Generalizing Kirchhoff laws for Signed Graphs
Lucas J. Rusnak, Josephine Reynes, Skyler J. Johnson, Peter Ye
GGeneralizing Kirchhoff laws for Signed Graphs
Lucas J. Rusnak a, ∗ , Josephine Reynes a , Skyler J. Johnson b , Peter Ye b a Department of Mathematics, Texas State University, San Marcos, TX 78666, USA b Mathworks, Texas State University, San Marcos, TX 78666, USA
Abstract
Kirchhoff-type Laws for signed graphs are characterized by generalizing transpedances through theincidence-oriented structure of bidirected graphs. The classical 2-arborescence interpretation of Tutte isshown to be equivalent to single-element Boolean classes of reduced incidence-based cycle covers, calledcontributors. A generalized contributor-transpedance is introduced using entire Boolean classes that natu-rally cancel in a graph; classical conservation is proven to be property of the trivial Boolean classes. Thecontributor-transpedances on signed graphs are shown to produce non-conservative Kirchhoff-type Laws,where every contributor possesses the unique source-sink path property. Finally, the maximum value of acontributor-transpedance is calculated through the signless Laplacian.
Keywords:
Signed graph, Laplacian, arborescence, transpedance, Kirchhoff.
1. Introduction and Background
We introduce and characterize Kirchhoff-type Laws for signed graphs by generalizing transpedances from[3]. This is accomplished by using the incidence-theoretic approach introduced in [13] to study hypergraphicLaplacians, and the incidence-path mapping families, called contributors , from [7] that generalize cycle coversto classify various hypergraphic characteristic polynomials similar to Sachs’ Theorem [1, 8]. It was shownin [14] that if all edges are size 2 these generalized cycle covers form Boolean lattices that generalize theMatrix-tree theorem. These Boolean families are naturally cancellative when G is a graph with the trivialsingle-element classes corresponding to spanning trees — providing “conservation” for the graphic KirchhoffLaws.Transpedances were introduced in [3] as a way to study the packing and cutting problem of dissecting arectangle into squares by translating the question into a networking potential problem. A graph is associatedto each dissection and its natural flow capacity is determined to be the tree-number via the Matrix-treeTheorem. Moreover, the size of the admissible squares are ordered second-cofactors of the Laplacian, anda combinatorial interpretation of Kirchhoff’s Laws via “spanning tree flows” is obtained for any source-sinkpair where edges are labeled by signed 2-arborescences. A brief introduction to transpedances appears in ∗ Corresponding author
Email address:
[email protected] (Lucas J. Rusnak)
Preprint submitted to Elsevier September 29, 2020 a r X i v : . [ m a t h . C O ] S e p ubsection 1.1. The original investigation into transpedances is credited with leading to Tutte’s investigationinto Graph-polynomials [16]. Non-conservative Kirchhoff-type Laws for directed graphs appear in [15], thealgebraic development of potential theory appears in [2], and a formulation using ported matroids appear in[6] that analyze signed contributions of spanning forests — in this paper, we identify non-forest contributorsin Boolean equivalence classes that produce non-conservative generalizations of Kirchhoff’s Laws.A signed graph is a generalization of a graph where each edge receives a sign + − D -contributor-transpedances , which include all the Boolean classes, not just the single element ones. Kirchhoff’s Degeneracyand Energy Reversal conditions are shown to immediately hold for them. The D -contributor-transpedancevalue is then calculated for an arbitrary edge and the Boolean classes are shown to vanish if they contain apositive circle. Thus, if G is a graph, then only the trivial contributors that correspond to 2-arborescencesremain.Section 4 proves that all D -contributor-transpedances possess a unique source-sink path property, andthe trivial classes used to label the edges sort spanning trees along their source-sink path. Kirchhoff’s Cycleand Vertex Conservation Laws are shown to be a property of the trivial Boolean classes, and conservationon non-cancellative Boolean classes (negative classes) cannot be guaranteed. The maximal contributor-transpedance problem is solved in Section 5 through the signless Laplacian and the permanent. This countholds for any oriented hypergraph, and a simple permanent version of Kirchhoff’s Laws via contributors isstated. Unfortunately, the techniques for a complete general hypergraphic transpedance version are limitedby: (1) the partial parallel edges that readily appear, and (2) the lack of Boolean nature of general classes.We hope the general class partial order introduced in Section 2 may serve to remedy this. Additionally, theconnection to Tutte-functions and ported matroids require further study to relate to the work in [5, 6]. A 2 -arborescence of G is a pair of disjoint rooted trees whose union spans G . Kirchhoff’s Laws with unitresistance has been shown to be equivalent to 2-arborescence counts whose values are commensurable withthe tree number of the graph [3]; non-unit resistance is simply a weighted version of this combinatorial result,while directed graphs produce a non-conservative version of Kirchhoff’s Laws [15]. Let u , u , w , w ∈ V ( G ) ,2nd define ⟨ u w , u w ⟩ be the number of 2-arborescences with one component rooted at u and containing w , and the other component rooted at u and containing w . v v v v v v Figure 1: All 2-arborescences of the graph of the form ⟨ v v , v v ⟩ . Given graph G with source u and sink u , the w w -transpedance of G is [ u u , w w ] = ⟨ u w , u w ⟩ − ⟨ u w , u w ⟩ . It was shown in [3, 15] that the value [ u u , w w ] is also the (ordered) second cofactor of the Laplacian of G .Let L G be the Laplacian of G , let L ( G ; u ,w ) be the u w -minor of L , let L ( G ; u u ,w w ) be the u w -minorof L ( G ; u ,w ) , and define L ( G ; u , w ) iteratively for vertex vectors u , w . Specifically, [ u u , w w ] is the valueof the u w -cofactor in the u w -minor using the positional sign of u w in L G and the positional sign of u w in L ( G ; u ,w ) . Example 1.1.1.
Since, in Figure 1, there are no -arborescences of the form ⟨ v v , v v ⟩ , the transpedancevalue [ v v , v v ] = is assigned to the edge between v and v . Note that transpedances are directional, so [ v v , v v ] can be regarded as the potential drop from v to v with source v and sink v . Thus, [ v v , v v ] would be − , but would arise from a different the set of -arborescences. Edge labeling by transpedances produces a combinatorial Kirchhoff’s Laws that are summarized as fol-lows:
Theorem 1.1.2 ([3, 15]).
Let G be a graph with tree number τ ( G ) , the following hold: (Degeneracy) [ u u , w w ] = [ u u , w w ] = , (Energy Reversal) [ u u , w w ] = −[ u u , w w ] = −[ u u , w w ] , (Cycle Conservation) [ u u , w w ] + [ u u , w w ] + [ u u , w w ] = , (Vertex Conservation) ∑ y ∶ y ∼ w l vy [ u u , w y ] = τ ( G ) δ u w − τ ( G ) δ u w ,where δ uw = if u = w , and is otherwise. Part (1) establishes that degenerate transpedances have a value of 0, part (2) is reversal of flow, (3) impliesboth path concatenation and cycle-conservation, and (4) is vertex-conservation, with the exception of thesource and sink where the edges have natural flow of τ ( G ) out of the source, and into the sink.3 xample 1.1.3. The transpedance labeling of the graph in Figure 1 with source v and sink v appears inFigure 2. The four -arborescences in Figure 1 are assigned to the directed adjacency between v and v . v v v v v v [ v v , v v ] = 11[ v v , v v ] = 4[ v v , v v ] = 4 [ v v , v v ] = 1 [ v v , v v ] = 1[ v v , v v ] = 1[ v v , v v ] = 3 Figure 2: A transpedance labeling of G with source v and sink v There are spanning trees, represented as the net inflow and outflow from v and v , respectively. It iseasy to check that the directed cycle sums relative to source v and sink v are zero. Also, the in/out vertexsums are also zero — with the exception of the source and sink, whose values are the tree number .1.2. Incidence Orientations and Signed Graphs An incidence hypergraph is a quintuple G = ( V, E, I, ς, ω ) consisting of a set of vertices V , a set of edges E , a set of incidences I , and two incidence maps ς ∶ I → V and ω ∶ I → E . An orientation of an incidencehypergraph G is a signing function σ ∶ I → {+ , − } , which produces a V × E integer incidence matrix H G .The Laplacian matrix of G is defined as L G ∶= H G H TG = D G − A G , where the degree matrix is the numberof incidences at a vertex, and the adjacency matrix has entries determined by the sign − σ ( i ) σ ( j ) , where i and j are the incidences of an adjacency [13]. A bidirected graph is an oriented hypergraph in which everyedge is a 2-edge. Bidirected graphs first appeared in integer programming [9], and later were shown to beorientations of signed graphs [17]. The edge labeling of a bidirected graph by the adjacency sign − σ ( i ) σ ( j ) is called a signed graph , and a graph can be regarded as a signed graph with all edges positive. v v v e e L G = ⎡⎢⎢⎢⎢⎢⎢⎢⎢⎣ − − − − ⎤⎥⎥⎥⎥⎥⎥⎥⎥⎦ Figure 3: An oriented hypergraph G and its Laplacian. A contributor of G is an incidence preserving map from a disjoint union of —→ P ’s with tail t and head h into G defined by c ∶ ∐ v ∈ V —→ P → G such that c ( t v ) = v and { c ( h v ) ∣ v ∈ V } = V . Due to the nature of theincidence-maps it is possible for a path to fold back on itself creating a backstep of the form v, i, e, i, v —4hese are the entries in the hypergraphic degree matrix. A contributor can be regarded as a permutationclone that is a generalized cycle covers similar to Sachs’ Theorem to determine characteristic polynomialcoefficients [8, 1, 7]; contributors naturally form Boolean lattices when G is a bidirected graph [14]. The setof contributors of an oriented hypergraph is denoted C ( G ) . Throughout, let U, W ⊆ V with ∣ U ∣ = ∣ W ∣ , whilea total ordering of each set will be denoted by u and w , respectively. Let C ( G ; u , w ) be the set of restricted contributors in G where c ( u i ) = w i , and two elements of C ( G ; u , w ) are said to be [ u , w ] -equivalent . Let ̂ C ( G ; u , w ) be the set obtained by removing the u → w mappings from C ( G ; u , w ) ; the elements of ̂ C ( G ; u , w ) are called the reduced [ u , w ] -equivalent contributors. To avoid confusion between an algebraic cycle and agraph component that forms a closed walk we refer to the graph images as circles , and backsteps will beconsidered separate from circles as they do not complete an adjacency. Example 1.2.1.
Figure 4 shows four contributors of the hypergraph G from Figure 3. The tail of each pathis labeled with a different shape and mapped to its corresponding vertex in G ; the heads are then mapped toagain cover the vertices. The bottom two contributors consist of all backsteps and are clones of the identitypermutation (but are distinct contributors). ⇒ id id (23) (123) v v v Figure 4: Contributor examples with associated permutations.
The contributors above each identity-clone are [ v , v ] -equivalent as there is a path mapping to the v v -adjacency in each contributor. Transpedances are second cofactors and these arise naturally as the coefficients of the degree-2 monomialsof the total minor polynomial for integer matrix Laplacians [10], where the coefficients are determined bysums of reduced contributors. Let χ P ( L G , x ) and χ D ( L G , x ) be the total minor polynomial as determinedby the permanent and determinant of X − L G , respectively, where the ij -entry of X is x ij . The total minorpolynomial is calculated as follows: Theorem 1.2.2 ([10], Theorem 3.1.2).
Let G be an oriented hypergraph with Laplacian matrix L G , then χ P ( L G , x ) = ∑ [ u , w ] ⎛⎜⎜⎜⎝ ∑ c ∈̂ C ( L ( G ) ; u , w ) sgn ( c )≠ (− ) nc ( c )+ bs ( c ) ⎞⎟⎟⎟⎠ ∏ i x u i ,w i , χ D ( L G , x ) = ∑ [ u , w ] ⎛⎜⎜⎜⎝ ∑ c ∈̂ C ( L ( G ) ; u , w ) sgn ( c )≠ (− ) ec ( ˇ c )+ nc ( c )+ bs ( c ) ⎞⎟⎟⎟⎠ ∏ i x u i ,w i .where ec ( ˇ c ) represents the number of even-cycles in the unreduced contributor of c , bs ( c ) represents thenumber of backsteps, and nc ( c ) represents the number of negative components. The hypergraph L ( G ) is the zero-loading of G and extends the hypergraph to a uniform hypergraph andassigns a weight of 0 to all new incidences. Thus, a reduced contributor exists in G if and only if it isnon-zero. As discussed in [10], the u → w maps need not exist in G as they are removed, but the maps mustbe allowed to exist a priori their removal, which is remedied by the zero-loading L ( G ) . To simplify notationlet ̂ C ≠ ( L ( G ) ; u , w ) be the set of non-zero reduced contributors in L ( G ) ; that is, the reduced contributorsthat reside in G . Example 1.2.3.
Consider the value [ v v , v v ] = along the top edge in Figure 2. To find this value usingthe total minor polynomial we first find all contributors where v ↦ v and v ↦ v , then remove thesetwo maps — these maps are allowed to exist in the zero-loading L ( G ) and are subsequently removed, theremaining objects need to exist in G to avoid mapping to . There is only one such reduced contributor thatlies in G , shown in Figure 5. v v v v v v v v v v v v Figure 5: Reduced contributors find coefficients of the total minor polynomial as generalized cycle covers.
Using the determinant signing function in Theorem 1.2.2, and assuming every edge is positive (as in agraph), we have ec ( ˇ c ) = since there are even circles in the non-reduced contributor, while nc ( c ) = and bs ( c ) = . Thus, the sign of the contributor is (− ) + + = . This is the value of the coefficient of x v v x v v as well as [ v v , v v ] , as depicted in Figure 2. We show that the contributor mappings produce a natural adjacency labeling for oriented hypergraphsand a non-conservative generalization of Kirchhoff’s Laws for signed graphs. Moreover, this process is notlimited to the determinant. If every adjacency is negative, then the permanent counts the total number ofcontributors for an edge; thus, providing a maximum value for potential on each edge.6 . Contributor Classes and Arborescences
Two contributors are said to be tail-equivalent if the image of their tail-incidences agree. Each identity-contributor in Figure 4 is tail-equivalent to the contributor above it, as they both enter the same edge, butcomplete to different permutations. Clearly, there is exactly one identity-contributor in each tail-equivalencyclass. The elements of a tail-equivalence class are partially ordered by c ≤ c ′ if (1) the set of circles of c iscontained in the set of circles of c ′ , or (2) the set of incidences are equal and c has more connected componentsthan c ′ . Thus, the identity-contributor, having the most components and an empty set of circles, is the leastelement of each poset, while the number of contributors on a single k -edge follow the Stirling numbers of thefirst kind. Two examples appear in Figure 6. Figure 6: Tail-equivalence classes from Figure 4.
The concept of tail-equivalence is a generalization of circle activation classes of bidirected graphs in [14],where extending a backstep into its unique directed adjacency was called unpacking , and folding a directedadjacency back into a backstep was called packing . These operations are well-defined and inverses in abidirected graph, while in an oriented hypergraph only packing is well-defined on larger edges. Contributorsthat were packing/unpacking equivalent were grouped into activation classes and ordered as new circlesappear.We adopt the convention of referring to a tail-equivalence class as an activation class when G is a bidirectedgraph. Let A( G ) denote a tail-equivalence class of G . As with restricted and reduced contributors we let A( u ; w ; G ) be the elements of tail-equivalency class A( G ) where u i ↦ w i , and ˆ A( u ; w ; G ) be the elements of A( u ; w ; G ) with u i ↦ w i removed for each i . From [14], the activation classes and their restricted subclasses(order ideals) of a bidirected graph are Boolean. Lemma 2.1.1 ([14], Lemma 3.6).
For a bidirected graph G , all activation classes of G are Boolean lat-tices. It was also shown in [10] that the reduced contributors in single element activation classes ˆ A ≠ ( u ; w ; L ( G )) are unpacking equivalent to k -arborescences. 7 heorem 2.1.2 ([10], Theorem 3.2.4). In a bidirected graph G the set of all elements in single-element ˆ A ≠ ( u ; w ; L ( G )) is unpacking equivalent to k -arborescences. Moreover, the i th component in the arbores-cence has sink u i , and the vertices of each component are determined by the linking induced by c − betweenall u i ∈ U ∩ W → U or unpack into a vertex of a linking component. Example 2.1.3.
Consider the graph from Figure 1 as an incidence-graph. Each identity-contributor hasno circles and the backsteps may be unpacked to produce new cycles. Since every edge contains a uniqueadjacency, the contributors are ordered by their circle sets. Moreover, the subclass of where v i ↦ v j is anorder ideal. Three activation classes appear in Figure 7 along with their v ↦ v subclasses highlighted. Thetop contributor in the rightmost figure in Figure 7 is a trivial v ↦ v subclass. Additionally, the removal ofthe v ↦ v map leaves a rooted spanning tree ( -arborescence). v v v v v v Figure 7: Three Boolean activation classes for the given graph and their v ↦ v activation subclass (darker). To see how a -arborescence is formed consider the middle activation class in Figure 7 where v ↦ v .Remove the v ↦ v map and then take the second order ideal induced by v ↦ v — this gives the middle figurein Figure 8. The removal of the v ↦ v mapping (and unpacking any backsteps) yields the -arborescenceon the right of Figure 8. ⇒ ⇒ Figure 8: A trivial [ v v , v v ] -reduced activation class unpacks into a 2-arborescence. We show that Tutte’s transpedances are actually statements about trivial, single-element, activationclasses. Since single-element trivial activation classes will appear repeatedly, let ̂ C ≠ ( L ( G ) ; u u , w w ) bethe non-zero elements of ̂ C ( L ( G ) ; u u , w w ) in trivial activation classes.8 .2. Contributor Arborescences The 2-arborescences that arise from trivial activation classes need not be the same as Tutte’s. A 2-arborescence for the transpedance calculation [ u u , w w ] will be called a Tutte- -arborescence , while a2-arborescence described as an element of ̂ C ≠ ( L ( G ) ; u u , w w ) will be called a contributor- -arborescence .Let F be a Tutte-2-arborescence in the calculation of [ u u , w w ] ; the sign of F (relative to [ u u , w w ] ), denoted sgn T ( F ) , is + ⟨ u w , u w ⟩ and − ⟨ u w , u w ⟩ . Tutte and contributor-2-arborescences are related via the Linking Lemma andthe number of cycles that are formed. Lemma 2.2.1.
There is a bijection between Tutte- -arborescences of the form [ u u , w w ] and contributor- -arborescences from ̂ C ≠ ( L ( G ) ; u u , w w ) . Proof.
Let u and u be the source and sink, respectively, and let w and w be two vertices. Part I:
Let F be a Tutte-2-arborescence for [ u u , w w ] . There are two cases based on sgn T ( F ) . Case 1 ( sgn T ( F ) = + ) : If sgn T ( F ) = +
1, then u and w are in one component, and u and w arein the other. Reverse the path from u and w and u and w within each component. Introduce edgesdirected u ↦ w and u ↦ w to complete two disjoint cycles. Note that these edges need not exist in G asthey exist in the 0-loading and will be removed in the reduced contributor. Next, pack all adjacencies awayfrom each cycle into backsteps and remove the u ↦ w and u ↦ w adjacencies. Since there are no morecircles, the resulting object is in ̂ C ≠ ( L ( G ) ; u u , w w ) . Case 2 ( sgn T ( F ) = − ) : If sgn T ( F ) = −
1, then u and w are in one component, and u and w are inthe other. This is identical to case 1, except the introduction of edges directed u ↦ w and u ↦ w formone cycle. Part II:
Let c ∈ ̂ C ≠ ( L ( G ) ; u u , w w ) and let ˇ c ∈ C ( L ( G ) ; u u , w w ) be the unreduced contributorfor c . Since c is in a trivial activation class, ˇ c must either (a) contain 2 circles with u ↦ w or u ↦ w belonging to different circles, or (b) contain 1 circle with u ↦ w and u ↦ w belonging to the same circle. Case 1 (Two-circles):
Suppose ˇ c has exactly 2-circles. First, unpack all backsteps of c , then re-introduce u ↦ w and u ↦ w to complete the two circles. Reverse the circle orientations and remove the adjacencies.The result is a Tutte-2-arborescence F of the form ⟨ u w , u w ⟩ and sgn T ( F ) = + Case 2 (One-circle):
Again, this is similar to case 1, except the adjacencies introduced form a singlecircle. The result is a Tutte-2-arborescence F of the form ⟨ u w , u w ⟩ , and sgn T ( F ) = − ◻ Example 2.2.2.
To see how a Tutte- -arborescence transforms into a circle-free reduced contributor, con-sider the top left Tutte- -arborescence from Figure 1 in the calculation for [ v v , v v ] . This Tutte- -arborescence appears on the left of Figure 9. ( a ) ⇒ ( b ) ⇒ ( c ) Figure 9: A Tutte-2-arborescence transforming into a reduced contributor.
The paths within each part of the arborescence are reversed in step ( a ) . The missing edge is added toproduce a unique (directed) circle in step ( b ) . Next, all edges connected to each circle via a path are packedinto backsteps away from each circle, producing the original restricted contributor. Finally, the introducededges are removed to produce the reduced contributor in step ( c ) . We have the following immediate corollaries.
Corollary 2.2.3.
Let F be a Tutte- -arborescence in the calculation of [ u u , w w ] and c F be its corre-sponding element in ̂ C ≠ ( L ( G ) ; u u , w w ) , then sgn T ( F ) = + if, and only if, ˇ c F has exactly two cycles, sgn T ( F ) = − if, and only if, ˇ c F has exactly one cycle. Corollary 2.2.4.
Let e be the edge between w and w . Introducing the w w -edge to any Tutte- -arborescence associated to [ u u , w w ] or a contributor- -arborescence associated to an element of ̂ C ≠ ( L ( G ) ; u u , w w ) produces a spanning tree in G ∪ e . Proof.
In either type of 2-arborescence w and w are in different components and each component is atree. If e is an edge of G a spanning tree of G is produced. If e does not exist in G , a spanning tree in G ∪ e is produced. ◻ Example 2.2.5.
Consider the graph in Figure 1. Two of the reduced contributors in ̂ C ≠ ( L ( G ) ; v v , v v ) that correspond to [ v v , v v ] appear on the left of Figure 10. The middle figures are obtained by unpackingbacksteps to produce a contributor- -arborescence. Finally, the introduction of the v v -edge yields a spanningtree. ⇒ ⇒⇒ ⇒ Figure 10: Trivial activation classes unpack into 2-arborescences, and those used for edge labeling produce spanning trees.
In the next Section Tutte-2-arborescences are replaced with entire contributor sets and the signs of thecontributors and the non-trivial classes are characterized.10 . Signed Graph Transpedances
We show that edge-labeling via signed contributors provide a generalization of transpedances andKirchhoff-type Laws to signed graphs via the coefficients of the degree-2 monomials x u w x u w from Theo-rem 1.2.2. The determinant-sign of a contributor c is taken from Theorem 1.2.2, where sgn D ( c ) = (− ) ec ( ˇ c )+ nc ( c )+ bs ( c ) . The contributor-based transpedance for the determinant, or
D-contributor-transpedance , is defined as [ u u , w w ] D = ∑ c ∈̂ C ≠ ( L ( G ) ; u u ,w w ) sgn D ( c ) , and consider the labeling of each w w -edge with the signed contributors from [ u u , w w ] D when w and w are adjacent. Example 3.1.1.
Again, consider the graph in Figure 1. The set of contributors that determine [ v v , v v ] D ,grouped into their activation classes, are shown in Figure 11. Figure 11: All activation classes for [ v v , v v ] D . We will see shortly that non-trivial classes sum to zero if all edges are positive, and only the trivial classeswill determine [ v v , v v ] D if G is a graph. There is a simple relationship between the signs of a Tutte-2-arborescence and their associated reducedcontributor.
Lemma 3.1.2.
Let F be a Tutte- -arborescence and c F be its corresponding element in ̂ C ≠ ( L ( G ) ; u u , w w ) , then sgn T ( F ) = (− ) ∣ V ∣ sgn D ( c F ) . Proof.
Tutte’s transpedances [ u u , w w ] are ordered second cofactors from the Laplacian L ( G ; u u ,w w ) ,and the Tutte-2-arborescences are the signed commensurable parts that sum to [ u u , w w ] . From Theorem1.2.2, the coefficient of x u w x u w is [ u u , w w ] D , and the reduced contributors are the signed commensu-rable parts that sum to [ u u , w w ] D , but the coefficient of x u w x u w is determined from X − L G . The twoadjacencies removed in each reduced contributor are mapped to x u w and x u w , while all ∣ V ∣ − (− ) ∣ V ∣− = (− ) ∣ V ∣ . ◻ D -contributor-transpedances. Lemma 3.1.3 (Contributor Degeneracy).
Let G be a signed graph with source u , sink u , and vertices w and w , then [ u u , w w ] D = [ u u , w w ] D = . Proof.
The set of reduced contributors for ̂ C ≠ ( L ( G ) ; u u , w w ) and ̂ C ≠ ( L ( G ) ; u u , w w ) are bothempty. The first would require two maps of the form u ↦ w and u ↦ w , and there cannot be two tails at u . The second would require two maps of the form u ↦ w and u ↦ w , and there cannot be two headsat w . ◻ Since unreduced contributors represent permutation clones in G , we may apply the Linking Lemma onreduced contributors to produce W U -paths as the circles are cut. This shows that Tutte’s energy reversalrule from Theorem 1.1.2 holds for D -contributor-transpedances. Lemma 3.1.4 (Contributor Energy Reversal).
Let G be a signed graph with source u , sink u , andvertices w and w , then [ u u , w w ] D = −[ u u , w w ] D = −[ u u , w w ] D . Proof.
We show the first equality, the second is similar.Consider c ∈ ̂ C ≠ ( L ( G ) ; u u , w w ) and reintroduce maps u ↦ w and u ↦ w to form the unreducedcontributor ˇ c . There are two cases depending if u , u , w , w belong to one or two circles in ˇ c . Case 1 (Two circles):
In this case we have that { u , w } and { u , w } are in disjoint circles in ˇ c . Remove u ↦ w and u ↦ w in ˇ c , and replace them with u ↦ w and u ↦ w to form a new non-zero unreducedcontributor ˇ c ′ in ̂ C ≠ ( L ( G ) ; u u , w w ) where { u , u , w , w } are in a single circle. Since c and c ′ havethe same adjacencies and backsteps, the sign difference between sgn D ( c ) and sgn D ( c ′ ) is determined by theeven circle structure of their unreduced contributors. If the original circles were both even, the new singlecircle is even; a loss of one even circle. If the original circles were both odd, the new circle is even; a gain ofone even circle. If the original circles have different parity, the new circle is odd; a loss of one even circle. Inany case sgn D ( c ) = − sgn D ( c ′ ) . Case 2 (One circle):
In this case we have that { u , u , w , w } are in a single circle ˇ c . Remove u ↦ w and u ↦ w in ˇ c , and replace them with u ↦ w and u ↦ w to form a new non-zero unreduced contributorˇ c ′ in ̂ C ≠ ( L ( G ) ; u u , w w ) where { u , w } and { u , w } are in disjoint circles. Since c and c ′ have thesame adjacencies and backsteps, the sign difference between sgn D ( c ) and sgn D ( c ′ ) is determined by theeven circle structure of their unreduced contributors. If the original circle is even, each new circle is odd. Ifthe original circle is odd, each new circle is even.This process is reversible, so we have the first equality. The second equality is similar. ◻ .2. Transpedance Evaluation With the sign adjustment in Lemma 3.1.2, we can immediately use the elements of ̂ C ≠ ( L ( G ) ; u u , w w ) in place of transpedances. However, we now characterize the placement of en-tire contributor families on edges. Let ˆ A ( u , w ; G ) be the set of all reduced activation classes of the formˆ A( u , w ; G ) , and let ˆ A − ( u , w ; G ) be the subset of ˆ A ( u , w ; G ) such that no element contains a positive circle.The activation class transversal consisting of maximal elements is denoted by M u , w , and M − u , w is the sub-set of maximal elements that are positive-circle-free. Since each activation class is Boolean (Lemma 2.1.1), D -contributor-transpedances have a simple presentation via the maximal element of each activation class. Theorem 3.2.1. If G is a signed graph, then [ u u , w w ] D = ∑ m ∈M −( u u ,w w ) sgn D ( m ) ⋅ ( ) η ( m ) where η ( m ) is the number of negative circles in maximal contributor m . Proof.
Let G be a signed graph with distinguished source u , sink u , edge w w , and total orderings u = ( u , u ) and w = ( w , w ) . Also, let ˆ A = ˆ A ( u , w ; G ) , and ˆ A = ˆ A( u , w ; G ) . Partition the D -contributor-transpedance value [ u u , w w ] D into activation classes as follows: [ u u , w w ] D = ∑ ˆ A∈ ˆ A ∑ c ∈ ˆ A sgn D ( c )= ∑ ˆ A∈ ˆ A − ∑ c ∈ ˆ A sgn D ( c ) + ∑ ˆ A∈ ˆ A ∖ ˆ A − ∑ c ∈ ˆ A sgn D ( c ) From Lemma 2.1.1, activation classes form Boolean lattices, and each sum is calculated separately.
Case 1 (No positive circles):
Let contributors c and c ′ only differ by a single negative circle, whichappears in c but not in c ′ . Let the length of this circle be (cid:96) . Packing this circle into backsteps will yield aloss of a single positive circle and a gain of (cid:96) backsteps. Case 1a ( (cid:96) is odd): If (cid:96) is odd, the sgn D ( c ′ ) is related to sgn D ( c ) as follows: sgn D ( c ′ ) = (− ) ec ( ˇ c )+( nc ( c )− )+( bs ( c )+ (cid:96) ) = (− ) ec ( ˇ c )+ nc ( c )+ bs ( c ) ⋅ (− ) (cid:96) − = sgn D ( c ) . Case 1b ( (cid:96) is even): If (cid:96) is even, packing also loses an even circle and sgn D ( c ′ ) is related to sgn D ( c ) asfollows: sgn D ( c ′ ) = (− ) ( ec ( ˇ c )− )+( nc ( c )− )+( bs ( c )+ (cid:96) ) = (− ) ec ( ˇ c )+ nc ( c )+ bs ( c ) ⋅ (− ) (cid:96) − = sgn D ( c ) . Since each element has the same sign and each activation class is Boolean there are 2 η ( m ) contributors,where m is the maximal contributor containing η ( m ) circles.13 ase 2 (Positive circle): Let contributors c and c ′ only differ by a single positive circle, which appearsin c but not in c ′ . Let the length of this circle be (cid:96) . Packing this circle into backsteps will yield a loss of asingle positive circle and a gain of (cid:96) backsteps. Case 2a ( (cid:96) is odd): If (cid:96) is odd, the sgn D ( c ′ ) is related to sgn D ( c ) as follows: sgn D ( c ′ ) = (− ) ec ( ˇ c )+ nc ( c )+( bs ( c )+ (cid:96) ) = (− ) ec ( ˇ c )+ nc ( c )+ bs ( c ) ⋅ (− ) (cid:96) = − sgn D ( c ) . Case 2b ( (cid:96) is even): If (cid:96) is even, packing also loses an even circle and sgn D ( c ′ ) is related to sgn D ( c ) asfollows: sgn D ( c ′ ) = (− ) ( ec ( ˇ c )− )+ nc ( c )+( bs ( c )+ (cid:96) ) = (− ) ec ( ˇ c )+ nc ( c )+ bs ( c ) ⋅ (− ) (cid:96) − = − sgn D ( c ) . Again, since each activation class is Boolean, there is a bijection between contributors with circle C and those without C via packing/unpacking. Thus, each activation class that contains a contributor with apositive circle will have those contributors sum to 0. Moreover, the remaining classes are determined by thesign of their maximal element. [ u u , w w ] D = ∑ ˆ A∈ ˆ A − ∑ c ∈ ˆ A sgn D ( c ) + = ∑ m ∈M − u ; w sgn D ( m ) ⋅ ( ) η ( m ) . ◻ Combining Lemmas 2.2.1, 3.1.2, and Theorem 3.2.1 we have the following interpretation of Tutte-transpedances:
Corollary 3.2.2 (Parity-Polarity Reversal). If G is a signed graph with all positive edges, then [ u u , w w ] = (− ) ∣ V ∣ [ u u , w w ] D . Proof.
If all edges are positive, by Theorem 3.2.1, the only non-cancellative terms are trivial reducedactivation classes. The bijection between 2-arborescence types in Lemma 2.2.1 combined with the signing inLemma 3.1.2 completes the proof. ◻ That is, Tutte’s edge-labeling via transpedances provides a natural orientation from source to sink, whilethe contributor version is reversed for graphs with an odd number of vertices.
Example 3.2.3.
If all edges are positive, the contributors for [ v v , v v ] D in Figure 11 produce a valueof + as the non-trivial classes sum to and there is an even number of vertices. This agrees with Tutte’s [ v v , v v ] . Example 3.2.4.
For a new example, consider the signed graph in Figure 12 with source v and sink v .To calculate the D-contributor-transpedance along edge v v we examine [ v v , v v ] D . The contributors inFigure 12 are the non-cancellative contributors as they do not contain positive circles. Since there are anodd number of vertices the value is negated relative to Tutte’s and the value is − . v v v v v Figure 12: Non-trivial reduced-contributors signed [ v v , v v ] D = − If all the edges were positive, the D -contributor-transpedance would have been − as the oriented -circleswould be positive. Also, observe that there are far more than contributors on edge v v , as the contributorsin Figure 13 always cancel as they repeat an adjacency, so the circle is always positive. Figure 13: Non-trivial reduced-contributors signed [ v v , v v ] D = − The sign between an edge does not matter when determining the D -contributor-transpedance on that edge,as the edge fails to appear in any contributor. Additionally, cycles may not cancel in their activation classas they would in a graph.
4. Adjacency Exchange and Kirchhoff ’s Laws
Contributor-transpedances satisfy their own general Degeneracy (Lemma 3.1.3) and Energy Reversal(Lemma 3.1.4) Kirchhoff-type laws. They are evaluated via activation classes (Theorem 3.2.1), with all pos-itive graphs related to Tutte-transpedances via Polarity reversal (Corollary 3.2.2). The Cycle Conservationand Vertex Conservation properties from Theorem 1.1.2 are now investigated by showing that transpedancesare contributor sorting along source-sink paths as a generalization of Corollary 2.2.4. However, the expecta-tion of conservation cannot be expected. 15 .1. Source-sink Pathing
Let G be a signed graph with source u , sink u , and vertices w and w . If w and w are adjacent, calltheir edge e . If w and w are not adjacent, regard G as a subgraph G ∪ e w w where edge e w w is addedbetween w and w . This is called the local-loading of G at { w , w } , and is related to the injective loadingproperties from [11, 10]; to simplify notation we will simply write G ∪ e w w with the understanding that e w w may exist in G . Let P( u u , w w ) be the set of u u -paths containing e w w in G ∪ e w w . Lemma 4.1.1.
A contributor c is in ̂ C ≠ ( L ( G ) ; u u , w w ) if, and only if, c contains a unique path P ∈P( u u , w w ) in G ∪ e w w . Proof.
Part I:
Let c ∈ ̂ C ≠ ( L ( G ) ; u u , w w ) , and ˇ c be its unreduced contributor. There are two casesdepending if u , u , w , w belong to one or two circles in ˇ c . Case 1 (Two circles):
In this case we have u , w and u , w are in disjoint circles in ˇ c . Remove u ↦ w and u ↦ w , reverse u → w , and introduce edge e to produce a u u -path P where w precedes w in P .Any additional circles and backsteps are external and may only extend the activation class. Case 2 (One circle):
In this case we have u , u , w , w in a single circle in ˇ c . Remove u ↦ w and u ↦ w , reverse u → w , and introduce edge e to produce a u u -path P where w precedes w in P . Anyadditional circles and backsteps are external and may only extend the activation class. Part II:
Let P ∈ P( u u , w w ) . There are two cases depending if w precedes w or w precedes w in P . Case 1 ( w precedes w ): Delete e w w , and introduce u ↦ w and u ↦ w . Reverse the u w -part of P , and do not reverse the u w -part of P to make two circles. Introduce backsteps/circles atall remaining vertices for form an unreduced contributor ˇ c . Remove u ↦ w and u ↦ w to get c ∈̂ C ≠ ( L ( G ) ; u u , w w ) . Pack/unpack as necessary to form activation classes. Case 2 ( w precedes w ): Delete e w w , and introduce u ↦ w and u ↦ w . Reverse the u w -part of P ,and do not reverse u w -part of P to make one circle. Introduce backsteps/circles at all remaining verticesto form an unreduced contributor ˇ c . Remove u ↦ w and u ↦ w to get c ∈ ̂ C ≠ ( L ( G ) ; u u , w w ) .Pack/unpack as necessary to form activation classes. ◻ We now have the immediate corollaries demonstrating that all contributors for a given transpedance arerelated to direct source-sink path property.
Corollary 4.1.2.
Let c ∈ ̂ C ≠ ( L ( G ) ; u u , w w ) . Every edge-adjacency appearing in c outside an activatedcircle is in one of the parts of the w i u j -paths. Moreover, these paths are oriented from w i to u j . Corollary 4.1.3. If w is a monovalaent vertex that is not a source or sink with supporting edge e w w ,then [ u u , w w ] = . We also have the following simple interpretation of Tutte’s transpedances.16 orollary 4.1.4.
Let G be a graph with source u and sink u . The edge labeling of G by transpedances [ u u , w w ] is equivalent to a sorting of spanning trees via adjacency swapping along the u u -path in G ∪ e . Proof.
Corollary 2.2.4 provides an interpretation of trivial activation classes as spanning trees, even fortranspedances not on adjacencies. Additionally, Corollary 3.2.2 shows that these are the only objects thatsurvive cancellation in the Boolean activation classes in a graph. Part 4 of Theorem 1.1.2 indicates the netinflow and outflow is the tree-number. ◻ Example 4.1.5.
The reduced contributors in trivial activation classes for Figure 2 appear on each edge inFigure 14 (left). A source-sink path is indicated on the right, and the associated unpacked contributors appearwith edge inserted to produce spanning trees to visualize both the spanning tree sorting and the unique pathingproperty.
Figure 14: Left: Contributors from trivial classes. Right: The associated spanning trees and unique paths.
Combining Theorem 3.2.1 and Lemma 4.1.1 we can discuss the Cycle and Vertex Conservation propertiesfrom Theorem 1.1.2. These conservation laws are byproducts of non-trivial Boolean classes vanishing in agraph, coupled with a natural matching of the elements in the trivial classes whose signs cancel. Negativeedges may produce non-vanishing Boolean classes as well as matched trivial classes of the same sign.
Lemma 4.1.6 (Contributor Cycle “Conservation”).
There is a matching between the elements of ̂ C ≠ ( L ( G ) ; u u , w w ) ∪ ̂ C ≠ ( L ( G ) ; u u , w w ) ∪ ̂ C ≠ ( L ( G ) ; u u , w w ) . Proof.
Let G be a signed graph with source u , sink u , and vertices w , w , and w . Additionally, let e w w , e w w , and e w w be the edges between their respective vertices, or the edge introduced to G if onedoes not exist.Consider c ∈ ̂ C ≠ ( L ( G ) ; u u , w w ) . From Lemma 4.1.1, let P be the unique u u -path in c made withthe inclusion of e w w so that P = P u ,w i ∪ e w w ∪ P w j ,u , where { i, j } = { , } . Since c is in a trivial class,there are no circles to activate, and from Corollary 4.1.2 vertex w must be linked to P u ,w i or P w j ,u by asequence of unpackings. Moreover, all backsteps outside of circle-activation unpack towards P , so there is aunique vertex w ′ that meets exactly one of P u ,w i or P w j ,u .17 u w w w P u w P w u w (cid:48) Assume w ′ meets P u ,w i , the case where w ′ meets P w j ,u is similar. Form the path P ′ ∶ u → w ′ → w . P ′ may contain w i if w ′ = w i but cannot contain e w w . Introducing edge e w ,w j forms a unique u u -path P ′′ = P ′ ∪ e w ,w j ∪ P w j ,u that uses exactly one of e w w , e w w , or e w w . Removing e w ,w i , reversingthe u w i part of P ′′ , and packing all non- P ′′ adjacencies away from P ′′ leaves a unique contributor c ′ ∈̂ C ≠ ( L ( G ) ; u u , w w j ) . u u w w w P (cid:48) P (cid:48)(cid:48) w (cid:48) Moreover, there is no corresponding contributor in ̂ C ≠ ( L ( G ) ; u u , w w i ) as e w ,w i does not form apath without using more than one of e w w , e w w , and e w w . ◻ Tutte’s Cycle Conservation in Theorem 1.1.2 is an immediate consequence of Lemma 4.1.6 as every edgeis positive and the matching converts between one and two circles, changing their signs. General signedgraphic conservation, however, cannot be guaranteed as (1) there may be negative edges between a trivial-class matching, and (2) the non-trivial classes need not cancel. Tutte’s Vertex Conservation in Theorem1.1.2 is also an immediate consequence of the following lemma and is easily seen in Figure 14 by followingthe contributor sorting along source-sink paths.
Lemma 4.1.7 (Vertex “Conservation”).
Let G be a signed graph with source u , sink u , and let v beanother vertex. If v ∉ { u , u } , then ∣ ⋃ x ∼ v ̂ C ≠ ( L ( G ) ; u u , xv )∣ = ∣ ⋃ y ∼ v ̂ C ≠ ( L ( G ) ; u u , vy )∣ . If v ∈ { u , u } , then ∣ ⋃ x ∼ v ̂ C ≠ ( L ( G ) ; u u , u x )∣ = ∣ ⋃ y ∼ v ̂ C ≠ ( L ( G ) ; u u , yu )∣ . Proof.
Let G be a signed graph with source u , sink u , and let v be another vertex. Consider ̂ C ≠ ( L ( G ) ; u u , xv ) and ̂ C ≠ ( L ( G ) ; u u , vy ) , where the edges e xv and e vy exist in G . Case 1 ( v ∉ { u , u } ): If c ∈ ̂ C ≠ ( L ( G ) ; u u , xv ) , using Lemma 4.1.1 consider the u u -paths thatcontains e xv . Since v is not the source or sink, each path must contain exactly one of the edges e vy for some y . From Corollary 4.1.2, all contributors in ̂ C ≠ ( L ( G ) ; u u , xv ) associated to a path containing both e xv and e vy must also have a corresponding element in ̂ C ( L ( G ) ; u u , vy ) .The argument is identical on the preceding edge when starting with ̂ C ( L ( G ) ; u u , vy ) . Case 2 ( v ∈ { u , u } ): If v is the source, there are no v -entrant edges in any u u -path. While, if v isthe sink, there are no v -salient edges in any u u -path. However, from Lemma 4.1.1 and Corollary 4.1.2,18ll contributors arise from u u -paths, therefore all trivial class contributors out of u have a correspondingcontributor in to u . ◻ If every edge of a signed graph is positive, not only do the non-trivial activation classes sum to zero,but the trivial ones in each matching above also cancel. Thus, conservation is guaranteed when G has allpositive edges. Corollary 4.1.8. If G has all positive edges, then the D-contributor-transpedances are both cycle and vertexconservative. It is clear that a graph with a single negative edge that is also between the source and sink is conservative,as that edge never appears in any contributor. It seems worthwhile to produce a complete characterizationof signed graphs that are conservative, even for fixed source and sinks.
5. Maximizing Transpedance, Permanents, and Signless Laplacians
Since contributor mappings are used in determining the characteristic and total minor polynomials [7, 10],and generalizations of the Matrix-Tree Theorem [14], we examine the net placement of contributors on agiven graph. The permanent of the oriented hypergraphic signless Laplacian was shown to count the numberof contributors, which occurs when every adjacency is negative.
Theorem 5.1.1 ([7], Theorem 4.3.1 part 1).
Let G be an oriented hypergraph with no isolated verticesor -edges with Laplacian matrix L G , then perm ( L G ) = ∣ C ( G )∣ if, and only if, every edge of G is extrovertedor introverted. As in prior sections, the previous theorem was a direct calculation on the Laplacian, while we makeuse of the coefficient of the total minor polynomial to keep track of the ordered minor placement. The permanental-sign of a contributor c is taken from Theorem 1.2.2, where sgn P ( c ) = (− ) nc ( c )+ bs ( c ) . The signless Laplacian can be used to count the number of reduced contributors for any oriented hypergraph.
Theorem 5.1.2. If G is an oriented hypergraph with all negative adjacencies, then ∑ c ∈̂ C ≠ ( L ( G ) ; u , w ) sgn P ( c ) = (− ) ∣ V ∣− k ∣̂ C ≠ ( L ( G ) ; u , w )∣ Proof.
Let k = ∣ U ∣ = ∣ W ∣ with U and W totally orderings u and w . Also let sgn P ( c ) = (− ) nc ( c ) + bs ( c ) bethe permanent signing function. We proceed with an inductive argument: Case 1 ( k = ): Observe that if c is a minimal (identity-clone) contributor, then nc ( c ) = bs ( c ) = ∣ V ∣ ,and the permanent sign of all minimal contributors is (− ) ∣ V ∣ . If c ′ is any contributor that can unpack into19nother covering contributor c ′′ containing a new cycle of length (cid:96) , then we have two cases based on (cid:96) ’sparity. Case 1a ( (cid:96) is even):
Unpack (cid:96) backsteps in c ′ to form a cycle of length (cid:96) in c ′′ . Since (cid:96) is even and all edgesare negative, we lose (cid:96) backsteps and gain 0 negative components. Since − (cid:96) + sgn P ( c ′ ) = sgn P ( c ′′ ) . Case 1b ( (cid:96) is odd):
Unpack (cid:96) backsteps in c ′ to form a cycle of length (cid:96) in c ′′ . Since (cid:96) is odd and all edgesare negative, we lose (cid:96) backsteps and gain 1 negative component. Since − (cid:96) + sgn P ( c ′ ) = sgn P ( c ′′ ) .Thus, all contributors have the same sign as their minimal contributor, and all minimal contributors havethe same sign, giving ∑ c ∈̂ C ≠ ( L ( G ) ; u , w ) sgn P ( c ) = (− ) ∣ V ∣ ∣̂ C ( G )∣ . Case ( k > ): In a contributor with all negative adjacencies, deleting a negative edge will swap the signof the component that contained the edge, thus changing nc ( c ) by one. Deleting a backstep will decrease bs ( c ) by one, which will flip the permanent sign of the total contributor. Since all contributors in ̂ C ( G ) havethe same permanent signing, the sign alternates with every edge or backstep that is removed. Thus, thepermanent counts of reduced contributors must be (− ) ∣ V ∣− k ∣̂ C ≠ ( L ( G ) ; u , w )∣ . ◻ We define the contributor based transpedance for the permanent, or
P-contributor-transpedance , to be [ u u , w w ] P = ∑ c ∈̂ C ≠ ( L ( G ) ; u u ,w w ) sgn P ( c ) . Corollary 5.2.1. If G is an oriented hypergraph with all negative adjacencies, then [ u u , w w ] P = (− ) ∣ V ∣ ∣̂ C ≠ ( L ( G ) ; u u , w w )∣ Proof.
From Theorem 5.1.2 the value of [ u u , w w ] P is equal to (− ) ∣ V ∣− ∣̂ C ≠ ( L ( G ) ; u u , w w )∣ , and ∣ V ∣ − ∣ V ∣ have the same parity. ◻ While a contributor-type transpedance can be defined for an arbitrary oriented hypergraph, the paralleladjacencies and the non-Boolean structure for tail-equivalence classes require further examination. However,the signless Laplacian provides a count on the total number of contributors, and a maximum possible valuefor a bidirected graph.A permanent version of Tutte’s transpedance Theorem follows.
Lemma 5.2.2 (P-Contributor Degeneracy).
Let G be a signed graph with source u , sink u , and dis-tinct vertices w , w and w , then [ u u , w w ] P = [ u u , w w ] P = , [ u u , w w ] P = [ u u , w w ] P = [ u u , w w ] P , There is a matching between the elements of ̂ C ≠ ( L ( G ) ; u u , w w ) ∪ ̂ C ≠ ( L ( G ) ; u u , w w ) ∪ ̂ C ≠ ( L ( G ) ; u u , w w ) . Let G be a signed graph with source u , sink u , and let v be another vertex. (a) If v ∉ { u , u } , then ∣ ⋃ x ∼ v ̂ C ≠ ( G ; u u , xv )∣ = ∣ ⋃ y ∼ v ̂ C ≠ ( G ; u u , vy )∣ . (b) If v ∈ { u , u } , then ∣ ⋃ x ∼ v ̂ C ≠ ( G ; u u , u x )∣ = ∣ ⋃ y ∼ v ̂ C ≠ ( G ; u u , yu )∣ . Proof.
The proofs are identical to the determinant case as they are the same set of objects. The onlyexception is even circles are not included in any signs. ◻ Acknowledgments
A portion of this research was supported by Mathworks at Texas State University. The authors wouldlike to thank Max Warshauer and Amelia Hu for their support and discussions.
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