Research Article

Johan Kok, Naduvath K. Sudev*, Kaithavalappil P. Chithra and Augustine Mary

Jaco-type graphs and black energy dissipation

DOI: 10.1515/apam-2016-0056

Received June 21, 2016; revised January 15, 2017; accepted January 20, 2017

Abstract: In this paper, we introduce the notion of an energy graph G of order n e N. Energy graphs are simple, connected and finite directed graphs. The vertices, labelled u\, u2,..., un, are such that (u/, Uj) i A(G) for all arcs (u/, Uj) with i > j. Initially, equal amount of potential energy is allocated to certain vertices. Then, at a point of time, these vertices transform the potential energy into kinetic energy and initiate transmission to head vertices. Upon reaching a head vertex, perfect elastic collisions with atomic particles take place and propagate energy further. Propagation rules apply which could result in energy dissipation. The total dissipated energy throughout the graph is called the black energy of the graph. The notion of the black arc number of a graph is also introduced in this paper. Mainly Jaco-type graphs are considered for the application of the new concepts.

Keywords: Energy graph, black energy, black arc number, black cloud, solid subgraph, Jaco-type graph MSC 2010: 05C07, 05C38, 05C75, 05C85

1 Introduction

For general notations and concepts in graphs and digraphs, see [1, 2, 5, 13]. Unless mentioned otherwise, all graphs in this paper are simple, connected and finite directed graphs (digraphs). When required, an undirected graph will be considered and then a digraph will be obtained according to specific vertex labelling and orientation of edges. If in a digraph G a directed path exists between the vertices ui and Uj, then the distance between the vertices is the minimum (shortest) length of a directed path between them and is denoted dG(ui, uj).

We observe that an undirected graph G of order n > 2 allows a vertex labelling, say ui, u2,..., un, and an orientation such that (ui, uj) i A(G) for all arcs (ui, uj) with i > j. This can be established as explained below.

Any arbitrary vertex can be labelled ui and all edges incident with ui can be orientated as out-arcs. Thereafter, all neighbours of ui can randomly be labelled u2, u,..., udG(ui). Sequentially, the resultant edges (if any) of vertex u2 can be orientated as out-arcs to its resultant neighbours, followed by similar orientation of the resultant edges of u3, u4,..., udG(ui). Proceeding iteratively, the graph G allows a vertex labelling and orientation as prescribed.

Johan Kok: Tshwane Metropolitan Police Department, City of Tshwane, South Africa, e-mail: kokkiek2@tshwane.gov.za Corresponding author: Naduvath K. Sudev: Centre for Studies in Discrete Mathematics, Vidya Academy of Science & Technology, Thalakkottukara, Thrissur, India, e-mail: sudevnk@gmail.com

Kaithavalappil P. Chithra: Naduvath Mana, Nandikkara, Thrissur, India, e-mail: chithrasudev@gmail.com

Augustine Mary: Department of Mathematics, Nirmala College For Women, Coimbatore, India, e-mail: marycbe@gmail.com

We recall that a digraph G is strongly connected if and only if there exists a directed path from every vertex to every other vertex in V( G). Otherwise, G is said to be weakly connected. Clearly the above mentioned process of labelling the vertices and assigning orientation to the edges of a given graph is well known and called topological ordering (see [3]). These graphs are all weakly connected. In this study the digraph thus obtained is called an energy graph.

In the digraph obtained above the vertex ui has d^(ui) = 0 and is called a source vertex, and the vertex un and possibly others with out-degree equal to zero are called sink vertices.

In view of the above definition on energy graphs, the following is an immediate result.

Lemma 1.1. An energy graph of order n > 2 has at least one source vertex and at least one sink vertex.

Proof. Firstly, that some energy graphs have exactly one source vertex follows from the topological ordering. Now, if the topological ordering begins with a number of non-adjacent vertices, say Z < n such vertices, then the vertices can randomly be labelled ui, u2,..., ue. Completing the vertex labelling and edge orientation as described, clearly results in Z source vertices. Therefore, an energy graph has at least one source vertex. In a similar way, we can establish the existence of at least one sink vertex. □

In general, a simple and connected digraph of order n > 2 does not necessarily allow a vertex labelling to render it an energy graph. The aforesaid is obvious for strongly connected digraphs. However, it can be observed that a simple, connected acyclic (or weakly connected) digraph always allows such vertex labelling. Henceforth, unless mentioned otherwise, we will consider weakly connected digraphs.

1.1 Black energy dissipation within an energy graph

The energy propagation model obeys the following propagation rules. An energy graph per se has zero mass. Each vertex ui is allocated a number equal to d+ (ui) of atomic particles, all of equal mass m > 0. All the atomic particles are initially at rest at their respective vertices.

All source vertices uk are initially allocated an equal amount of potential energy of % > 0 joules. The allocated potential energy is shared equally amongst the atomic particles. At time t = 0, all source vertices simultaneously ignite kinetic energy, and only one atomic particle per out-arc transits an out-arc towards the head vertex. Hence, when transition ignites for each source vertex uk with d+(uk) = Z, we have Z • 2 mv2 = % joule.

Regardless of the true speed v of an atomic particle, the time to transit along an arc is considered to be 1 time unit. Therefore, the minimum travelling time t for some atomic particle to reach a head vertex uj from ui is equal to the distance between ui and uj, i.e., t = dG(ui, uj).

Upon one or more, say Z' of the first atomic particles reaching a head vertex uj simultaneously, the atomic particles merge into a single atomic particle of mass Z'm and engage in a perfect elastic collision with the atomic particles allocated to the head vertex uj. Clearly these atomic particles transited from one or more source vertices. The energy distance, denoted by e-dG(ui, uj), is the minimum over all distances from source vertices to uj. A directed path corresponding to e-dG(ui, uj) is called an energy path. For the perfect elastic collision, the laws of conservation of momentum and total energy apply. The kinetic energy of all atomic particles reaching the head vertex uj later than t = e-dG(ui, uj), together with the mass-energy equivalence mc2 (c being the speed of light), dissipate into the surrounding universe. We call an in-arc of a vertex uj that does not lie on some energy path from some source vertex uj a black arc. The total sum of all dissipated energy after propagation has completed is called the black energy of the energy graph and denoted by EG.

The kinetic energy reaching a sink vertex ut is stored (capacitated) as potential energy of %'ut joule. By the conservation of momentum and total energy, it follows that

Eg = ^ mc2 • d+(ui) + ^ %uk - ^ t - (total dissipated mass-energy equivalence).

i=i (uk a source vertex) (ut a sink vertex)

To find a closed-form expression, the last term must be determined for an energy graph G.

2 Black energy dissipation within Jaco-type graphs

For ease of the introductory analysis, we consider graphs with well-defined vertex labelling and well-defined orientation. The families of graphs which naturally offer this research avenue are Jaco graphs and Jaco-type graphs.

The concept of linear Jaco graphs was introduced in [7] and initially studied in [7, 8]. Further studies on these graph classes have been reported in [6, 10, 11] and following these studies some other significant papers have been published.

In [10], it is reported that a linear Jaco graph Jn (x) can be defined as the graphical embodiment of a specific sequence. The introductory research (see [10]) dealt with non-negative and non-decreasing integer sequences. This observation has opened a wide scope for the graphical embodiment of countless other interesting integer sequences. These graphs are broadly termed as Jaco-type graphs and the notions of finite and infinite Jaco-type graphs are as given below.

Definition 2.1 ([10]). For a non-negative and non-decreasing integer sequence {an}, the infinite Jaco-type graph, which is denoted by Jm({an}), is a directed graph with vertex set V(Jtx>({an})) = {u/ : i e N} and arc set A(Jm({an})) c {(u/, uj) : i, j e N, i < j} such that (u/, uj) e A(Jm({an})) if and only if i + a/ > j.

Definition 2.2 ([10]). For a non-negative and non-decreasing integer sequence {an}, the family of finite Jaco-type graphs is the set of finite subgraphs of the infinite Jaco-type graph Jm({an}), n e N. A Jaco-type graph (a member of the family or an element of the set) is denoted by Jn({an}).

Jaco graphs and Jaco-type graphs can be used for modelling many theoretical and practical problems. In this paper, we discuss some of the applications of these types of graph classes.

Lemma 2.3. Finite Jaco-type graphs are energy graphs.

Proof. In view of Definitions 2.1 and 2.2, it can be noted directly that a finite Jaco-type graph has both a unique source vertex ui and a primary sink vertex un. Furthermore, (u/, uj) i A(Jn({an})) if i > j. Hence, every finite Jaco-type graph is an energy graph. □

An interesting directed complement graph exists, namely, the graph G' with arcs (n, m) such that n < m and (n, m) i A(G). Hence, all pairs (n, m) such that an + n < m belong to G', so a strict order on N is defined.

Ordered sets obtained in this way have a minimal type. This means that the sets are infinite and no proper initial segment is finite. Conversely, a poset P has minimal type if and only if its elements can be enumerated in a sequence{vn}n£N, and there is a sequence {an}n£N of positive integers such that vn < vm if n + an < m. Also an interval order is an ordered set P in which the poset Q made of the direct sum of two 2-element chains does not embed. If moreover, P does not embed the poset made of the direct sum of a 3-element chain and a 1-element chain, then P is a semi-order. Also, for interval orders both the predecessor and the successor order are total quasi-order (see [4, 12])1. This implies that a graph G is a Jaco-type graph with enumeration {vn }n if and only if the order attached to the enumeration is the predecessor order that corresponds to the order G' on the directed complement as defined above. Finally, an order is minimal if and only if it extends the directed complement of a Jaco-type graph.

2.1 Jaco-type graph for the sequence of natural numbers

For purpose of notation, all sequences will be labelled si, i e N. The definition of the infinite Jaco-type graph corresponding to the captioned sequence can be derived from Definition 2.1. We have that the graph Jm(si) is defined by VJm(si)) = {u/ : i e N}, AQm(si)) c {(u/, uj) : i, j e N, i < j} and (u/, uj) e AQm(si)) if and only

1 This information was communicated to us by the referee.

if 2i > j. Note that a finite Jaco-type graph Jn(s1) in this family is obtained from Jm(s1) by lobbing off all vertices uk (with incident arcs) for all k > n.

The arrival times of the atomic particles at a head vertex uj will be stringed and shall be denoted by Uj - (t1, t2,..., td-(Uj}) with tj < tl for i < i, l < dG(uj). For the Jaco-type graph Js(si), we find that (see Figure!) ui ~ (0), u2 ~ (i), U3 ~ (2), U4 ~ (2,3},hence( | % + mc2) joules transiting along arc (u3, U4) dissipate as black energy. Then we have us - (3, 3), so two atomic particles merge to collide perfectly at us with % joules. Then u6 - (3, 3, 4), hence % joules collide perfectly at u6, and (77 % + mc2) joules dissipate as black energy. At vertices u7 and u8, the arrival times are stringed as u7 - (3, 4, 4) and u8 - (3,4,4,4), respectively. At uj and u8, energy amounting to (J2% + 478% + 2mc2) and (jj% + 478% + 8% + 3mc2) joules, respectively, dissipate as black energy. Only (8% + mc2) joules reach the sink vertex u8. The energy graph has a total of (8%+9mc2) joules potential energy and mass-energy equivalence capacitated within the graph. Hence, a total of (8% + 7mc2) joules dissipated as black energy. Figure 1 depicts J8(s1).

2.2 Jaco-type graph for the Fibonacci sequence

The definition of the infinite Jaco-type graph corresponding to the Fibonacci sequence, s2 = {fn}, f0 = 0, f1 = 1, f2 = 1, fn = fn-1 + fn-2, n = 1, 2, 3,... can be derived from Definition 2.1. We have that the graph JOT(s2) is defined by V(JOT(s2)) = {uj : i € N}, A JOT(s2)) c {(uj, uj) : i, j € N, i < j} and (uj, uj) € A J«,(s2)) if and only if i + fi > j.

For the Jaco-type graph J12(s2), we find (see Figure 2) u1 ~ (0), u2 ~ (1), u3 ~ (2), u4 ~ (3), us - (3, 4), u6 - (4, 4), uj - (4, 4, S), u8 - (4, S, S), u9 - (4, S, S, S), uw - (4, S, S, S, S), un - (S, S, S, S, S) and u12 - (S, S, S, S, S, 6). This implies that only (H3% + Smc2) joules of kinetic energy capacitate at the sink vertex u12. The energy graph has a total of (^% + 21mc2) joules potential energy and mass-energy equivalence allocated within the graph. This means that (UJ % + 12 mc2) joules dissipate into black energy. Figure 2 depicts J12(s2).

2.3 Number of black arcs of certain graphs

For a simple connected graph G in general, the number of black arcs depend on the orientation of a corresponding energy graph. In general an energy graph for a given graph G is not unique. We denote an orientation of a graph by ^(G). We denote by b^(G}(G) the number of black arcs with respect to a given ^(G). We define the black arc number b'(G) = min^(G}{b^(G}(G)}. First we present a perhaps obvious, but useful lemma.

Lemma 2.4. If the stringed arrival times of atomic particles at vertex uj in an energy graph G is given by uj ~ {ti, t2,..., te, te+i, te+2,..., td-G (uj)) and ti = t2 = ■■■ = te < te+i < te+2 < ■■■ < Vd-G (uj), then dG (uj) - Z in-arcs of uj dissipate black energy.

Proof. Because ti = t2 = ■■■ = te < te+i < te+2 < ■■■ < ud-(u,), the atomic particles at vertices u/, i < i < Z, simultaneously arrive first at vertex uj and they merge to collide perfectly with the atomic particles allocated to uj. In terms of the rules of the energy propagation model, all other atomic particles along the other in-arcs of uj dissipate black energy. □

To illustrate the use of Lemma 2.4, we apply it to star graphs, paths and cycles. First, an obvious corollary.

Corollary 2.5. We have b'(G) = 0 if and only if E(G) = 0.

Note that for an energy graph G, the portion of black energy resulting from atomic particles dissipating their mass-energy equivalence is b'(G) ■ mc2 joules.

Proposition 2.6. For star graphs Sin, n > 3, for paths Pn, n > 2, and for cycles Cn, n > 3, the following hold:

(i) b'(Si,n) = 0 and E(Si,n) = 0 for all y(Si,n).

(ii) b'(Pn) = 0 and E(Pn) = 0.

(iii) We have

Proof. Part (i): Consider a star graph Sin, n > 3, and first orientate all edges as out-arcs from the central vertex. Label the central vertex as ui and randomly vertex label the pendant vertices u/, 2 < i < n + i. Clearly, at t = i a single atomic particle has transited an arc to reach a corresponding pendant sink vertex. All arcs were transited simultaneously and hence u/ ~ {i), 2 < i < n + i. Hence, b'(Si>n) = 0 and thus E(Si>n) = 0. Now take the inverse orientation and the reasoning remains the same.

Finally, without loss of generality, consider the first orientation of the star graph and inverse the orientation of any number of Z < n arcs. Now the star digraph has Z source vertices. At t = i, Z atomic particles will merge at ui for a perfect collision to transfer Z ■ % joules to the n - Z atomic points allocated to ui. So ui ~ {i, i, i,..., i) (Z entries). The aforesaid kinetic energy will equally divide amongst n - Z atomic particles, each transited an arc to reach a corresponding pendant sink vertex uj. For each sink vertex, we have Vj ~ {2). The total energy is conserved, hence E(Si>n) = 0, implying b'(Si>n) = 0. The result holds for all y(Siin), n > 3.

Part (ii): Consider any path Pn, n > 2, and label the vertices to have the path uiu2 ■■■ un and orientate with arcs (u/, uj), i < i < n - i, j = i + i. Clearly, on igniting propagation of energy and atomic particle mass, we have u/ ~ {i - i), i < i < n. Since a single atomic particle arrives at each vertex uj, j = i, no black edges exist. Therefore, b'(Pn) = 0, which implies E(Pn) = 0.

Part (iii): Consider a cycle Cn, n > 4, with n being even. Let the vertices of Cn be located along the circumference of a circle with vertex ui at the exact-top, and un at the exact-bottom of the circle. From ui label the vertices located anticlockwise consecutively u2, u3,..., un, and orientate all corresponding edges including unun, anticlockwise. Similarly, from ui label the vertices located clockwise consecutively un+i, un+2,..., un-i, and orientate all corresponding edges, including un-iun, clockwise. Clearly, this construction of an energy cycle does not contradict generality. Hence, on igniting propagation of energy and atomic particle mass, we have u/ ~ {i - i), i < i < 2,and un~ {i), i < i < 2 - i. Therefore, un ~ {j, j).In view of Lemma 2.4, no black arc exists. Therefore, b'(Cn) = 0 implies E(Cn) = 0 if n is even.

The vertex labelling and orientation of a cycle Cn, where n > 3 and is odd, follows similar to the previous case with the anticlockwise vertex labels running through u2, u3,..., u[ n j, and the clockwise vertex labels running through u|_nj+i, u[aj+2,..., un-i. Note that the arc (un-i, un) results in un ~ (Lf J, T21). So exactly one arc results in a black arc with black energy (2% + mc2) joules. Therefore, b'(Cn) = i implies E = i% + mc2 because the initial amount of potential energy capacitated at ui is % joules. □

Theorem 2.7. An acyclic graph G of order n > 2 has b'(G) = 0 and therefore E(G) = 0.

b' (Cn)

0 if n is even,

1 if n is odd,

and E(Cn)

0 if n is even,

i% + mc2 if n is odd.

Proof. An acyclic graph G is a bipartite graph. Partition V(G) into disjoint subsets X, Y and without loss of generality, orientate each edge uw, u € X, w € Y, to be the arc (u, w). Label the vertices in X as u1, u2,..., u|X| and those in Y as W|X+, W|X|+2,..., W|X|+|Y|. Clearly, the resultant vertex labelled digraph G is an energy graph. Furthermore, at t = 1 all atomic particles transit along an arc from a corresponding source vertex in X to a sink vertex in Y. Therefore, b'(G) = 0. It immediately follows that E(G) = 0. □

Corollary 2.8. A simple connected graph G of order n > 2 which has no odd cycle has b'(G) = 0, and therefore

E(G) = 0.

Proof. It is well known that a graph which has no odd cycle is bipartite. The result then follows from the proof of Theorem 2.7. □

An energy graph G with E(G) = 0 (b'(G) = 0) is said to satisfy the law of conservation of total energy. A characterisation of these graphs follows in the next result.

Theorem 2.9. An energy graph G of order n > 2 satisfies the law of conservation of total energy if and only if G has no odd cycle.

Proof. If G has no odd cycle, then the result follows from Corollary 2.8.

Assume that G satisfies the law of conservation of total energy and has an odd cycle. Hence, E(G) = 0. However, since m > 0 and % > 0 and the order n € N is finite, a vertex of the odd cycle receives or propagates some joules of energy from in-arcs or along out-arcs, respectively. From Proposition 2.6 (iii), it follows that for all arcs along the odd cycle, at least one such arc (ui, uj) is a black arc in respect of the odd cycle as an induced subgraph of G. Thus, at least (1 mc2 + e) joules dissipate as black energy along the arc (ui, uj) during exhaustive propagation and b'(G) > 1. The aforesaid is a contradiction to the assumption. Therefore, the result holds. □

The Jaco-type graphs Jn(s1), Jn(s2) clearly have well-defined black arcs and therefore it is possible to determine the number of black arcs. The black arc number can be determined through the Jaco-type black arc algorithm which is presented next.

Definition 2.10. In a finite Jaco-type graph G of order n > 2 and for a vertex uj € V(G), the out-open neighbourhood is the set of head vertices of uj, and is denoted by H( uj). The set of vertices of the induced subgraph (H(uj)) is called the black cloud of uj.

2.4 Jaco-type black arc algorithm

Note that the algorithm is described informally and requires iterative refinement for computer application. Step 1: For a finite Jaco-type graph G, set i = 1, B0(G) = 0 and let Gi = G. Go to Step 2. Step 2: Set j = i, consider a vertex uj € V(Gj), and determine (H(uj)). Go to Step 3.

Step 3: Let Bj (G) = Bj-1(G) u E((H(uj))). If j = n - 1, go to Step 4. Else, let i = j + 1 and Gi = Gj - E((H(uj))).

Go to Step 2. Step 4: Set b*(G) = |Bn-1(G)| and exit.

We call the set Bn-1(G) the blackcloudof the graph G. Note that b*(G) = |Bn-1(G)| < | U;n=1 E((H(uj)))|.

Theorem 2.11. For a finite Jaco-type graph G the Jaco-type black arc algorithm is well defined and it converges.

Proof. Clearly, Step 1 is unambiguous and finite. Clearly, Step 2 is unambiguous and since G is finite the range j < n - 1 in Step 3 is well defined and finite which implies that Step 2 converges. Furthermore, since the out-neighbourhood of any vertex is well defined and the arcs are unambiguous in a simple digraph, the induced subgraph (H(uj)) is well defined. Since Gj is finite, determining (H(uj)) converges. Therefore, Step 3 converges. Step 4 is unambiguous and finite. Hence, the Jaco-type black arc algorithm is well defined and it converges. □

Theorem 2.12. The Jaco-type black arc algorithm determines all the black arcs of a finite Jaco-type graph G of order n > 2.

Proof. At any time ti which corresponds to the i-th iteration, the vertex ui € V(Gi) is under consideration. If any two out-neighbours of ui, say the vertices uk and ue, k < are adjacent, then the arc (uk, ue) exists. No energy transmission will be possible from uk to ue at time ti+1, hence all the energy allocated at the vertex uk will dissipate into black energy at ti+1. Therefore, the Jaco-type black arc algorithm determines all black arcs of G. □

Application 2.13. Applying the Jaco-type black arc algorithm on Js(s1) and J12(s2), we have

®7^Js(S1)) = {(u3, u4), (u5, u6), (u5, u7), (u5, u8), (u6, u7), (u6, u8), (u7, u8)}, ®11CJ12(S2)) = {(u4, u5), (u6, u7), (u6, u8), (u6, u9), (u6, u10), (u7 , u8), (u7, u9), (u7 , u10), (u8, u9), (u8, u10), (u9 , u10), (u11, u12)}.

Therefore, b'(J8(s1)) = 7 and b'(J12(s2)) = 12.

The notion of primitive holes in graphs has been introduced in [9], where the primitive degree dG (u) of a vertex u € V(G) was defined. Let the graphs Gi, 1 < i < n - 1, be those resulting from the Jaco-type black arc algorithm. Denote the underlying graph of G by G*. The next theorem represents b'(G) in terms of the primitive degrees found in Jaco-type graphs.

Theorem 2.14. For a finite Jaco-type graph G of order n > 2, we have b'(G) = ^n-1 hpG, (ui).

Proof. Lemma 2.3 states that a Jaco-type graph is an energy graph. From Definitions 2.1 and 2.2, it follows directly that each arc of the black cloud (H(ui)) is an edge of a primitive hole of G* with the common vertex ui. So the number of black edges associated with ui € V(G*) equals dPG.(ui). Hence, the result b'(G) = I"-! hPG' (ui) is settled. □

The subgraph G - Un-1 E((H(ui))) = G - Bn-1(G) is called solid subgraph of the propagating graph G and is denoted by Gs. For a given energy graph G (given orientation), the corresponding solid subgraph is unique. We now have a useful lemma with trivial proof.

Lemma 2.15. The number of arcs, | Bn-1 (G)|, equals the number of atomic particles which dissipated into mass-energy equivalence when propagation is exhausted. Similarly, the number of arcs, | A( Gs )|, equals the number of atomic particles which remained within the energy graph when propagation is exhausted.

Application 2.16. For the Jaco-type graph J8(s1), the number of atomic particles remained is only

|A J8(s1))| - b-J8(s1)) = 16 - 7 = 9 = |A(J8(s1))|, and the number of atomic particles dissipated is 7. In J12(s2), only

|AJ12(s2))| - b-J12(s2)) = 33 - 12 = 21 = |A(Js2(s2))| atomic particles remained and 12 are dissipated.

3 Application to binary code, gray code and modular arithmetic Jaco-type graphs

For a binary code of bit width n, the the binary combinations are always finite and an even number of combinations (that is, 2n numbers) exist. The binary combinations will be tabled so that consecutive rows map to consecutive decimal representations bi = i - 1, where i = 1, 2, 3,..., 2n of the binary code. Table 1 serves as an example of the convention for a binary code of bit width 3.

Contrary to the general finite Jaco-type graphs Jn({an}), n e N the graphical embodiment of the binary codes, denoted by Sn, n e N, is defined finitely on exactly 2n+1 vertices u\, u2, u3,..., u2n, u2n+1,..., u2n+1. We define d+ (u,) = bi + 1,1 < i < 2n and d+ (u,) = 2n - k, j = 2n + k, k = 1, 2, 3,..., 2n.

It follows easily that Sn - J2n+1 (s1). See Figure 1 for S2 - Js(s1).Thedisconnected graph obtained from the union of t > 2 copies of Sn has vertices labelled such that the corresponding vertices in the i-thcopy, i > 2, are u(i-1)2n+1 +j, 2 < i < t, 1 < j < 2n+1. Through easy sequential counting and indexing known results, the finite Jaco-type graph J2n+1 (s1) can be extrapolated. For example, for Ut copies Sn, with Sn the graphical embodiment of the binary code with bit width n, we have J(Ut copies Sn) = {up+j : up e J(J2n+1 (s1)), j = 0,1,..., (t - 1)2n+1}, IKUt copies Sn)| = t -Win+1 (S1 ))| and b'(Ut copies Sn) = t ■ b'(J2n+1 (S1).

The analysis for Gray codes follows similarly to that for binary codes, since the Gray combinations can be mapped onto exactly the same decimal representations.

3.1 Jaco-type graph for sequences modulo k

It is well known that for N0 and n, k e N, k > 2, modular arithmetic allows an integer mapping with respect to mod k as follows: 0 ^ 0 = m0, 1 ^ 1 = m1, 2 ^ 2 = m2,..., k - 1 ^ k - 1 = mk-1, k ^ 0 = mk, k + 1 ^ 1 = mk+1,____Let s3 = {an}, an = n (mod k) = mn. Consider the infinite root-graph JTO(s3), and define d+(ui) = mi for i = 1, 2,____

From the aforesaid definition, it follows that the case k = 1 will result in a null (edgeless) Jaco-type graph for all n e N. For the case k = 2, the Jaco-type graph for n > 2 and even is the union of 2 copies of directed P2. For the case k = 3, the Jaco-type graph is a directed tree, hence an acyclic graph G, and therefore b'(G) = 0. The smallest mod k Jaco-type graph which has a black arc is J4 (s3) for k = 4. Figure 3 depicts J12(s3) for k = 5.

We note that a Jaco-type graph Jn (s3) has [ ^ 1 sink vertices. The sink vertices resulting from the modular function is called mod-sink vertices. It follows from the Jaco-type black arc algorithm that b'(J12(s3)) = 5. A total of (24% + 2mc2), (214% + mc2) and (12% + 2mc2) joules reach the sink vertices v5, v10 and v12, respectively. A total of (% + 6mc2) joules dissipate as black energy along black arcs. Therefore, a total of (12% + 14mc2) joules are capacitated within the graph.

Lemma 3.1. A mod-sink vertex uj of the Jaco-type graphs Jn (s3), k > 3, has d-(uj) = L 2 J-

Proof. Consider any mod-sink vertex uj of the Jaco-type graph Jn (s3), n > k > 4. Clearly, the minimum t for which the arc (ut, uj) exists is Lf J. Hence, the result. □

For the case k = 4, 4 < n < 35, applying the Jaco-type black arc algorithm iteratively leads to the next result.

n 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19

b-(Jn (5з)) 1 1 1 2 3 3 4 5 5 6 6 7 7 8 8 9

n 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35

b-(Jn (S3)) 10 11 11 12 12 13 14 14 15 15 16 16 17 18 18 19

Table 2. Black arc number of Jn(sj) when n > 4 and k = 5.

Proposition 3.2. For s3 = {an}, an = n (mod 4), for the Jaco-type graph Jniss), we have

b'(Jn(Sj)) = [2J - 1.

Alternative,

b'(Jn(sj))

i for all n = 3 + 2i, i = 0,1, 2,..., i + 1 for all n > max{Z : Z < n and Z = 3 + 2i, i = 0,1, 2,

Proof. For the first case, it is trivially true that b'(Jn(S3)) < b"(Jn+1(S3)). Applying the Jaco-type black arc algorithm to both J4(s3) and J5(s3) results in the only black arc (u3, u4) for both, hence b"(/4(s3)) = b"(/5(s3)) = 1 = [2J - 1 = L 5 J - 1. Thus, the result holds for the integer pair, n = 4, 5. Assume the results holds for the integer pair n = t, t + 1, t € N. Then b'Jt(S3)) = L2J - 1 = LtJrJ - 1 = b'(Jt+i(s3)).

Observe that if t is even, then L 2 J - 1 = L J - 1. Since, for t" = t + 2, which could have been the initial even integer in N by the induction assumption, we have that L tr J - 1 = (L 2 J + 1) - 1, which implies b'Jv(s3)) = b'Jt(s3)) + 1 = b'Jt+1(s3)) + 1 = L 1-=т J - 1 = b"(Jt»+1(s3)). Hence, the result holds for the integer pair t" and t" + 1 or, to put in another way, it holds for the pair of Jaco-type graphs Jt» (s3) and Jt»+1(s3) with b'Jt»(s3)) = L2J - 1 = L^J - 1 = b'Jt>>+1(s3)). The general result, for all n > 4, n € N, follows by mathematical induction.

Alternatively, for all n = 3 + 2i, i = 0,1, 2,..., it follows from the first case that LЩ J -1 = L2 + 2 J -1 = '.

Since 3 + 2i, i = 0,1, 2..., is odd, the integer n > max{£ : Z < n and Z = 3 + 2i, i = 0,1, 2,...}, which implies n = Z + 1 and even. Therefore, from the first case it follows that b'(Jn(s3)) = i + 1. □

If k = 5, applying the Jaco-type black arc algorithm iteratively for n > 4, we get Table 2.

The black clouds are

S3CWS3)) = {(U3 , U4)}, B4J5 (S3)) = {(U3 , U4)}, ®5Je(S3)) = {(U3 , U4)}, ®6(J7(S3)) = {(U3 , U4), (U6, U7)}, ®7(J8(S3)) = {(U3 , U4), (U6, U7), (U7, Us)}, ®8(J9(S3)) = {(U3 , U4), (U6, U7), (U7, Us)}, ®9^J10(S3)) = {(U3 , U4), (U6, U7), (U7, Us), (U9, U10)}, ®10^11(S3)) = {(U3 , U4), (U6, U7), (U7, Us), (U9, U10), (U9 , U11)}, B11J12(S3)) = {(U3 , U4), (U6, U7), (U7, Us), (U9, U10), (U9 , U11)}, B12 J13 (S3)) = {(U3 , U4), (U6, U7), (U7, Us), (U9, U10), (U9 , U11), (U12 , U13)}, ®13^J14(S3)) = {(U3 , U4), (U6, U7), (U7, Us), (U9, U10), (U9 , U11), (U12 , U13)}, ®14^J15(S3)) = {(U3 , U4), (U6, U7), (U7, Us), (U9, U10), (U9 , U11), (U12 , U13), (U14, U15)}, ®15^J16(S3)) = {(U3 , U4), (U6, U7), (U7, Us), (U9, U10), (U9 , U11), (U12 , U13), (U14, U15), (U14, U16)}, ®16^J17(S3)) = {(U3 , U4), (U6, U7), (U7, Us), (U9, U10), (U9 , U11), (U12 , U13), (U14, U15), (U14, U16)}, B17^J1s(S3)) = {(U3 , U4), (U6, U7), (U7, Us), (U9, U10), (U9 , U11), (U12 , U13), (U14, U15), (U14, U16), (U17 , U1s)}, B1s^J19(S3)) = {(U3 , U4), (U6, U7), (U7, Us), (U9, U10), (U9 , U11), (U12 , U13), (U14, U15), (U14, U16), (U17 , U1s)},

®19(J20(S3)) = {("3 , "4), ("6, "7), ("7 , us), ("9 , "10), ("9 , "11), ("12 , "13), ("14, "15), ("14, "16), ("17, "18), ("19, "20)},

B20(J21(S3)) = {("3 , "4), ("6, "7), ("7 , "8), ("9 , "10), ("9 , "11), ("12 , "13), ("14, "15), ("14, "16), ("17, "18), ("19 , "20), ("19 , "21)},

B21 (J22(S3)) = {("3 , "4), ("6, "7), ("7 , "8), ("9 , "10), ("9 , "11), ("12 , "13), ("14, "15), ("14, "16), ("17, "18), ("19 , "20), ("19 , "21)},

B22(J23 (S3)) = {("3 , "4), ("6, "7), ("7 , "8), ("9 , "10), ("9 , "11), ("12 , "13), ("14, "15), ("14, "16), ("17, "18),

("19 , "20), ("19 , "21), ("22, "23)}, B23 (J24(S3)) = {("3 , "4), ("6, "7), ("7 , "8), ("9 , "10), ("9 , "11), ("12 , "13), ("14, "15), ("14, "16), ("17, "18),

("19 , "20), ("19 , "21), ("22, "23)}, B24(J25 (S3)) = {("3 , "4), ("6, "7), ("7 , "8), ("9 , "10), ("9 , "11), ("12 , "13), ("14, "15), ("14, "16), ("17, "18),

("19 , "20), ("19 , "21), ("22, "23), ("24, "25)}, B25 (J26(S3)) = {("3 , "4), ("6, "7), ("7 , "8), ("9 , "10), ("9 , "11), ("12 , "13), ("14, "15), ("14, "16), ("17, "18),

("19 , "20), ("19 , "21), ("22, "23), ("24, "25), ("24, "26)}, ®26(J27(S3)) = {("3 , "4), ("6, "7), ("7 , "8), ("9 , "10), ("9 , "11), ("12 , "13), ("14, "15), ("14, "16), ("17, "18), ("19 , "20), ("19 , "21), ("22, "23), ("24, "25), ("24, "26)},

B27(J28(S3)) = {("3 , "4), ("6, "7), ("7 , "8), ("9 , "10), ("9 , "11), ("12 , "13), ("14, "15), ("14, "16), ("17, "18),

("19 , "20), ("19 , "21), ("22, "23), ("24, "25), ("24, "26), ("27, "28)},

®28(J29(S3)) = {("3 , "4), ("6, "7), ("7 , "8), ("9 , "10), ("9 , "11), ("12 , "13), ("14, "15), ("14, "16), ("17, "18),

("19 , "20), ("19 , "21), ("22, "23), ("24, "25), ("24, "26), ("27, "28)} B29(J30(S3)) = {("3 , "4), ("6, "7), ("7 , "8), ("9 , "10), ("9 , "11), ("12 , "13), ("14, "15), ("14, "16), ("17, "18),

("19 , "20), ("19 , "21), ("22, "23), ("24, "25), ("24, "26), ("27, "28), ("29, "30)}, B30(J31(S3)) = {("3 , "4), ("6, "7), ("7 , "8), ("9 , "10), ("9 , "11), ("12 , "13), ("14, "15), ("14, "16), ("17, "18),

("19 , "20), ("19 , "21), ("22, "23), ("24, "25), ("24, "26), ("27, "28), ("29, "30)}, B31C/32(S3)) = {("3 , "4), ("6, "7), ("7 , "8), ("9 , "10), ("9 , "11), ("12 , "13), ("14, "15), ("14, "16), ("17, "18),

("19 , "20), ("19 , "21), ("22, "23), ("24, "25), ("24, "26), ("27, "28), ("29, "30), ("31, "32)}, B32CJ33 (S3)) = {("3 , "4), ("6, "7), ("7 , "8), ("9 , "10), ("9 , "11), ("12 , "13), ("14, "15), ("14, "16), ("17, "18), ("19 , "20), ("19 , "21), ("22, "23), ("24, "25), ("24, "26), ("27, "28), ("29, "30), ("31, "32), ("32, "33)},

B33 (J34(S3)) = {("3 , "4), ("6, "7), ("7 , "8), ("9 , "10), ("9 , "11), ("12 , "13), ("14, "15), ("14, "16), ("17, "18), ("19 , "20), ("19 , "21), ("22, "23), ("24, "25), ("24, "26), ("27, "28), ("29, "30), ("31, "32), ("32, "33)},

B34(J35 (S3)) = {("3 , "4), ("6, "7), ("7 , "8), ("9 , "10), ("9 , "11), ("12 , "13), ("14, "15), ("14, "16), ("17, "18), ("19 , "20), ("19 , "21), ("22, "23), ("24, "25), ("24, "26), ("27, "28), ("29, "30), ("31, "32), ("32 , "33), ("34, "35)}.

Figure 4 depicts J18(S3).

4 Application of a general black arc algorithm

A general black arc algorithm is not known yet. However, in the event of finding such an algorithm, the complexity of application to energy graphs with cut vertices can be simplified through partitioned application.

Figure 4. The graph7i8(s3).

It is known that a vertex ut e V(G) is a cut vertex of G if and only if the set of arcs A(G) can be partitioned into subsets Ai(G) and A2(G) and the induced arc-subgraphs (Ai(G)) and (A2(G)} have only the vertex ut in common.

Theorem 4.1. Let the energy graph G have c cut vertices and the subsets Ai(G), A2(G),..., A^(G) ofA(G) form the maximal partition of A(G) so that all pairs of distinct induced arc-subgraphs (At(G)) and {Aj(G)), where i, j e {1,2,..., €}, have at most one vertex in common. Then b'(G) = Ya=i b'((At (G))).

Proof. Clearly, an induced arc-subgraph (At (G)) has no cut vertex, otherwise the arc partitioning is not a maximum. Furthermore, for those distinct pairs of induced arc-subgraphs which have a vertex u in common, such u is a cut vertex of G. It is observed that for any induced arc-subgraph (At(G)), at least one induced arc-subgraph (Aj(G)) exists such that a common vertex u e V(G) between the arc-induced subgraphs exists, otherwise G is disconnected. Consider any distinct pair of arc-induced subgraphs (At(G)) and (Aj(G)) which share the common vertex u.

Case 1: If u is a source vertex in G, then u is a source vertex in (At(G)) and (Aj(G)), respectively. Hence, during the time interval [0,1) of propagation, no arc is rendered as black arc.

Case 2: If u is a sink vertex in G, then u is a sink vertex in (At(G)) and (Aj(G)), respectively. Let t' and t" be the total (exhaustive) propagation time through (At(G)) and (Aj(G)), respectively. Since a sink vertex can never initiate further propagation, it cannot propagate kinetic energy or particle mass which could dissipate. Hence, during the time interval [0, t') of propagation, the arcs in (At (G)) rendered as black arcs are identical to those found in G. Similar reasoning applies to (Aj(G)) over the propagation period [0, t").

Case 3: If u is an intermediate (d£(u) > 0, dG(u) > 0) vertex, then the independence of both propagation and the creation of black arcs is due to the fact that (ui, uk) $ A(G) for u\ e V((At(G))) and uk e V((Aj(G))). Therefore, the black arcs resulting from propagation through (At(G)) during [0, t') and those resulting from propagation through (Aj(G)) during [0, tff) are independent in each induced arc-subgraph, and therefore identical to those found in G.

Therefore, b'(G) = Xf=i b'((Ai(G))), completing the proof. □

5 Conclusion

There is a wide scope for further research in respect of total dissipated black energy and determining the black arc number for different classes of graphs. The Jaco-type black arc algorithm poses the challenge of complexity analysis. Such analysis will contribute to theoretical computer science. Determining the amount of kinetic energy which dissipate from an energy graph G is an open problem. Studying the properties of solid subgraphs is also open. Some other open problems we have identified during our study are the following.

Problem 1. Verify whether the solid subgraph of an energy graph G is connected.

Problem 2. Find a closed formula for b'(Jn(s3)), where s3 = {mod 5 sequence}, either in terms of the black arc number or in terms of appropriate vertex indices.

Problem 3. Describe a black arc algorithm for a general energy graph.

A valid observation made by the referee is that the sequence of positive integers need not necessarily be non-decreasing. This observation and the perspective from a set theory point of view, as mentioned in Section 2, open further research.

Acknowledgment: The authors of this article gratefully acknowledge the critical and constructive comments of the anonymous referee, which significantly improved the content and presentation of this article.

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