THE LEAST POSSIBLE EIGENVALUE OF A
SUPER LINE MULTIGRAPH
THOMAS ZASLAVSKY
Not long ago, Bagga, Beineke, and Varma [1] defined the super line multigraph of a simple
graph Γ = (V, E) to be the graph Mr (Γ) whose vertex set is Pr (E), the class of r-element
subsets of the edge set, and with an adjacency R ∼ R0 (where R, R0 ∈ Pr (E)) for every edge
pair (e, f ) with e ∈ R and f ∈ R0 such that e and f are adjacent in Γ. Thus, the number
of edges joining R and R0 in Mr (Γ) is the number of such ordered edge pairs (e, f ). The
simplest example is M|E| (Γ), which has a single vertex but may have many loops. The super
line multigraph generalizes the line graph, which equals M1 (Γ), but there is the significant
difference that when r ≥ 2 the super line multigraph is always connected, as long as Γ has
fewer than r components consisting of a single edge.
The adjacency matrix A(Mr (Γ)), unlike that of the simplified super line graph (where the
multiple edges are reduced to single edges), has good eigenvalue properties;
in particular,
q−1
Bagga and Ferrero [2] proved that its least eigenvalue is not less than −2 r−1 , where q := |E|
and 1 ≤ r ≤ q. This fact is an elementary extension of a familiar property of the line graph
L(Γ); but Bagga, Ellis, and Ferrero went deeper by showing exactly when this minimum
really is an eigenvalue. Recall that an eigenvalue of a graph is defined as an eigenvalue of its
adjacency matrix.
Theorem 1 (Bagga, Ellis, and Ferrero [3]). For 1 ≤ r ≤ q, −2 q−1
is an eigenvalue of
r−1
Mr (Γ) if and only if Γ contains either an even circle or a pair of edge-disjoint odd circles in
the same connected component.
This necessary and sufficient condition can be interpreted as saying that the even-circle
matroid of Γ is dependent. This matroid, introduced by Tutte [9] and then Doob [6], is the
linear dependence matroid of the columns of the unoriented incidence matrix
of Γ. (These
q−1
terms will be defined.) Thus, I wondered whether the multiplicity of −2 r−1 might be equal
to the nullity of the even-circle matroid, which is a positive number if and only if the columns
of the unoriented incidence matrix are dependent. This turns out to be true. Thus, defining
p := |V | and b(Γ) := the number of bipartite components of Γ, we have:
Theorem 2. For 1 ≤ r ≤ q, −2 q−1
is an eigenvalue of Mr (Γ) with multiplicity q −p+b(Γ).
r−1
It is most natural to prove this result in the context of signed and bidirected graphs. A
bidirected graph (a concept due to Jack Edmonds) is a graph in which each end of each edge
has been oriented, either into or away from its incident vertex. Thus, each edge has two
arrows, one at each end. We indicate the arrow at the end (v, e) where e is incident with v
by a sign: η(v, e) = +1 if the arrow points into v, and −1 if the opposite. Thus, a bidirection
Date: November 19, 2010.
2000 Mathematics Subject Classification. Primary 05C50; Secondary 05B22, 15A18.
Key words and phrases. Super line graph, adjacency matrix, smallest eigenvalue, incidence matrix, bidirected graph, signed graph.
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is a sign on each end of each edge. We allow a bidirected graph B (read B as “Beta”) to
have multiple edges and loops. We write |B| for the underlying graph.
A signed graph Σ = (Γ, σ) is a graph Γ in which every edge e has a positive or negative
sign, σ(e) [7]. A bidirection of Γ, that is, a bidirected graph B whose underlying graph is Γ,
implies an edge signature σB by the formula σB (e) = −η(v, e)η(w, e) if e is an edge whose
endpoints are v and w; this gives us a signed graph ΣB := (Γ, σB ). In a signed graph, a
circle (or “circuit” or “cycle”) is positive if the product of its edges is positive, and negative
otherwise. A subgraph or edge set is called balanced if every circle in it is positive. The
adjacency matrix A(Σ) is defined to have in position (v, w) the number of positive edges
joining v and w, less the number of negative edges joining them. We write A(B) for the
adjacency matrix of A(ΣB ).
The incidence matrix of a bidirected graph B is the V × E matrix H(B) (read H as “Eta”)
whose (v, e) entry ηve equals η(v, e) if v and e are incident, and 0 if they are not. The
incidence matrix of B is what in [10, Section 8A] is called “an” incidence matrix of the
signed graph ΣB . Let b(B) denote the number of balanced components of ΣB , including
isolated vertices.
Lemma 3. The incidence matrix H(B) has rank p − b(B).
Proof. Combine Theorems 5.1(j) and 8B.1 of [10].
The net degree of a vertex v in B is the number of entering edge ends less the number of
departing ends at v, or in symbols,
X
d±
(v)
=
ηve .
B
e∈E
(Thus, a positive loop adds 0 and a negative loop adds ±2 to the net degree.) The net degree
±
vector of B is the 1 × p vector d±
B := dB (v) v∈V .
Now we define super line multigraphs. Formally, Mr (Γ), the super line multigraph of index
r of a simple graph Γ = (V, E), is a graph whose vertex set is Pr (E) and whose edges are the
quintuples [R, e, v, f, R0 ], where R, R0 ∈ Pr (E), R 3 e 3 v ∈ f ∈ R0 , e 6= f , and we regard
[R, e, v, f, R0 ] and [R0 , f, v, e, R] as the same edge. (v is redundant but it seems desirable to
keep it in the notation.) This graph may have loops: each adjacent pair {e, f } of edges in Γ
such that e, f ∈ R gives a loop at R in Mr (Γ). M1 (Γ) is the line graph L(Γ).
The super line multigraph of index r, Mr (B), of a bidirected graph B without loops is
a bidirected graph whose underlying graph is the super line multigraph of |B|, that is,
|Mr (B)| = Mr (|B|). Note that M1 (B) is the line graph Λ(B) defined in [13, 14] and A1 is
its adjacency matrix, which equals 2I − H(B)T H(B) [11, 13, 14]. To avoid complications we
do not allow the base graph B to have loops, although its super line multigraph may have
loops. An edge end has the form (R, [R, e, v, f, R0 ]), where e, f are adjacent edges of B and
v is the common vertex of e and f in B. Putting
ηr (R, [R, e, v, f, R0 ]) := η(v, e)
defines the bidirection ηr of Mr (B).
In case r = 1, R and R0 are redundant because R = {e} and R0 = {f }; thus an edge of
the line graph can be written [e, v, f ] and, in a sense, an edge of Mr (B) is an edge of the line
graph grafted onto vertices R, R0 . The important point is that the bidirection of the edge
[R, e, v, f, R0 ] in Mr (B) is the same as its bidirection in the line graph M1 (B), and therefore
the sign σr [R, e, v, f, R0 ] in Mr (B) is the same as the sign σ1 [e, v, f ] in M1 (B).
2
Define B to be Eulerian if each vertex has equal indegree and outdegree; i.e., the number of
incoming edge ends = the number of outgoing edge ends at each vertex. (Eulerian bidirected
graphs are characterized as edge-disjoint unions of certain irreducible bidirected graphs in
[5].)
Theorem
4. Let B be a bidirected graph. For 1 ≤ r ≤ q, every eigenvalue of Mr (B) is
q−1
≤ 2 r−1 .
q
For 1 ≤ r < q, 2 q−1
is
an
eigenvalue
of
M
(B)
with
multiplicity
− p + b(B).
r
r
r−1
q−1
For r = q, 2 r−1 = 2 is an eigenvalue of Mq (B) with multiplicity 1 if B is Eulerian and
0 if it is not.
Proof. The demonstration that 2 q−1
is the largest possible eigenvalue follows the lines of
r−1
the clever proof in [3]. We write Ur for the qr × q incidence matrix between r-subsets of E
and edges in E; that is, the (R, e) entry is 1 if e ∈ R and otherwise is 0. We also write
Ar := A(Mr (B)).
Lemma 5. Ar = Ur A1 UrT .
Proof. The (R, R0 ) entry in the product matrix equals the sum of all the (e, f ) entries of
A1 for which e ∈ R and f ∈ R0 . These entries are the signs σr [R, e, v, f, R0 ] of the edges
in Mr (B), which equal the signs σ1 [e, v, f ] of the corresponding edges in M1 (B). Thus, the
(R, R0 ) element of the product equals the sum of the entries of A1 in rows indexed by R and
columns indexed by R0 .
The definition of Ar is that its (R, R0 ) entry is the sum of the signs σr [R, e, v, f, R0 ] of the
edges joining R to R0 . These edges correspond to the edges [e, v, f ] of the line graph such
that e ∈ R and f ∈ R0 and their signs equal σ1 [e, v, f ]. Thus, the (R, R0 ) element of Ar also
equals the sum of the entries of A1 in rows indexed by R and columns indexed by R0 .
It follows that the matrices are equal.
q−1
Lemma 6. Ar = 2 r−1 Ur UrT − Ur H(B)T H(B)UrT .
Proof. The line-graph adjacency matrix A1 equals 2I − H(B)T H(B) [14]. Apply Lemma
5.
Since Ur H(B)T H(B)UrT is a matrix times its own transpose, its eigenvalues
are nonnegative.
q−1
It follows from Lemma 6 that all the eigenvalues of Ar are ≤ 2 r−1 .
q
T
T
The multiplicity of 2 q−1
is
the
nullity
of
U
H(B)
H(B)U
,
which
equals
−rk Ur H(B)T .
r
r
r−1
r
Our next task is to find this rank.
Let us first dispose of the case r = q. Here Uq = 1 1 · · · 1 so Uq H(B)T is the net
degree vector d±
B . The net degree vector equals the zero vector if and only if B is Eulerian.
Thus, rk Uq H(B)T is 0 in that case and 1 otherwise. The case r = q follows.
When r = 1, U1 = I so U1 H(B)T has rank rk H(B), which by Lemma 3 equals p − b(B).
That solves the case r = 1.
In general the rank of Ur H(B)T cannot be greater than that of H(B)T . It is our task to
prove that it is not less, when 1 ≤ r < q. We employ a well known fact about Ur .
Lemma 7. rk Ur = q if 1 ≤ r < q.
Proof. The standard proof is so simple, and harmonizes so
well with our matrix theme,
that
q−1
q−2
T
I repeat it here. Ur Ur is the q × q matrix with c := r−1 on the diagonal and b := r−2 off
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it. The matrix reduces by row operations to


1 | 1 1 ··· 1
 |


.
0 |
(c − b)I 
|
When 1 ≤ r < q, c and b are never equal; therefore, rk Ur = q.
The property we need of Ur H(B)T is that its rank equals that of H(B)T . We know this for
r = 1; we prove it for larger r < q by induction. It will suffice to prove the row rank does
not decrease as r increases. Row N denotes the row space of a matrix N .
Lemma 8. For row spaces we have Row Ur H(B)T ⊆ Row Ur+1 H(B)T if 1 ≤ r < q − 1.
Proof.PLet ηe denote the eth column of the incidence matrix. A row of Ur H(B)T has the
form e∈R ηe , where R ∈ Pr (E), so a vector in Row Ur H(B)T has the form
X X
ηe
αR
e∈E
R3e
where R ranges over Pr (E) and α = (αR : R ∈ Pr (E)). Similarly, a vector in Row Ur+1 H(B)T
has the form
X X
ηe
βR0
e∈E
R0 3e
0
where R ranges over Pr+1 (E) and β = (βR0 : R ∈ Pr+1 (E)). If we can solve the equation
X
X
αR
βR0 =
0
R0 3e
R3e
for β given any α, the lemma is proved. However, the left-hand side is Ur+1 β, and as Ur+1
has full column rank q, the equation Ur+1 β = γ is solvable given any γ ∈ RE .
Proposition 9. dim Row Ur H(B)T = p − b(B) for all r = 1, 2, . . . , q − 1.
Proof. We know this is true for r = 1. The lemma implies that dim Row Ur H(B)T is weakly
increasing with r, but it cannot get any bigger than dim Row H(B)T = rk H(B) = p −
b(B).
We now know the rank of Ur H(B)T . Theorem 4 follows at once.
In the case r = q, A(Mq (B)) is a 1 × 1 matrix (a). Thus, Theorem 4 says a ≤ 2 with
equality iff B is Eulerian. Combinatorially, a is twice the number of positive loops less the
number of negative loops, so the number of negative loops in Mr (B) is never smaller than
the number of positive loops, unless B is Eulerian.
Proof of Theorem 2. The incidence matrix we need here is the unoriented incidence matrix
of Γ, which has +1 in row v, column e if v and e are incident, and is otherwise zero. This is
the incidence matrix of the bidrected graph (Γ, +), in which η(v, e) = + for every incident
vertex-edge pair. The edge signs are consequently all negative; that is, Σ(Γ,+) = −Γ, the allnegative signature of Γ. A balanced circle in −Γ is an even circle, so a component is balanced
if and only if it contains no odd circles, i.e., it is bipartite. The rank of the incidence matrix
is thus p − b(Γ).
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The even-circle matroid of Γ is the matroid whose minimal dependent sets are the even
circles and the edge sets which are the union of two odd circles that have exactly one vertex
in common (odd tight handcuffs) or are the union of two vertex-disjoint odd circles and a
minimal connecting path (odd loose handcuffs). (The latter two are the minimal connected
edge sets with cyclomatic number 2, not containing an even circle.) Thus I call this the
even-circle matroid of Γ. The unoriented incidence matrix of Γ has less than full column
rank if and only if the even-circle matroid contains a (minimal) dependent set.
Corollary 10. −2 q−1
is an eigenvalue of Mr (Γ) if, and only if, Γ contains an even circle
r−1
or an odd handcuff.
Proof. We require three facts. For any graph Γ, A(−Γ) = −A(Γ). The all-positive bidirection
of a connected graph is not Eulerian unless q = 0. The eigenvalues of a disconnected graph
are those of its connected components.
Proof of Theorem 1. The essential point is that Bagga, Ellis, and Ferrero’s criterion is equivalent to that of the corollary. The non-obvious part of proving this is to show that the
existence of two edge-disjoint odd circles in one component implies the existence of an odd
handcuff or an even circle. If the two odd circles have more than one common vertex, one
can decompose their union into circles by the Euler–Hierholzer theorem [8, Theorem ??];
then it is easy to find an even circle or an odd handcuff.
I have restricted the treatment to bidirections of graphs with no loops. Loops are more
complicated but can be handled in much the same way. The treatment of line graphs in [14]
may serve as a guide.
References
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Dept. of Mathematical Sciences, Binghamton University (SUNY), Binghamton, NY 139026000, U.S.A.
E-mail address: [email protected]
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