On fat Hoffman graphs with smallest eigenvalue at least 3

Transcription

1 On fat Hoffman graphs with smallest eigenvalue at least 3 Qianqian Yang University of Science and Technology of China Ninth Shanghai Conference on Combinatorics Shanghai Jiaotong University May 24-28, 2017 (Joint work with Jack Koolen and Yan-Ran Li.) 1 / 24

2 1 Motivation 2 Main tool Hoffman graphs Representation of Hoffman graphs Special matrix 3 Result Special ( )-graphs Weighted graphs 2 / 24

3 In 1976, Cameron et al. showed that If G is a connected graph with smallest eigenvalue at least 2, then 1 G is a generalized line graph; or 2 G has at most 36 vertices. Denote A(G) the adjacency matrix of G. G is a generalized line graph, that is, there exists a matrix N such that A(G) + 2I = N T N, where N ij Z for all i, j. P.J. Cameron, J.-M. Goethals, J.J. Seidel and E.E. Shult, Line graphs, root systems, and elliptic geometry, J. Algebra 43 (1976), no. 1, / 24

4 Questions If G is a connected graph with smallest eigenvalue at least 3, then G is??? Suppose there exists a matrix N such that then G is??? A(G) + 3I = N T N, where N ij Z, Definition A graph G is called integrally representable of norm 3, if there exists a map ϕ : V (G) Z m for some positive integer m such that 3, if x = y; ϕ(x), ϕ(y) = 1, if x y; 0, otherwise. 4 / 24

5 Question What are the connected integrally representable graphs of norm 3? 5 / 24

6 Example Let G be the graph, then there are two integral representations ϕ 1 and ϕ 2 of G of norm 3, where ϕ 1 ( ) ϕ 1 ( ) ϕ 2 ( ) ϕ 2 ( ) f 1 f 1 f 2 f 3 f 4 f 5 f 1 f 1 f 2 f 2 f 1 f 1 f 2 f 2 f 3 f 3 f 4 f 4 f 5 f 5 g ϕ1 g ϕ2 6 / 24

7 Definition A Hoffman graph h is a pair (H, µ) of a graph H = (V, E) and a labeling map µ : V {f, s} satisfying the following conditions: 1 every vertex with label f is adjacent to at least one vertex with label s; 2 vertices with label f are pairwise non-adjacent. slim vertex f 1 f 2 fat vertex f 1 f 2 f 3 f 4 f 5 7 / 24

8 Definitions and notations f 1 f 2 h V f (h) = {f 1, f 2 }, V s (h) = {, }, N f h () = {f 1, f 2 }, Nh s() = { }, N f h () = {f 1, f 2 }, Nh s() = { }, h is 2-fat, slim graph of h is. V f (h) (V s (h)) the set of fat (slim) vertices of h. N f h (x) (N h s (x)) the set of fat (slim) neighbors of x in h; If every slim vertex has a fat neighbor, we call h fat; If every slim vertex has at least t fat neighbors, we call h t-fat. The slim graph of a Hoffman graph h = (H, µ) is the subgraph of H induced by V s (h). 8 / 24

9 Definition For a Hoffman graph h and a positive integer m, a mapping φ : V (h) Z m such that 3 if x = y and x, y V s (h); 1 if x = y and x, y V φ(x), φ(y) = f (h); 1 if x y; 0 otherwise, is called an integral representation of norm 3 of h. x1 x2 x1 ϕ 2 ( ) ϕ 2 ( ) f1 f2 f3 f4 f5 f1 x2 f2 φ( ) φ( ) φ(f 1 ) φ(f 2 ) G g ϕ2 Figure 1 9 / 24

10 Definition For a Hoffman graph h and a positive integer m, a mapping ψ : V s (h) Z m such that x1 x1 x2 ϕ 2 ( ) ϕ 2 ( ) x 3 N f f1 h (x) f2 if x = y; 1 1 ψ(x), 1 ψ(y) 1= 1 N f h (x, y) if x y; f1 f2 f3 f4 f5 x2 N f h (x, y) otherwise, G is called an integral reduced representation Figure 1 of norm 3 of h. g ϕ2 φ( ) φ( ) φ(f 1 ) φ(f 2 ) x1 x2 ϕ 2 ( ) ϕ 2 ( ) 1 1 f 1 f 2 f 3 1 f 4 1 f f1 f2 f3 f4 f5 f1 f 1 f 2 x1 x2 f2 ψ( ) ψ( ) G g ϕ2 Figure 2 f 1 f 2 10 / 24

11 Result Condition: Let G be a connected graph with an integral representation of norm 3. Conclusion: G is the slim graph of a fat Hoffman graph with an integral (reduced) representation of norm 3. Question: What are all fat Hoffman graphs with an integral (reduced) representation of norm 3? 11 / 24

12 Definition The special matrix of a Hoffman graph h is defined as follows: for any x, y V s (h), Sp(h) (x,y) = N f h (x), if x = y; 1 N f h (x, y), if x y; N f h (x, y), if x y. Eigenvalues of h are the eigenvalues of its special matrix. The smallest eigenvalue of h is denoted by λ min (h). f 3 f 4 f 5 f 1 f 2 ( 2 1 special matrix: 1 2 λ min (g ϕ2 ) = 3 ) g ϕ2 12 / 24

13 Let h be a Hoffman graph with an integral reduced representation ψ of norm 3. For any x, y V s (h), Sp(h) (x,y) = ψ(x), ψ(y) = N f h (x), if x = y; 1 N f h (x, y), if x y; N f h (x, y), if x y. 3 N f h (x) if x = y; 1 N f h (x, y) if x y; N f h (x, y) otherwise. A Hoffman graph h has an integral reduced representation of norm 3. Equivalently, there exists a matrix N, such that Sp(h) + 3I = N T N, where N ij Z for all i, j. It also implies that all Hoffman graphs with an integral reduced representation of norm 3 has smallest eigenvalue at least / 24

14 Definition Let µ be a real number with µ 1. A Hoffman graph h is called µ-saturated, if 1 λ min (h) µ; 2 no fat vertex can be attached to h in such a way that the resulting Hoffman graph has smallest eigenvalue at least µ. ( 3)-saturated ( 3)-saturated 14 / 24

15 Definition A Hoffman graph h is called decomposable, if there exists a partition {Vs 1 (h), Vs 2 (h)} of V s (h), such that ( V s 1 (h) Vs 2 (h) ) Vs 1 (h) 0 Sp(h) = Otherwise, h is indecomposable. V 2 s (h) 0. ( 3 0 Sp(g ϕ1 ) = 0 3 f 1 f 2 ) f 1 f 2 f 3 f 4 f 5 g ϕ1, decomposable 15 / 24

16 What are fat Hoffman graphs, which have an integral (reduced) representation of norm 3? What is the family H of Hoffman graphs? H is the family of fat, ( 3)-saturated, indecomposable integrally (reduced) representable Hoffman graphs of norm / 24

17 H: fat, ( 3)-saturated, indecomposable, integrally (reduced) representable of norm 3 Definition The special graph of a Hoffman graph h is the edge-signed graph S(h) := (V (S(h)), E + (S(h)), E (S(h))), where V (S(h)) = V s (h) and E + (S(h)) = { {x, y} x, y V s (h), x y, sgn(sp(h) (x,y) ) = + }, x 3 E (S(h)) = { {x, y} x, y V s (h), x y, sgn(sp(h) (x,y) ) = }. The special ε-graph of h is the graph S ɛ (h) = (V s (h), E ɛ (S(h))) for ɛ {+, }. x 3 + x 3 x1 x2 x3 x 1 x Sp(h) = x x 3 x 3 h + x1 x2 x3 Figure 1 S(h) x 3 S (h) 17 / 24

18 H: fat, ( 3)-saturated, indecomposable, integrally (reduced) representable of norm 3 Jang et al. showed that x 3 xn 2 x n 1 x n x 3 xn 2 x n 1 x n If a Hoffman graph h H has n slim vertices, then its special ( )-graph S (h) isx n one of the following: x n x 3 xn 2 x n 1 x n A n : 1 x 3 xn 2 x n 1 x n, Ã n 1 : D n : x 3 xn 3 x n 2 x n 1 x 3 x 4 xn 2 x n 1 x n x n 2 3 n 1 xn 3 n 2 x n 1 1 x 3 xn 3 x n 2 x n, Dn 1 : x 3 xn 3 x n 2 x n 1 x n 3 x 34 xn 3 xn 2 x n 2 n 1 x n 1 x n x 3 x 4 xn 2 x n 1 x n 1 n x 3 x 4 xn 3 x n 2. x n, x 3 x 4 xn 2 x n 1 x x n 1 x n 1 x 3 x 4 xn 3 x n 2 x n x 3 xn 3 x n 2 x n 1 H.J. Jang, J. Koolen, A. Munemasa and T. Taniguchi, On fat Hoffman x n graphs with smallest eigenvalue at least 3, Ars Math. Contemp. 7 (2014), no. 1, x n 1 x n 1 x 3 xn 3 x n 2 x 3 x 4 xn 3 x n 2 x n x n 18 / 24

19 H: fat, ( 3)-saturated, indecomposable, integrally (reduced) representable of norm 3 Definition A weighted graph is a pair (G, w) of a graph G and a weight function w : V (G) Z 0. The weighted special ( )-graph of a Hoffman graph h is the weighted graph (S (h), w), where S (h) is the special ( )-graph of h and w(x) := N f h (x). Question What are the necessary and sufficient conditions for a weighted graph to be the weighted special ( )-graph of a Hoffman graph in H? 19 / 24

20 H: fat, ( 3)-saturated, indecomposable, integrally (reduced) representable of norm 3 A leaf of a graph G is a vertex of valency 1 in G. A t-leaf is a leaf with its unique neighbor having valency t. Necessary conditions Let (G, w) be a weighted special ( )-graph of a Hoffman graph in H, which has n slim vertices where n 6. 1 G is isomorphic to A n, Ã n 1, D n or D n 1 and w(x) {1, 2} for any x V (G). 2 For, V (G), if and w() = w() = 2, then one of them is a leaf of G. 3 If is a 3-leaf of G, then w() = 1. 4 If is a 2-leaf of G with w() = 1, then there exists a unique 4-path x 3x 4 containing in G. Moreover, w() = 2, w(x 3) = w(x 4) = 1. 5 If G is isomorphic to Ãn 1 and w(x) = 2 for some x V (G), then there exist at least 3 vertices having weight 2 in (G, w). 20 / 24

21 H: fat, ( 3)-saturated, indecomposable, integrally (reduced) representable of norm 3 Given a weighted graph (G, w) which satisfies the above necessary conditions and has at least 6 vertices, we will construct a Hoffman graph h (G,w) H, h (G,w) has (G, w) as its weighted special ( )-graph for each fat vertex f of h (G,w), the induced subgraph of S (h (G,w) ) on N s h (G,w) (f) is connected. We call the Hoffman graphs satisfying the second condition generic. 21 / 24

22 H: fat, ( 3)-saturated, indecomposable, integrally (reduced) representable of norm 3 Construction For each pair of vertices x and y of V (G), they share exactly one common fat neighbor in h (G,w) if one of the following conditions is satisfied: There exists a path P = x... x py in G such that w(x i) = 1 for i = 1,..., p. Note that the value of p can be zero as well. Both x and y are 3-leaves of G and they have one common neighbor in G. For each pair of vertices x and y of V (G), they are adjacent in h (G,w) if one of the following conditions is satisfied: N f h (G,w) (x, y) = 1 and x y in G; N f h (G,w) (x, y) = 0 and there exists a vertex z with w(z) = 2 in (G, w) such that x and y are adjacent to z in G, except the case where x or y is a 2-leaf of G with weight 1; N f h (G,w) (x, y) = 0, one of them is a 2-leaf with weight 1 in (G, w) and the distance of x and y is 3 in G. 22 / 24

23 H: fat, ( 3)-saturated, indecomposable, integrally (reduced) representable of norm 3 How to obtain the family H Weighted graph (G, w), which satisfies the necessary conditions Generic Hoffman graph h (G,w), which has (G, w) as weighted special ( )-graph identify fat vertices identify fat vertices Hoffman graph h H Note that the for a Hoffman graph h 1 H, two fat vertices f 1 and f 2 of h 1 can be identified, if and only if N s h 1 (f 1, f 2) = and for any fat vertex f V f (h 1) {f 1, f 2}, N s h 1 (f) (N s h 1 (f 1) N s h 1 (f 2)) / 24

24 Thank you for your attention! 24 / 24