Construction of networks by associating with submanifolds of almost Hermitian manifolds
aa r X i v : . [ m a t h . G M ] F e b Appeared in Fundamenta Informaticae XXX(X) : 1–17 (2021). 1Available at IOS Press through https://doi.org/10.3233/FI-2021-0001
Construction of networks by associating with submanifolds ofalmost Hermitian manifolds
Arif Gursoy * Department of MathematicsEge UniversityBornova, 35100 Izmir, [email protected]
Abstract.
The idea that data lies on a non-linear space has brought up the concept of manifoldlearning [1] as a part of machine learning and such notion is one of the most important researchfield of today. The main idea here is to design the data as a submanifold model embedded in ahigh-dimensional manifold. On the other hand, graph theory is one of the most important researchareas of applied mathematics and computer science. As a result, many researcher investigate newmethods for machine learning on graphs. From the above information, it is seen that the theoryof submanifolds and graph theory have become two important concepts in machine learning andnowadays, the geometric deep learning research area using these two concepts is emerged [2]. Bycombining these two fields, this article aims to present the relationships between submanifolds ofcomplex manifolds with the help of graphs. We give relations among holomorphic submanifolds,totally real submanifolds, CR-submanifolds, slant submanifolds, semi-slant submanifolds, hemi-slant submanifolds and bi-slant submanifolds in almost Hermitian manifolds in terms of graphtheory.
Keywords:
Graph theory, digraphs, almost Hermitian manifold, submanifolds, manifold learn-ing, machine learning, artificial intelligence
Address for correspondence: Department of Mathematics, Science Faculty, Ege University, Bornova, 35100, Izmir, Turkey * Corresponding Author
A. Gursoy / Construction of networks by associating with submanifolds of almost Hermitian manifolds
1. Introduction
Graph theory can be used to model computer networks, social networks, communications networks,information networks, software design, transportation networks, biological networks, etc. So thistheory is applicable in many real-world mathematical modeling. Therefore, this theory is the mostactive areas of mathematical research.On the other hand, one of the most active research area of differential geometry is the submanifoldtheory of complex manifolds. A submanifold of an almost Hermitian manifold is characterized bythe behavior of tangent space of the submanifold of almost Hermitian manifold under the complexstructure of the ambient manifold. In this way, we have various submanifolds titled as holomorphic,totally real, CR, slant, semi slant, hemi-slant, bi-slant for almost Hermitian manifolds. In fact, thetheory of submanifolds of almost Hermitian manifolds is still main active area of complex differentialgeometry, see:[3, 4, 5] for recent results.Manifold learning method is one of the most exciting developments in machine learning recently.Manifold learning has been applied in utilizing semi-supervised learning [1]. Moreover, Vakulenkoand Radulescu have used the theory of invariant and inertial manifold to prove the realization of pre-scribed dynamics by networks in patterning by centralized gene networks [6]. Furthermore, manifoldsalso play an important role in public health. Fiorini has defined the Riemannian manifold, which is iso-morphic to traditional information geometry Riemannian manifold, for noise reduction in theoreticalcomputerized tomography providing many competitive computational advantages over the traditionalEuclidean approach [7]. Besides, Monti et. al. have introduced a general framework, geometric deeplearning, enabling to design of convolutional deep architectures on manifolds and graphs [2].Also, Carriazo and Fernandez [8] have constructed a relation between slant surface and graphtheory. Later, they have related graph theory with vector spaces of even dimension [9, 10]. Their workwas restricted to slant submanifolds. We believe that further use of graph theory is possible in thetheory of submanifolds.By considering vast literature of graph theory and submanifold theory, one expects more relationsbetween these research areas. In this direction, the aim of this paper is to examine the relation amongvarious submanifolds of almost Hermitian manifolds by using graph theory. We note that our approachis different from the approach considered in [8] and [9]. They only considered adapted frame ofslant surface and they used them to characterize CR-submanifolds by means of trees. Later theyhave extended this approach for weakly associated graph. In this paper, we give relations betweensubmanifolds of Hermitian manifolds in terms of graph theory notions.
2. Preliminaries
In this section, we are going to recall certain notions used graph theory to be used in this paper from[11, 12, 13, 14, 15, 16]. For those who are not familiar with the theory of graphs (especially for readersworking with the submanifolds theory), we specifically recall the basic definitions from graph theory.A graph G = ( V, E ) consists of a nonempty set V of vertices and a set E of edges. Each edgehas either one or two vertices connected with it, called its endpoints. An edge connects its endpoints.Two distinct vertices u, v in a graph G are called adjacent (or neighbors) in G if there is an edge e . Gursoy / Construction of networks by associating with submanifolds of almost Hermitian manifolds between u and v , where the edge e is called incident with the vertices u and v and e connects u and v . The set of all neighbors of a vertex v of G = ( V, E ) is denoted by N ( v ) . If A ⊂ V , we denote by N ( A ) the set of all vertices in G that are adjacent to at least one vertex in A . The degree of a vertexin a graph is the number of edges incident with it. The degree of the vertex v is denoted by d ( v ) and d ( v ) = | N ( v ) | . The graph theory can be divided into two branches as undirected and directed graphs.[16]A directed graph (digraph) D is a finite nonempty set of objects called vertices together with a setof ordered pairs of distinct vertices of D called directed edges or arcs. For a digraph D = ( V, A ) ,the vertex set of D is denoted by V ( D ) or simply V and the arc set of D is denoted by A ( D ) or A . Each arc is an ordered pair of vertices. The arc ( u, v ) is said to start at u and end at v . The in-degree of a vertex v, d − ( v ) , is the number of edges which end at v . The out-degree of v , d + ( v ) , is thenumber of edges with v as their initial vertex. Also, for a vertex v ∈ V ( D ) , N − D ( v ) and N + D ( v ) arerespectively called out-neighbors and in-neighbors where N − D ( v ) = { u | ( u, v ) ∈ A ( D ) , u ∈ V ( D ) } and N + D ( v ) = { u | ( v, u ) ∈ A ( D ) , u ∈ V ( D ) } . [14, 11, 17, 16]In a digraph D = ( V, A ) , given a pair of vertices u and v , whether or not there is a path from u to v in the digraph is useful to know. The transitive closure of D is to construct a new digraph, D ∗ = ( V, A ∗ ) , such that there is an arc ( u, v ) in D ∗ if and only if there is a path from u to v in D .[15]A walk W = x a x a x ...x k − a k − x k is a sequence of vertices x i and arcs a j in D such thatthe tail and head of a i is x i and x i +1 for every i ∈ [ k − , respectively. The set of vertices and arcs ofthe walk W are denoted V ( W ) and A ( W ) , respectively. W is denoted without arcs as x x ...x k andshortly ( x , x k ) -walk. If x = x k then W is a closed walk, and otherwise w is an open walk. If W isan open walk, the vertices x and x k are end-vertices and named as the initial and the terminal vertexof W , respectively. The length of a walk is the number of its arcs and the walk W above has length k − . [11]A trail is a walk in which all arcs are distinct. W is called a path, if the vertices of a trail V ( W ) ⊂ V ( D ) are distinct. If the vertices x , x , ..., x k − are distinct, k ≥ and x = x k , then W is a cycle.The longest path in D is a path of maximum length in D . [11] Proposition 2.1. [11] Let D be a digraph and let x, y be a pair of distinct vertices in D . If D hasan ( x, y ) -walk W , then D contains an ( x, y ) -path P such that A ( P ) ⊆ A ( W ) . If D has a closed ( x, x ) − walk W , then D contains a cycle C through x such that A ( C ) ⊆ A ( W ) .An oriented graph is a digraph with no cycle of length two [11]. For a digraph D , the UnderlyingGraph of D is the undirected graph engendered utilizing all vertices in V ( D ) , and superseding all ofthe arcs in A ( D ) with undirected edges. [13]If a digraph D has an ( x, y ) -walk, then the vertex y is reachable from the vertex x . Every vertex isreachable from itself specifically. By Proposition 2.1, y is reachable from x if and only if D contains an ( x, y ) -path. If every pair of vertices in digraph D is mutually reachable then D is strongly connected(or shortly strong). A strong component of digraph D is a maximal induced strong subdigraph in D .If D , ..., D t are the strong components of D , then precisely V ( D ) ∪ ... ∪ V ( D t ) = V ( D ) . If adigraph D is not strongly connected and if the underlying graph of D is connected, then D is said tobe weakly connected. [11, 17] A. Gursoy / Construction of networks by associating with submanifolds of almost Hermitian manifolds
Pseudograph is a graph having parallel edges and loops, and multigraph is a pseudograph with noloops. If every pair of distinct vertices are adjacent in a multigraph then the multigraph is complete.A multigraph H is called as p − partite if there is a partition into p sets V ( H ) = V ∪ V ∪ ... ∪ V p where V i ∩ V j = Ø for every i = j . In particular, when p = 2 the graph is called a bipartite graph.A bipartite graph B is denoted by B = ( V , V ; E ) . If the edge ( x, y ) is in p − partite multigraph H where all x ∈ V i , y ∈ V j for i = j then H is complete p − partite. [11]A digraph D = ( V, A ) is symmetric if arc ( x, y ) ∈ A implies arc ( y, x ) ∈ A . A matching M is anarc set having no common end-vertices and loops in D . Also, the arcs of M are independent if M is amatching. If a matching M implicates the highest number of arcs in D , then M is maximum. Besides,a maximum matching is perfect if it has | A ( D ) | arcs. A set Q of vertices in a directed pseudograph H is independent if there are no arcs between vertices in Q . The independence number of H is the sizeof the independent set having maximum cardinality in H . A coloring of a digraph H is a partition of V ( H ) into disjoint independent sets. The minimum number of independent sets in the coloring of H is the chromatic number of H . A simple directed graph is a digraph that has no multiple arcs or loops.if a digraph contains no cycle, then it is acyclic and called acyclic digraph. [11]The eccentricity e ( v ) of a vertex v is the distance from v to the farthest vertex from itself. Theradius ( rad ) of D is the minimum eccentricity, and the diameter ( diam ) is the maximum eccentricity.Besides, a vertex v is central if e ( v ) = rad ( D ) , and v is peripheral if e ( v ) = diam ( D ) . [18]Let D = ( V, A ) be a digraph, V ( D ) = n and S ⊂ V ( D ) . S is a dominating set of D if eachvertex v ∈ V ( D ) − S is dominated by at least a vertex in S . A dominating set of D having thesmallest cardinality is called the minimum dominating set of D . Also, the cardinality of the minimumdominating set is called the domination number of D [19, 20]Let r be a root vertex in D . A directed spanning tree T starting from r is a subdigraph of D such that the undirected form of T is a tree and there is a directed unique ( r, v ) -path in T for each v ∈ V ( T ) − r . [11]The vertex-integrity of a digraph D is defined by I ( D ) = min {| F | + m ( D − F ) : F ⊆ V ( D ) } ,where m ( D − F ) indicates the maximum order of a strong component of D − F . If I ( D ) = | F | + m ( D − F ) then F is called as an I -set of D . In addition, the arc-integrity of a digraph D , shortly I ′ ( D ) , is described as the minimum value of {| F | + m ( D − F ) : F ⊆ A ( D ) } . The set F is called asan I ′ -set of D if I ′ ( D ) = | F | + m ( D − F ) . [21] Proposition 2.2. [21] If S is a subdigraph of D then I ( S ) ≤ I ( D ) and I ′ ( S ) ≤ I ′ ( D ) .
3. Construction of digraphs by relations among submanifolds of almostHermitian manifolds
Let ( M, g ) be a Riemannian manifold. ( M, g ) is called an almost Hermitian manifold if there is a(1,1) tensor field on M such that J = − I , where I is the identity map on the tangent bundle of M ,and g ( J X, J Y ) = g ( X, Y ) for vector fields X, Y on M . Moreover if J is parallel with respect toany vector field X , then ( M, J, g ) is called a Kaehler manifold [22]. There are various submanifoldsof an almost Hermitian manifold based on the behavior of the tangent space of the submanifold at apoint under the almost complex structure J . Let N be a submanifold of an almost Hermitian manifold . Gursoy / Construction of networks by associating with submanifolds of almost Hermitian manifolds v v v v Figure 1. Digraph D built by submanifolds holomorphic, CR, anti-invariant and slant and T p N the tangent space at a point p belongs to N . Then if T p N is invariant with respect to J p for any point p , then N is called holomorphic (or complex) submanifold [22]. We denote the normalspace at p by T p N ⊥ . A submanifold of an almost Hermitian manifold is called an anti-invariantsubmanifold if J T p N ⊆ T p N ⊥ [22]. As a generalization of holomorphic submanifold and anti-invariant submanifolds, a submanifold M of a Kaehler manifold N is called CR-submanifold [23]if there are two orthogonal complementary distributions D and D such that D is invariant withrespect to J and D is anti-invariant with respect to J for every point p ∈ M . It is clear that if D = { } , then a CR-submanifold becomes an anti-invariant submanifold. If D = { } , then M becomes a holomorphic submanifold. Another generalization of holomorphic submanifolds and anti-anti-invariant submanifolds is slant submanifolds. Let N be a submanifold of an almost Hermitianmanifold M . The submanifold N is called slant [24] if for each non-zero vector X tangent to N theangle θ ( X ) between J X and T p N is a constant, i.e, it does not depend on the choice of p ∈ M and X ∈ T p N . θ is called the slant angle. It is clear that if θ ( X ) = 0 then N becomes a holomorphicsubmanifold. If θ ( X ) = π/ , N becomes an anti-invariant submanifold. We will use the v , v , v ,and v to represent the submanifolds holomorphic, CR, anti-invariant and slant, respectively.Digraph D = ( V, A ) has four vertices, V ( D ) = { v , v , v , v } , and four arcs, A ( D ) = { ( v , v ) , ( v , v ) , ( v , v ) , ( v , v ) } in Fig. 1. D has the maximum length of one as the longest path. D has 2 vertices ( v and v ) which are not reachable. Topological sort of D is v − v − v − v . rad ( D ) = 1 , the radius of D is v → v . diam ( D ) = 1 , the diameter of D is the same as theradius. Also, in D , there is no center vertex, but two peripheral vertices such as v and v . Theorem 3.1.
For a digraph D constructed by the four submanifolds holomorphic, CR, anti-invariantand slant considering as the vertices v , v , v , and v , respectively,i D is a bipartite digraph as well as a complete bipartite digraph.ii D has a perfect matching.iii The independence number of D is 2.iv The chromatic number of D is 2.v D has no directed spanning tree. A. Gursoy / Construction of networks by associating with submanifolds of almost Hermitian manifolds vi The domination number of D is 2. Proof: i There exists a partition V and V of V ( D ) into two partite sets for the submanifolds in D : V = { v , v } and V = { v , v } . Owing to V ( D ) = V ∪ V and V ∩ V = Ø , then D is abipartite digraph.Besides, for every submanifold, x ∈ V , y ∈ V , a connection from x to y (i.e. an arc ( x, y ) ) is in D . Therefore, D is a complete bipartite digraph.ii There is a matching M = { ( v , v ) , ( v , v ) } ⊂ A ( D ) in D . Each element (arc or connectionbetween two submanifolds) in M is independent, i.e. no common vertices, and M is maximum.Also, M is perfect so that | M | = | A ( D ) | . It is obvious that D has a perfect matching.iii The subset e V = { v , v } ⊂ V ( D ) is one of the independent sets having maximum cardinalityand the size of maximum independent submanifolds set is two. This also means that there is norelation between submanifolds v and v . Then, the independence number of D is two.iv V = { v , v } and V = { v , v } are two subsets of V ( D ) . V i ( i = 1 , are all independent setsproviding the minimum number of cardinality at the same time. Hence, the minimum number ofindependent sets of D is two. Then, the chromatic number of D is two.v There is no root vertex where a subdigraph T of D contains a directed path from the root to anyother vertex in V ( D ) . Then, D has no directed spanning tree.vi There is a subset e V = { v , v } ⊂ V ( D ) that including minimum cardinality of vertices in D .Considering this subset, for each vertex v ∈ e V and u ∈ V ( D ) − e V , (v,u) is an arc in D . Thedomination number is two, because of no smaller cardinality of dominating sets in D . ⊓⊔ Corollary 3.2.
In the submanifold network represented by D in Fig. 1, the submanifolds, CR ( v ) and slant ( v ) , cannot be derived by the other submanifolds, because the in-degree of these vertices(submanifolds) are zero in D , d − ( v ) = d − ( v ) = 0 . In addition, whereas CR and slant subamnifoldscannot be mutually derived as between holomorphic ( v ) and anti-invariant ( v ) , holomorphic andanti-invariant submanifolds can be derived separately from CR and slant from N − D ( v ) = N − D ( v ) = { v , v } .We now recall the notion of hemi-slant submanifolds of an almost Hermitian manifold. Let M bean almost Hermitian manifold and N a real submanifold of M . Then we say that N is a hemi-slantsubmanifold [25, 26] if there exist two orthogonal distributions D ⊥ and D θ on N such that1. T N admits the orthogonal direct decomposition
T N = D ⊥ ⊕ D θ .2. The distribution D ⊥ is an anti-invariant distribution, i.e., J D ⊥ ⊂ T M ⊥ . . Gursoy / Construction of networks by associating with submanifolds of almost Hermitian manifolds v v v v v Figure 2. Digraph D built by submanifolds in D and the hemi-slant submanifold
3. The distribution D θ is slant with slant angle θ .It is easy to see that if D ⊥ = { } , N becomes a slant submanifold with a slant angle θ . If D θ = { } , then N becomes an anti-invariant submanifold. Moreover if θ = 0 , then N becomes a CR-submanifold. Furthermore, if D ⊥ = { } and θ = 0 , then N becomes a holomorphic submanifold. Wedenote hemi-slant submanifolds by v .Digraph D = ( V, A ) is an extension of D , and has five vertices, V ( D ) = { v , v , v , v , v } ,and seven arcs, A ( D ) = { ( v , v ) , ( v , v ) , ( v , v ) , ( v , v ) , ( v , v ) , ( v , v ) , ( v , v ) } in Fig. 2. D has the maximum length of two as the longest path. It has 2 vertices ( v and v ) which are notreachable. Topological sort of D is v − v − v − v − v . rad ( D ) = 1 , the radius of D is v → v . diam ( D ) = 1 , the diameter of D is the same as the radius. Also, there is no center vertex but threeperipheral vertices such as v , v and v . Theorem 3.3.
For the digraph D created by adding the hemi-slant submanifolds as vertex v to the D ,i D is a three-partite digraph.ii The maximum matching is 2.iii The independence number is 2.iv The chromatic number is 3.v D has no directed spanning tree.vi The domination number is 2. Proof: i There exists a partition V = { v , v } , V = { v } and V = { v , v } of V ( D ) . These threesubsets are three partite sets because of following attributes: V ( D ) = S i =1 V i and V i ∩ V j = Ø ( i, j = 1 , , and i = j ). Then, D is a three-partite digraph. A. Gursoy / Construction of networks by associating with submanifolds of almost Hermitian manifolds ii There is an arc subset M = { ( v , v ) , ( v , v ) } in D , and | M | = 2 . In M , there is no commonvertices and loops, that is M is a matching. Also, there is no arc subset having greater cardinalitythan M . Therefore, M is maximum matching in D .iii The maximum independent set and independence number of D is the same as D . See Theorem3.1-iii.iv The minimum number of disjoint independent sets of D is three: V = { v , v } , V = { v } and V = { v , v } . Then, chromatic number of D is three.v No root vertex that contains a directed path from the root to any other vertex in V ( D ) . Then, D has no directed spanning tree.vi There is a subset e V = { v , v } ⊂ V ( D ) . Considering this subset, that including the minimumcardinality of vertices in D as a dominating set, for each vertex v ∈ e V and u ∈ V ( D ) − e V , (v,u)is an arc in D . Clearly, the domination number is two. ⊓⊔ Corollary 3.4.
In the submanifold network represented by D in Fig. 2, the submanifolds, slant ( v ) and hemi-slant ( v ) , cannot be derived by the other submanifolds, because d − ( v ) = d − ( v ) = 0 in D . Also, holomorphic ( v ) and anti-invariant ( v ) submanifolds can be derived separately by CR ( v ) , slant and hemi-slant since N − D ( v ) = N − D ( v ) = { v , v , v } .We now recall the notion of semi-slant submanifolds of an almost Hermitian manifold. Let M bean almost Hermitian manifold and N a real submanifold of M . Then we say that N is a semi-slantsubmanifold [27] if there exist two orthogonal distributions D and D θ on N such that1. T N admits the orthogonal direct decomposition
T N = D ⊕ D θ .2. The distribution D is an invariant distribution, i.e., J ( D ) = D .3. The distribution D θ is slant with slant angle θ .It is easy to see that if D = { } , N becomes a slant submanifold with a slant angle θ . If D θ = { } ,then N becomes a holomorphic submanifold. Moreover if θ = π , then N becomes a CR-submanifold.Furthermore, if D = { } and θ = π , then N becomes an anti-invariant submanifold. We denote semi-slant submanifolds by v .Digraph D = ( V, A ) is another extension of D , and has five vertices, V ( D ) = { v , v , v , v , v } ,and seven arcs, A ( D ) = { ( v , v ) , ( v , v ) , ( v , v ) , ( v , v ) , ( v , v ) , ( v , v ) , ( v , v ) } in Fig. 3. D has the maximum length of two as the longest path. It has a vertex ( v ) which is not reachable.Using transitive closure, D has only one new direct connection such as v → v . Topological sort of D is v − v − v − v − v . rad ( D ) = 1 , the radius of D is v → v . diam ( D ) = 2 , the diameterof D is v → v → v . Also, in D , there are two center vertices as v and v , and one peripheralvertex as v . Theorem 3.5.
For the digraph D created by adding the semi-slant submanifolds as vertex v to the D , . Gursoy / Construction of networks by associating with submanifolds of almost Hermitian manifolds v v v v v Figure 3. Digraph D built by submanifolds in D and the semi-slant submanifold i D is a three-partite digraph.ii The maximum matching is 2.iii The independence number is 2.iv The chromatic number is 3.v D has a directed spanning tree.vi The domination number is 2. Proof: i There exists a partition V = { v , v } , V = { v , v } and V = { v } of V ( D ) as three partitesets in D , and the subsets provide following properties: V ( D ) = S i =1 V i and V i ∩ V j = Ø ( i, j = 1 , , and i = j ). In that case, D is a three-partite digraph.ii There is an arc subset M = { ( v , v ) , ( v , v ) } in D , and | M | = 2 . Because of no commonvertices and no loops in M , M is a matching. Furthermore, M has the maximum cardinality sothat M is the maximum matching in D .iii The maximum independent set and independence number of D is the same as D . See Theorem3.1-iii.iv The minimum number of disjoint independent sets of D is three: V = { v , v } , V = { v , v } and V = { v } . It follows that the chromatic number of D is three.v D has a unique directed spanning tree of length 4 and rooted at v such as in Fig. 4. It also meansthat there is a transformation from submanifolds v to all other submanifolds in D .vi There is a subset e V = { v , v } ⊂ V ( D ) . According to this subset, that having the minimumcardinality, and for each vertex v ∈ e V and u ∈ V ( D ) − e V , (v,u) is an arc in D , the dominationnumber is two. A. Gursoy / Construction of networks by associating with submanifolds of almost Hermitian manifolds v v v v v Figure 4. Directed spanning tree in D v v v v v v Figure 5. Digraph D built by submanifolds in D and the hemi-slant submanifold ⊓⊔ Corollary 3.6.
In the submanifold network represented by D in Fig. 3, while no submanifolds can betransformed to semi-slant ( v ) submanifold since N − D ( v ) = ∅ , all other submanifolds (holomorphic ( v ) , CR ( v ) , anti-invariant ( v ) and slant ( v ) ) can be obtained from semi-slant submanifold becauseof existence of a directed spanning tree with a root vertex v (Fig. 4).Digraph D = ( V, A ) has six vertices, V ( D ) = { v , v , v , v , v , v } , and 10 arcs, A ( D ) = { ( v , v ) , ( v , v ) , ( v , v ) , ( v , v ) , ( v , v ) , ( v , v ) , ( v , v ) , ( v , v ) , ( v , v ) , ( v , v ) } in Fig. 5. D has the maximum length of two as the longest path. It has 2 vertices ( v and v ) which are not reach-able. Using transitive closure, D has only one new direct connection such as v → v . The topo-logical sort of D is v − v − v − v − v − v . rad ( D ) = 1 , the radius of D is v → v . diam ( D ) = 2 , the diameter of D is v → v → v . Also, in D , there are three center vertices as v , v and v , and one peripheral vertex as v . Theorem 3.7.
For the digraph D created by adding the hemi-slant submanifolds as vertex v to the D ,i D is a three-partite digraph.ii D has a perfect matching. . Gursoy / Construction of networks by associating with submanifolds of almost Hermitian manifolds iii The independence number is 2.iv The chromatic number is 3.v D has no directed spanning tree.vi The domination number is 2. Proof: i There exists a partition V = { v , v } , V = { v , v } and V = { v , v } of V ( D ) as threesubsets, and these subsets provide that V ( D ) = S i =1 V i and V i ∩ V j = Ø ( i, j = 1 , , and i = j ).Under these conditions, D is a three-partite digraph.ii There is an arc subset M = { ( v , v ) , ( v , v ) , ( v , v ) } in D , and | M | = 3 . On conditions thatno common vertices and no loops in M and | M | = | A ( D ) | , M is perfect matching that’s why D has a matching also perfect.iii The maximum independent set and the independence number of D is the same as D . SeeTheorem 3.1-iii.iv The minimum number of disjoint independent sets of D is three: V = { v , v } , V = { v , v } and V = { v , v } . Then, the chromatic number of D is three.v No root vertex that contains a directed path from the root to any other vertex in V ( D ) . Then, D has no directed spanning tree.vi There is a subset e V = { v , v } ⊂ V ( D ) . According to this subset, that having the minimumcardinality, and for each vertex v ∈ e V and u ∈ V ( D ) − e V , (v,u) is an arc in D so that thedomination number is two. ⊓⊔ Corollary 3.8.
In the submanifold network represented by D in Fig. 5, semi-slant ( v ) and hemi-slant ( v ) submanifolds cannot be obtained by any other submanifolds because d − ( v ) = d − ( v ) = 0 .Besides, no submanifolds can be derived from holomorphic ( v ) and anti-invariant ( v ) submanifoldssince N − D ( v ) = N − D ( v ) = ∅ .We now recall the notion of bi-slant submanifolds of an almost Hermitian manifold. Let M bean almost Hermitian manifold and N a real submanifold of M . Then we say that N is a bi-slantsubmanifold [25] if there exist two orthogonal distributions D θ and D θ on N such that1. T N admits the orthogonal direct decomposition
T N = D θ ⊕ D θ .2. The distributions D θ and D θ are slant distributions with slant angles θ and θ . A. Gursoy / Construction of networks by associating with submanifolds of almost Hermitian manifolds v v v v v v v Figure 6. Digraph D built by submanifolds in D and the bi-slant submanifold It is easy to see that if D θ = { } (or D θ = { } ), N becomes a slant submanifold with a slant angle θ . If θ = θ = θ = { } , then N becomes a holomorphic submanifold. If θ = θ = θ = π ,then N becomes an anti-invariant submanifold. Moreover if θ = π and θ = 0 , then N becomes aCR-submanifold. Furthermore, θ = π and θ = 0 , then N becomes a hemi-slant submanifold andsemi-slant submanifold, respectively. We denote bi-slant submanifolds by v .Digraph D = ( V, A ) has seven vertices, V ( D ) = { v , v , v , v , v , v , v } , and 12 arcs, A ( D ) = { ( v , v ) , ( v , v ) , ( v , v ) , ( v , v ) , ( v , v ) , ( v , v ) , ( v , v ) , ( v , v ) , ( v , v ) , ( v , v ) , ( v , v ) , ( v , v ) } in Fig. 6. D has the maximum length of threeas the longest path. It has a vertex ( v ) which is not reachable. Using transitive closure, D has fivenew direct connections such as v → v , v → v , v → v , v → v and v → v . Topological sortof D is v − v − v − v − v − v − v . rad ( D ) = 1 , the radius of D is v → v . diam ( D ) = 2 ,the diameter of D is v → v → v . Also, in D , there are three center vertices as v , v and v , andtwo peripheral vertices as v and v . Theorem 3.9.
For the digraph D created by adding the bi-slant submanifolds as vertex v to the D ,i D is a three-partite digraph.ii The maximum matching is 3.iii The independence number is 3.iv The chromatic number is 3.v D has a directed spanning tree.vi The domination number is 3. Proof: i There is a partition V = { v , v } , V = { v , v , v } and V = { v , v } of V ( D ) as three subsets,and these subsets support that V ( D ) = S i =1 V i and V i ∩ V j = Ø ( i, j = 1 , , and i = j ). Then, D , containing the subsets, is actually a three-partite digraph. . Gursoy / Construction of networks by associating with submanifolds of almost Hermitian manifolds ii M = { ( v , v ) , ( v , v ) , ( v , v ) } is an arc subset in D , and | M | = 3 . According to this, M , thatincludes no common vertices and no loops, is a matching. Since no other subset greater cardinalitythan M , D has a maximum matching called M .iii The subset e V = { v , v , v } is an independent set having maximum cardinality. It also meansthat there is no direct relationship between any two elements, i.e. submanifolds, in e V . Then, theindependence number of D is three, because | e V | = 3 .iv The minimum number of disjoint independent sets of D is three: V = { v , v } , V = { v , v , v } and V = { v , v } . According to that, three different colors are needed to coloring D and that’swhy the chromatic number of D is three.v D has a directed spanning tree of length 6 and root at v such as in Fig. 7. It also means thatthere is a transformation from submanifolds v to all other submanifolds in D at most two-step. v v v v v v v Figure 7. A directed spanning tree in D vi There is a subset e V = { v , v , v } ⊂ V ( D ) . According to this subset, that having the minimumcardinality, and for each vertex v ∈ e V and u ∈ V ( D ) − e V , ( v, u ) is an arc in D . The dominationnumber is three. ⊓⊔ Corollary 3.10.
In the submanifold network represented by D in Fig. 6, all other submanifolds canbe derivated from bi-slant ( v ) submanifold since v is the root vertex of the directed spanning tree of D and N + D ( v ) = { v , v } in Fig. 7. Also, no submanifolds can be transformed to bi-slant because N − D ( v ) = ∅ .Digraph D = ( V, A ) has also seven vertices as well as D , V ( D ) = { v , v , v , v , v , v , v } ,and 12 arcs, A ( D ) = { ( v , v ) , ( v , v ) , ( v , v ) , ( v , v ) , ( v , v ) , ( v , v ) , ( v , v ) , ( v , v ) , ( v , v ) , ( v , v ) , ( v , v ) , ( v , v ) , ( v , v ) , ( v , v ) } in Fig. 8. D has the maximum length of three as thelongest path. It has a vertex ( v ) which is not reachable. Using transitive closure, D has four newdirect connections such as v → v , v → v , v → v and v → v . Topological sort of D is v − v − v − v − v − v − v . rad ( D ) = 1 , the radius of D is v → v . diam ( D ) = 2 , thediameter of D is v → v → v . Also, in D , there are four center vertices as v , v , v and v , andone peripheral vertex as v . A. Gursoy / Construction of networks by associating with submanifolds of almost Hermitian manifolds v v v v v v v Figure 8. Digraph D built by D with arcs ( v , v ) and ( v , v ) Theorem 3.11.
For the digraph D created by adding two more relations from semi-slant to holomor-phic and from hemi-slant to slant as arcs to the D ,i D is a three-partite digraph.ii The maximum matching is 3.iii The independence number is 3.iv The chromatic number is 3.v D contains a directed spanning tree.vi The domination number is 2. Proof: i See Teorem 3.9-i.ii See Teorem 3.9-ii.iii See Teorem 3.9-iii.iv See Teorem 3.9-iv.v D has a directed spanning tree having the same structure as in Fig. 7. See Theorem 3.9-v.vi There is a subset e V = { v , v } ⊂ V ( D ) . According to that, the subset has the minimumcardinality while dominating all other vertices, and for each vertex v ∈ e V and u ∈ V ( D ) − e V , ( v, u ) is an arc in D . The domination number is two. ⊓⊔ Corollary 3.12.
In the most comprehensive submanifold network represented by D in Fig. 7, justtwo submanifolds, holomorphic ( v ) and anti-invariant ( v ) , are not generative since d + ( v ) = d + ( v ) =0 . Besides, bi-slant ( v ) is the most productive submanifold owing to transforming to all other sub-manifolds. . Gursoy / Construction of networks by associating with submanifolds of almost Hermitian manifolds Using the seven submanifolds, named as holomorphic, CR, anti-invariant, slant hemi-slant, semi-slant and bi-slant, it is constructed six digraphs, called D , D , D , D , D and D , whose verticesare submanifolds and arcs are connections among submanifolds from one to another. Theorem 3.13.
Let D be a digraph which indicates digraphs from { D , D , D , D , D , D } . D provides the following properties:i Simple directed graph.ii Directed acyclic graph.iii Weakly connected. Proof: i In digraph D , there is no more than one relationship between any two submanifolds and no trans-formations from a submanifold to itself. According to that, D is a simple directed graph.ii Given a transition list among submanifolds such as v v ...v k , meaning that v is the source sub-manifold and v k is the sink submanifold. Because D doesn’t have any transition list including thesame submanifold is both source and also sink, D is acyclic. That’s why D is a directed acyclicdigraph.iii D has one pair of submanifolds as a relation at least that they can not mutually be transformedfrom one to another submanifold. Hence, D is not strongly connected. However, when D , thatconsidered as without direction of transformations, is connected, named connectedness of under-lying graph because there are no isolated submanifolds. For this reason, D is weakly connected. ⊓⊔ Corollary 3.14.
Among all digraphs D , D , D , D , D , and D , the digraph D has• the maximum vertex-integrity, and• the maximum edge-integrityas well as the maximum size by Proposition 2.2.
4. Conclusion
Manifold learning plays an important role in analyzing data lying on a non-linear space as a part ofmachine learning. Moreover, the geometric deep learning yields using the concepts of manifolds andgraphs together in building convolutional deep structures. In this paper, using holomorphic subman-ifolds, anti-invariant submanifolds, CR-submanifolds, slant submanifolds, semi-slant submanifolds,hemi-slant submanifolds and bi-slant submanifolds in almost Hermitian manifolds, it is given rela-tions among them, six different digraphs are created as a network of these submanifolds, and main A. Gursoy / Construction of networks by associating with submanifolds of almost Hermitian manifolds properties of them are first examined in terms of digraphs. We note that there is a much wider classthat includes slant submanifolds. This class was first defined in [28] by Etayo as quasi-slant submani-folds. Later, these submanifolds were called pointwise slant submanifolds in [29] by Chen and Garay.Although we have excluded such submanifolds in this article, our next research will be to examine theconnections between these submanifolds and graph theory.
References [1] Zheng N, Xue J. Statistical learning and pattern analysis for image and video processing. Springer Science& Business Media, 2009.[2] Monti F, Boscaini D, Masci J, Rodola E, Svoboda J, Bronstein MM. Geometric deep learning on graphsand manifolds using mixture model cnns. In: Proceedings of the IEEE Conference on Computer Vision andPattern Recognition (CVPR 2017), 2017 pp. 5115–5124.[3] Aquib M, Lee JW, Vˆılcu GE, Yoon DW. Classification of Casorati ideal Lagrangian submanifolds in com-plex space forms.
Differential Geometry and its Applications , 2019. (63):30–49.[4] Lee J, Vˆılcu GE. Inequalities for generalized normalized δ -Casorati curvatures of slant submanifolds inquaternionic space forms. Taiwanese Journal of Mathematics , 2015. (3):691–702.[5] Vˆılcu GE. An optimal inequality for Lagrangian submanifolds in complex space forms involving Casoraticurvature. Journal of Mathematical Analysis and Applications , 2018. (2):1209–1222.[6] Vakulenko S, Radulescu O. Flexible and robust patterning by centralized gene networks.
FundamentaInformaticae , 2012. (4):345–69.[7] Fiorini RA. Computerized tomography noise reduction by CICT optimized exponential cyclic sequences(OECS) co-domain.
Fundamenta Informaticae , 2015. (2–3):115–34.[8] Carriazo A, Fern´andez L. Submanifolds associated with graphs.
Proceedings of the American MathematicalSociety , 2004. (11):3327–3336.[9] Carriazo Rubio A, Fern´andez LM, Rodr´ıguez Hidalgo A. Submanifolds weakly associated with graphs.
Proceedings of the Indian Academy of Sciences. Mathematical sciences , 2009. (3):297–318.[10] Boza L, Carriazo A, Fern´andez LM. Graphs associated with vector spaces of even dimension: A link withdifferential geometry.
Linear Algebra and its Applications , 2012. (1):60–76.[11] Bang-Jensen J, Gutin GZ. Digraphs: theory, algorithms and applications. Springer Science & BusinessMedia, 2008.[12] Belmonte R, Hanaka T, Katsikarelis I, Kim EJ, Lampis M. New results on directed edge dominating set. arXiv preprint , 2019. arXiv:1902.04919.[13] Bondy JA, Murty USR. Graph theory with applications. Macmillan London, 1976.[14] Chartrand G, Lesniak L, Zhang P. Graphs & digraphs. Chapman and Hall/CRC, 2010.[15] Cormen TH, Leiserson CE, Rivest RL, Stein C. Introduction to algorithms. MIT press, 2009.[16] Rosen KH, Krithivasan K. Discrete mathematics and its applications. McGraw-Hill, 2013.[17] Sedgewick R, Wayne K. Algorithms. Addison-Wesley, 2015. 4th Edition. . Gursoy / Construction of networks by associating with submanifolds of almost Hermitian manifolds [18] Chartrand G, Tian S. Distance in digraphs. Computers & Mathematics with Applications , 1997. (11):15– 23. doi:10.1016/S0898-1221(97)00216-2.[19] Lee CW. Domination in digraphs. Journal of the Korean Mathematical Society , 1998. (4):843–853.[20] Pang C, Zhang R, Zhang Q, Wang J. Dominating sets in directed graphs. Information Sciences , 2010. (19):3647–3652.[21] Vandell RC. Integrity of digraphs.
Dissertations , 1996.[22] Yano K, Kon M. Structures on Manifolds.
World Scientific , 1985. doi:10.1142/0067.[23] Bejancu A. Geometry of CR-submanifolds. Springer Science & Business Media, 2012.[24] Chen BY. Geometry of slant submanifolds. Katholieke Universiteit Leuven, 1990.[25] Carriazo A. Bi-slant immersions. In: Proceedings ICRAMS 2000, Kharagpur, India, 2000 pp. 88–97.[26] Sahin B. Warped product submanifolds of Kaehler manifolds with a slant factor.
Annales Polonici Math-ematici , 2009. :207–226.[27] Papaghiuc N. Semi-slant submanifolds of a Kaehlerian manifold. Ann. St. Univ. Iasi. tom. , 1994. :55–61.[28] Etayo F. On quasi-slant submanifolds of an almost Hermitian manifold. Publicationes Mathematicae-Debrecen , 1998. (1–2):217–223.[29] Chen B, Garay O. Pointwise slant submanifolds in almost Hermitian manifolds. Turkish J. Math. , 2012. (4):630–640.[30] Murathan C, S¸ ahin B. A study of Wintgen like inequality for submanifolds in statistical warped productmanifolds. Journal of Geometry , 2018.109