremal w.r.t. Randić energy The following result is obtained in [10]. Theorem 5. [10] Let G be a connected graph of order n and P be the absolute value of the determinant of the Randić matrix R(G). Then 1 + (n 1) (n 2) P 2/(n 1) + 2 R 1 1 RE 1 + (n 1) (2 R 1 1) (2) with equality holding in both of these inequalities if and only if G = K n. In Theorem 5, the characterization of equality cases is not true. For G = K 1 rk2 (r 2), n = 2r + 1; both the equality holds in (2). Here we want to characterize the extremal graphs in Theorem 5 when = n 1. For this we need the following result: Theorem 6. Let G be a connected graph of order n with maximum degree = n 1. Then ρ 1 1 = ρ 2 1 = = ρ n 1 1 if and only if G = K n or G = K 1 r K2 with n = 2r + 1 (r 2).

4 -80- Proof. For G = K n, by Lemma 1, we have ρ 1 1 = ρ 2 1 = = ρ n 1 1 = 1 n 1. For G = K 1 rk2 (r 2), then by (1), again we get ρ 1 1 = ρ 2 1 = = ρ n 1 1 = 1 2. Conversely, let ρ 1 1 = ρ 2 1 = = ρ n 1 1 = a, (say). We have to prove that G = K n or G = K 1 r K2 with n = 2r + 1 (r 2). If a = 0, then ρ i = 1 for i = 1, 2,..., n 1. Therefore, by n ρ i = n, we get a contradiction. Otherwise, a > 0. i=1 Then the graph G has two or three distinct normalized Laplacian eigenvalues. If G has two distinct normalized Laplacian eigenvalues, then by Lemma 1, G = K n. Otherwise, G has three distinct normalized Laplacian eigenvalues, that is, ρ 1 (= 1 + a), ρ n 1 (= 1 a) and 0. Note that ρ 1 + ρ n 1 = 2. Now we assume that the vertex v has the maximum degree = n 1. Using (i) and (ii) in Lemma 2 with ρ 1 + ρ n 1 = 2, for any vertex u V (G) {v}, we have ˆd u = t d 2 u (ρ 1 ρ n 1 1)d u (3) ˆλ uv = t d u (n 1) (4) where t = ρ1 ρn 1. From the definition of ˆd 2m u and ˆλ uv, one can easily see that ˆd u = ˆλ uv + 1 n 1. Using (3) and (4) in the above relation, we get t d 2 u (ρ 1 ρ n t n t) d u 1 n 1 = 0. Hence for any vertex u V (G) {v}, its degree will satisfy the following equation: t x 2 (ρ 1 ρ n t n t) x 1 n 1 = 0. (5) This implies that there are at most two distinct degrees of vertices except vertex v in the graph G. Let x 1 and x 2 be the roots of the equation (5). Then 1 x 1 x 2 = t (n 1) < 0. (6) Thus one of the roots is negative, which cannot be a vertex degree. Hence the graph G v is regular. By Lemma 3, G = K 1 rks (r 2) with eigenvalues 1, 1 + 1, 0. Since s s ρ 1 + ρ n 1 = 2, we have s = 2, that is, G = K 1 rk2 with n = 2 r + 1 (r 2). This completes the proof of the theorem.

5 -81- Therefore we conclude the following: Theorem 7. Let G be a connected graph of order n and P be the absolute value of the determinant of the Randić matrix R(G). Then 1 + (n 1) (n 2) P 2/(n 1) + 2 R 1 1 RE(G) 1 + (n 1) (2 R 1 1). (7) If the maximum degree is equal to n 1, then both the equalities hold in (7) if and only if G = K n or G = K 1 r K2 with n = 2r + 1 (r 2). Proof. The two inequalities in (7) have been proven by Maden [10]. Moreover, these two inequalities are equalities if and only if θ 2 = θ 3 = = θ n, that is, ρ 1 1 = ρ 2 1 = = ρ n 1 1. By Theorem 6, if = n 1, then both the equalities hold in (7) if and only if G = K n or G = K 1 r K2 with n = 2r + 1 (r 2). A strongly regular graph with parameters (n, r, λ, µ), denoted by SRG(n, r, λ, µ), is an r-regular graph on n vertices such that for every pair of adjacent vertices there are λ vertices adjacent to both, and for every pair of non-adjacent vertices there are µ vertices adjacent to both. In the above Theorem 7, we obtained the characterization of extremal graphs for = n 1. We now give a conjecture for the case n 2. Conjecture 8. Let G be a connected graph of order n > 2 with maximum degree n 2 and P be the absolute value of the determinant of the Randić matrix R(G). If RE(G) = 1 + (n 1) (n 2) P 2/(n 1) + 2 R 1 1 or RE(G) = 1 + (n 1) (2 R 1 1) then G = SRG(n, d, d2 d n 1, d2 d n 1 ) or G = K 1 r K2 with n = 2r + 1 (r 2). The following corollary is obtained in [10]: Corollary 9. [10] Let G be a connected graph of order n and P be the absolute value of the determinant of the Randić matrix R(G). Then 1 + (n 1) (n 2) P 2/(n 1) + n RE(G) 1 + (n 1) with equality holding in both of these inequalities if and only if G = K n. ( ) n δ δ (8)

6 -82- Again, in Corollary 9, the characterization of equality cases are not true. In the right side, one of the present authors [6] gave the proof of the equality holding if and only if G = K n or G = SRG(n, δ, δ2 δ, δ2 δ n 1 n 1 ). Moreover, these characterizations of extremal graphs are also satisfying for the left hand side equality. Thus we arrive at the following: Corollary 10. Let G be a connected graph of order n and P be the absolute value of the determinant of the Randić matrix R(G). Then 1 + (n 1) (n 2) p 2/(n 1) + n ( ) n δ RE(G) 1 + (n 1) δ with equality holding in both of these inequalities if and only if G = SRG(n, δ, δ2 δ n 1, δ2 δ n 1 ) or G = K n. Theorem 11. [10] Let G be a connected bipartite graph of order n and P be the absolute value of the determinant of the Randić matrix R(G). Then 2 + (n 2) (n 3) P 2/(n 2) + 2 R 1 2 RE(G) 2 + (n 2) (2 R 1 2) (9) with equality holding in both of these inequalities if and only if G = K p, q with n = p + q. In the above theorem, the characterization of extremal graphs is not true. By the proof of the Theorem 2.21 in [10], one can easily check that both equalities hold in (9) if and only if G is a bipartite graph with 2 R 1 2 ρ 2 1 = ρ 3 1 = = ρ n 1 1 =. n 2 If ρ 2 = 1, then G = K p, q with n = p + q. Otherwise ρ 2 > 1. Since G is a bipartite graph, by Lemma 4, one can easily see that G is a bipartite graph with normalized Laplacian spectrum 2 R R 1 2 S(G) = 2, 1 ±,, 1 ±, 0. n 2 n 2 }{{} n 2 2 Remark 1. For odd n, both equalities hold in (9) if and only if G = K p, q with n = p + q. Using the bounds of R 1 in the above Theorem 11, Maden gave a corollary about the bounds on the Randić energy of bipartite graphs. But the characterization of extremal graphs is not true. Here we revise this result as follows:

7 -83- Corollary 12. Let G be a connected bipartite graph of order n and P be the absolute value of the determinant of the Randić matrix R(G). Then 2 + (n 2) (n 3) P 2/(n 2) + n 2 RE(G) 2 + (n 2) ( ) n 2δ with equality holding in both of these inequalities if and only if G = K ν, ν or G is the incidence graph of a symmetric 2-(ν, δ, 2δ2 2δ )-design where n = 2 ν and ν > δ. n 2 δ Proof. The proof of the inequalities is already done in [10]. From this proof, we conclude that the equality holds on each side if and only if G is a d-regular bipartite graph with 2 R 1 2 n 2d ρ 2 1 = ρ 3 1 = = ρ n 1 1 = = n 2 nd 2d. If ρ 2 = 1, then G is a complete bipartite graph. Since G is regular, we have G = K ν, ν with n = 2 ν. Otherwise ρ 2 > 1. In this case, G is a connected d-regular bipartite graph n 2d with 4 distinct normalized Laplacian eigenvalues 2, 1 +, 1 n 2d and 0. Now nd 2d nd 2d we consider L = I n 1A. Then G has 4 distinct eigenvalues d, n d 2d 2, n d 2 d 2 d n 2 n 2 and d. By [4] (see, p. 166), G is the incidence graph of a symmetric 2-(ν, k, λ)-design, (ν > k > λ > 0), with n = 2 ν with 4 eigenvalues k, k λ, k λ, k. Thus we have k = d and λ = 2 d2 2 d n 2 with ν > d. This completes the proof. Acknowledgements: The first author is supported by the National Research Foundation funded by the Korean government with Grant no. 2013R1A1A References [1] M. Cavers, The normalized Laplacian matrix and general Randi c index of graphs. Ph.D. Thesis, Univ. Regina, [2] M. Cavers, S. Fallat, S. Kirkland, On the normalized Laplacian energy and general Randić index R 1 of graphs, Lin. Algebra Appl. 433 (2010) [3] F. K. Chung, Spectral Graph Theory, Am. Math. Soc., Providence, [4] D. Cvetković, M. Doob, H. Sachs, Spectra of Graphs Theory and Applications, Barth, Heidelberg, 1995.

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