mates for the Grassmann graphJ n d with q( , ) n≥ + ≥d 3 6or with n= + ≥d 1 3 respectively were given by E.R. van Dam and J.H.Koolen, and by E.R. van Dam, W.H. Haemer, J.H. Koolen and E. Spence respectively. From the view point that J n d is the point graph of the projective q( , ) incidence structures , ; wonder whether similar situations hold for the bilinear forms graphsH n d due to the close q( , ) relationship between the projective incidence structures and the attenuated spaces,.
Some necessary backgrounds and some proposal are collected for references.
[DK05] E.R. van Dam and J.H.Koolen, A new family of distance-regular graphs with unbounded diameter, Invent. Math. 162, 189-193 (2005)
[DH 06] E.R. van Dam, W.H. Haemer, J.H. Koolen and E. Spence, Characterizing distance-regularity of graphs by the spectrum, JCT A 113 (2006) 1805-1820.
3-1 some known facts about the Grassmann graphJ n d q( , ) Let V =Fqn, consider the semilinear incidence structures
1 , ;
c. two points are collinear if and only if they meet in an (d− -dimensional subspace. 1) d. the point graph is the Grassmann graph J n d with the adjacency matrix of q( , )
2. for the incidence structure 2 , ;
c. two points are collinear if and only if they together generate a d+1 subspace, and hence they meet in an (d− -dimensional subspace. 1)
3-2 cospectral mates and distance-regular mate of the Grassmann graphJ n d q( , )
3-2 a. A non-distance-regular cospectral mate ofJ n d q( , ) in terms of the line graphs of some incidence structures:
[DH 06] E.R. van Dam, W.H. Haemer, J.H. Koolen and E. Spence, Characterizing distance-regularity of graphs by the spectrum, JCT A 113 (2006) 1805-1820.
1. the Grassmann graphJ n d is the line graph of the incidence structureq( , ) I n d : q( , ) LetV =Fqn, consider the semilinear incidence structure
( , ) , )
with the point-line incidence matrix N,
1. the point graph of I n d with adjacent matrix q( , ) 1
a. the point graph remains the same, and
Remark: Ifn≥2d, many (if not all) of the constructed cospectral graphs are not distance-regular,
for example: there is a d-dimensional subspace W that intersectsH (say) in a1 (d− dimensional 2) subspace U and not contained in H for each i. i
Question: Do similar arguments work for the bilinear forms graphs in terms of the attenuated spaces?
There is a correspondence between bipartite regular graphs with 5 eigenvalues and so called partial geometric designs. Examples of the latter are transversal designs, and these form the key to the construction of graphs cospectral with distance-regular antipodal covers of complete bipartite graphs.
The incidence structure between the two biparts of such a cover is a (square) resolvable transversal deigns (also called a symmetric net). A transversal design is a design of points and lines, such that all blocks have the same size, each point is in the same number of blocks, and such that the points can be partitioned into groups, such that each block intersects each group in one point, and such that two points from different groups meet in a constant numberμpoints.
Lemma [Lemma 9.3.2, BCN 269]
3-2 b: a non-transitive distance-regular mate of the Grassmann graphJq(2e+1, )e
[DK05] E.R. van Dam and J.H.Koolen, A new family of distance-regular graphs with unbounded diameter, Invent. Math. 162, 189-193 (2005)
The case ( , )n d =(2e+1, )e
Let V =Fq2e+1 with a fixed hyperplane H, consider the semilinear incidence structure
3 V , 1 2; #
2 1 L H
e
⎡ ⎤
= ⎢⎣ − ⎥⎦ , the line B∈ is incident to the e-dimensional subspaces of H containing B. L2
Note that in the semilinear incidence structure 3 V , 1 2; # L L
3. two points are collinear if and only if they meet in an (e− -dimensional subspace. 1) 4. its point graph is the Grassmann graph Jq(2e+1, )e with 1
6. its the line graph (or called the block graph), with 1 1
becauseNN andt N N have the same nonzero eigenvalues. t Theorem [ DK05]
Let G be the graph with vertex set
all(e+ -dimensional subspaces of V not contained in H, together with 1) the(e− -dimensional subspaces of H, where 1)
1. two vertices of the 1st kind are adjacent if they intersect in an e-dimensional subspace;
2. a vertex of the 1st kind is adjacent to a vertex of the 2nd kind if the first contains the second;
3. two vertices of the 2nd kind are adjacent if they intersect in an(e− dimensional subspaces. 2) Then G is distance-regular with the same parameters as that of the Grassmann graphJq(2e+1, )e , not vertex-transitive and hence not isomorphic toJq(2e+1, )e .
1. G is distance - regular
a. a graph cospectral with a distance-regular graph Γ with diameter e is itself distance regular if for every vertex the number of vertices at distance e is the same as in Γ ; b. sincek in G is indeed the same as in the Grassmann graph, and hence G is e
distance-regular; c. the parameters of a distance-regular graph follows from its spectrum, G has the same
parameters asJq(2e+1, )e . 2. G is not vertex-transitive:
Question: Do similar arguments work forJq(2 , )e e ? (H dq +2, ),d H dq( +1, )d orH d d ? q( , )
3-3 the bilinear forms graphH n d q( , )
⎣ ⎦ lines in the semilinear incidence structure;
2. each line is incident to q points, and each point is incident to n
3. two points are collinear if and only if they meet in an (d− -dimensional subspace. 1) 4. the point graph of π1 is the bilinear forms graph H n d with q( , ) adjacency graph where N is the point-line incidence matrix of π1.
.
3-4 candidates for distance-regular mate of the bilinear forms graphsH n d q( , ) (all need to be further checked)
3-4 a: Searching for cospectral mates of the the bilinear forms graphs
3-4 b: Searching for distance regular mates of the the bilinear forms graphs The case ( , )n d =(d+1, )d
Consider the semilinear incidence structure
( )
2 d( ,V W), d 1(H W, ); #
π = ℑ ℜ ℑ∪ − .
(The condition B⊄H considered in ℜ needs double checks, one check point is that adjacency matrix where N is the point-line incidence matrix of π2.
5. the line graph G of π2 is defined on the vertex set consisting of all (d+ -dimensional 1) subspaces of V not contained in H, together with the (d− -dimensional subspaces of H 1) meeting trivially with W, where
1. two vertices of the 1st kind are adjacent if they intersect in an e-dimensional subspace;
2. a vertex of the 1st kind is adjacent to a vertex of the 2nd kind if the first contains the 2nd;
3. two vertices of the 2nd kind are adjacent if they intersect in an (d− dimensional subspaces. 2) Claim: the line graph G is distance-regular with the same parameters as that of the bilinear form graphH dq( +1, )d , not vertex-transitive and hence not isomorphic toH dq( +1, )d .
3.5 A non-distance-regular cospectral mate of ( , )J n d E.R. van Dam, W.H. Haemers, J. H. Koolen, E. Spence,
Journal of Combinatorial Theory Series A 113 (2006) 1805-1820
A constructions of cospectral mates in terms of switching tool by Godsil and McKay:
Theorem [Godsil switching G82]Let G be a graph and let ∏ = { ,D C C1, 2,...,Cm} be a partition
Question: Does Godsil switching preserve the walk-regularity of graphs?
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