![](/files/happy5.png)
eigenvalues
The sum of the two largest eigenvalues ★★
Author(s): Gernert
Problem Let
be a graph on
vertices and let
be the eigenvalues of
. Is
?
![$ G $](/files/tex/b8e7ad0330f925492bf468b5c379baec88cf1b3d.png)
![$ n $](/files/tex/ec63d7020a64c039d5f6703b8fa3ab7393358b5b.png)
![$ \lambda_1 \ge \lambda_2 \ge \ldots \ge \lambda_n $](/files/tex/4dab458778057be2d191e80cde57ee36191958e9.png)
![$ G $](/files/tex/b8e7ad0330f925492bf468b5c379baec88cf1b3d.png)
![$ \lambda_1 + \lambda_2 \le n $](/files/tex/a383bcff06b594eb75570de2cd8813be94854126.png)
Keywords: eigenvalues; spectrum
Alon-Saks-Seymour Conjecture ★★★
Author(s): Alon; Saks; Seymour
Conjecture If
is a simple graph which can be written as an union of
edge-disjoint complete bipartite graphs, then
.
![$ G $](/files/tex/b8e7ad0330f925492bf468b5c379baec88cf1b3d.png)
![$ m $](/files/tex/ddaab6dc091926fb1da549195000491cefae85c1.png)
![$ \chi(G) \le m+1 $](/files/tex/0aefe0f752ac272776eec5c61c7d1a1822a3d224.png)
Keywords: coloring; complete bipartite graph; eigenvalues; interlacing
Fowler's Conjecture on eigenvalues of (3,6)-polyhedra ★★
Author(s): Fowler
Conjecture Let
be the graph of a
-polyhedron with
vertices. Then the eigenvalues of
can be partitioned into three classes:
,
(where
is nonnegative for
), and
.
![$ G $](/files/tex/b8e7ad0330f925492bf468b5c379baec88cf1b3d.png)
![$ (3,6) $](/files/tex/54cca356ae4bc518fc2bb0e473f1368b61415fcc.png)
![$ 2k + 4 $](/files/tex/bb5d192e8d6a16e90cc773b79783d49620f76ede.png)
![$ G $](/files/tex/b8e7ad0330f925492bf468b5c379baec88cf1b3d.png)
![$ K = \{3, -1, -1, -1\} $](/files/tex/acf0918d0c9caeff482349970a3636db0ddfca51.png)
![$ P = {x_1, ..., x_k\} $](/files/tex/e613a55348a12a5a8c95a79d7689bfa7ab7435bd.png)
![$ x_i $](/files/tex/dce4936db6220b56450615964eb030778cb2790f.png)
![$ i = 1, \dots , k $](/files/tex/b2dc2586f78a0a5695206b04bab6ea68dbb0ff46.png)
![$ N = - P $](/files/tex/f936c56cf996ce092f80ad0718fb169d863bf1fe.png)
Keywords: (3,6)-polyhedron; eigenvalues
![Syndicate content Syndicate content](/misc/feed.png)