Stepwise transmission irregular graphs

Applied Mathematics and Computation 371 (2020) 124949

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Applied Mathematics and Computation journal homepage: www.elsevier.com/locate/amc

Stepwise transmission irregular graphs Andrey A. Dobrynin a,b,∗, Reza Sharafdini c a b c

Laboratory of Topology and Dynamics, Novosibirsk State University, Novosibirsk, 630090, Russia Sobolev Institute of Mathematics, Siberian Branch of the Russian Academy of Sciences, Novosibirsk, 630090, Russia Department of Mathematics, Persian Gulf University, Bushehr 7516913817, Iran

a r t i c l e

i n f o

Article history: Received 11 June 2019 Revised 18 November 2019 Accepted 29 November 2019

MSC: 05C12 05C05 05C07 05C90 05C35

a b s t r a c t The distance d(u, v) between vertices u and v of a connected graph G is defined as the number of edges in a shortest path connecting them. The transmission of a vertex v of G is the sum of distances from v to all the other vertices of G. A graph is stepwise transmission irregular (STI) if the transmissions of any two of its adjacent vertices differ by exactly one. Some basic properties of STI graphs are established and infinite families are constructed. © 2019 Elsevier Inc. All rights reserved.

Keywords: Graph distance Topological index Transmission

1. Introduction All graphs considered in this paper are undirected, connected, without loops and multiple edges. The vertex and edge sets of a graph G are denoted by V(G) and E(G), respectively. The order of a graph is its number of vertices. By distance d(u, v) between vertices u, v ∈ V(G) we mean the standard distance of a simple graph G, i.e., the number of edges on a shortest path connecting these vertices in G. The degree of a vertex v, denoted by deg(v ), is the number of edges incident with v. The transmission, σ (v), of vertex v ∈ V(G) is defined as the sum of distances from v to all the other vertices of G,  i.e., σ (v ) = u∈V (G ) d (v, u ). Transmissions can be considered as the main contributions in the constructions of numerous distance-based invariants and topological indices of abstract and molecular graphs. For instance, a half of the sum of vertex transmissions gives the Wiener index that has found important applications in chemistry [14,24]. Relationships between vertex transmissions are also the subject of studies. Several types of transmission irregularities are known. A graph is called transmission irregular if its vertices have pairwise different transmissions. Various properties of transmission irregular graphs are studied in [1–3,6–8]. The number of different vertex transmissions is known as the Wiener complexity of a graph [2]. Some topological indices can be calculated using an irregularity measure of vertex transmissions. For example, the Wiener index of trees and the Szeged index of bipartite graphs can be expressed through differences (σ (u ) − σ (v ))2 of all edges (u, v) [10,11]. ∗

Corresponding author at: Laboratory of Topology and Dynamics, Novosibirsk State University, Novosibirsk, 630090, Russia. E-mail address: [email protected] (A.A. Dobrynin).

https://doi.org/10.1016/j.amc.2019.124949 0 096-30 03/© 2019 Elsevier Inc. All rights reserved.

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28 Fig. 1. STI graphs with 13 vertices.

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Fig. 2. Smallest STI graphs.

In this paper, we consider graphs G in which equality |σ (u ) − σ (v )| = 1 holds for every edge (u, v) of G. Such graphs will be called stepwise transmission irregular (STI graphs). As an illustration, two STI graphs are shown in Fig. 1. Transmissions are listed near vertices. This paper is organized as follows: the second section is devoted to the basic properties of SIT graphs; several infinite families of STI graphs are described in the third section; in the fourth section, properties of some topological indices of STI graphs are considered. 2. Basic properties of STI graphs All STI graphs having n ≤ 7 vertices are shown in Fig. 2. Transmissions are indicated near vertices of graphs. It can be noted that these graphs are bipartite. The stepwise change of transmissions by traversing paths of a graph implies its bipartiteness. Proposition 1. Every STI graph is bipartite. Proof. Consider an arbitrary cycle v1 , v2 , . . . , vk and go along this cycle. Let p1 and m1 be the numbers of +1 and −1 which we add/subtract to/from vertex transmissions during visiting vertices v1 → v2 → · · · → vk → v1 . Then σ (v1 ) + p1 − m1 = σ (v1 ) and, therefore, p1 = m1 . This implies that the number of edges of the cycle is equal to p1 + m1 , i. e., the cycle has even length.  It is clear that sets of vertices of odd and even transmissions form two maximal independent sets of a graph. Therefore, a bipartite part of a STI graph contains vertices with the same parity of transmissions. For an edge (u, v) of a graph G, denote nu = |{w ∈ V (G ) : d (w, u ) < d (w, v )}| and nv = |{w ∈ V (G ) : d (w, v ) < d (w, u )}|. Since an n-vertex bipartite graph G does not contain odd cycles, n = nu + nv . The next well-known result is useful in considerations of bipartite graphs (see, for example, [13]). Lemma 1. If G is a bipartite graph, then σ (u ) − σ (v ) = nv − nu for every edge (u, v) of G and n = nu + nv . Let u be a pendant vertex of an arbitrary graph and v be a neighboring vertex of u. Then σ (v ) − σ (u ) = n − 2. If a STI graph G has a pendant vertex, then n = 3 and G∼ =P3 . Therefore, the path P3 is the unique STI graph with pendant vertices. Proposition 2. The number of vertices of a STI graph is odd. Proof. Let (u, v) be an arbitrary edge of a STI graph G with n vertices. By Lemma 1, |nu − nv | = |σ (v ) − σ (u )| = 1. Therefore, n = nu + nv is odd.  Obviously, bipartite regular graphs always have even number of vertices. Corollary 1. There are no regular STI graphs. Denote by Vo and Ve the numbers of vertices of odd and even degree of a graph, so = |Vo| and se = |Ve |. Corollary 2. The number of vertices with odd (even) transmissions of a STI graph is even (odd).

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Fig. 3. STI graph Lk of diameter 3.

  Proof. Let G be a STI graph of order n. Since v∈Vo σ (v ) + v∈Ve σ (v ) = 2W (G ), the number of vertices with odd transmission, so , is even. By Proposition 2, se = n − so is an odd number.  The parity of the number of edges of STI graphs depends on the parity of vertex degrees and transmissions. Denote by Uo the set of vertices with odd transmissions of a graph. Proposition 3. The number of edges of a STI graph is odd if and only if the number of vertices having both an odd degree and a transmission is odd. Proof. Let G be a STI graph with m edges. Then we can write

m=



[ σ ( u ) − σ ( v ) ]2 =

(u,v )∈E (G )

=





deg(v )σ (v )2 − 2

v∈V (G )



σ ( v )σ ( u )

(u,v )∈E (G )

deg(v )σ (v )2 − 2 (. . . ).

v∈Vo ∩Uo

The quantities σ (v)2 and σ (v) have the same parity.



Let G = K(n−1 )/2,(n+1 )/2 be the complete bipartite graph of odd order n. It is easy to verify that G is a STI graph. Indeed, suppose that (u, v) ∈ E(G) and vertex u belongs to the largest part of G. Then the difference of the vertex transmissions is σ (u ) − σ (v ) = (3n − 3 )/2 − (3n − 5 )/2 = 1. This gives an upper bound for the number of edges of STI graphs. Proposition 4. For a STI graph with n vertices and m edges, m ≤ (n2 − 1 )/4. STI graphs do not contain cut vertices. Proposition 5. STI graphs are 2-connected except the path P3 . Proof. Suppose that a STI graph G with n ≥ 4 vertices has a cut vertex v. Let G consists of two subgraphs A and B that have the unique common vertex v. Since G is bipartite and has no pendant vertices, a = |V (A )| ≥ 4 and b = |V (B )| ≥ 4. Let (v, u) and (v, w) be edges of A and B, respectively. If the edges are considered in A or B, we will use notation nAv , nAu , etc. We have σ (u ) − σ (v ) = nv − nu = (b + nAv ) − nAu = ±1 and σ (w ) − σ (v ) = nv − nw = (a + nBv ) − nBw = ±1. Then a + b = (nAu − nAv ) + (nBw − nBv ) ± 1 ± 1 ≤ (a − 2 ) + (b − 2 ) + 2 < a + b. This implies that G has no cut vertices.  A degree-based measure of edge irregularity was introduced in [17]. A graph G is called stepwise irregular if | deg(u ) − deg(v )| = 1 for every edge (u, v) of G. Proposition 6. Let G be a graph of diameter 2 with n vertices. Then G is a STI graph if and only if G is stepwise irregular. Proof. For an edge (u, v), we can write σ (u ) = deg(u ) + 2(n − deg(u ) − 1 ) and σ (v ) = deg(v ) + 2(n − deg(v ) − 1 ). Then σ (u ) − σ (v ) = deg(v ) − deg(u ).  3. Infinite families of STI graphs All STI graphs of small order can be found by computer calculations. For example, there are 1, 1, 3, 7, 18, 87, 1171 STI graphs with 3, 5, 7, 9, 11, 13, 15 vertices, respectively. Here we present several infinite families of STI graphs. The first family contains STI graphs of order 2k + 1 and diameter 3. Let graph Lk be obtained from the complete bipartite graph Kk,k by deleting a perfect matching and attaching a new vertex v to one part of vertices as shown in Fig. 3. Because of symmetry of Lk , it is sufficient to check vertex transmissions for edges (v, u) and (u, w). We have σ (v ) = 3k, σ (u ) = 3k + 1, and σ (w ) = 3k + 2. Therefore, Lk is a STI graph. Three infinite families contain STI graphs shown in Fig. 4. Graphs Gk , Hk , and Fk have order 2k + 3, 2k + 7, 3(2k + 1 ), and diameter k + 1, k + 2, and 2k + 1, respectively. Proposition 7. Gk , Hk , and Fk are STI graphs for all k ≥ 1.

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Fig. 4. STI graphs Gk , Hk , and Fk of large diameter.

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Fig. 5. STI graphs based on the prism and the Möbius ladder.

Proof. Because of symmetry of graphs in Fig. 4, it is sufficient to check difference between vertex transmissions for several edges. Denote by σ n the transmission of a vertex in the simple cycle of even order n, σn = n2 /4. For graph Gk , we have σ (vi ) = σ2k+2 + i, σ (u ) = σ2k+2 + 2, and σ (w ) = σ2k+2 + k + 1. Then σ (vi ) − σ (vi−1 ) = σ (u ) − σ (v1 ) = σ (w ) − σ (vk ) = 1, i = 1, 2, . . . , k. Therefore, Gk is a STI graph. For vertex transmissions of graph Hk , we can write σ (vi ) = σ2k+4 + 2k − i + 4, σ (u ) = σ2k+4 + 2(k + 1 ) + 2, σ (x ) = σ2k+4 + (k + 2 ) + 2, and σ (y ) = σ2k+4 + (k + 1 ) + 4, i = 1, 2, . . . , k. Therefore, σ (vi−1 ) − σ (vi ) = σ (u ) − σ (v1 ) = σ (y ) − σ (vk ) = σ (y ) − σ (x ) = 1. Hence, Hk is a STI graph. Consider vertices of graph Fk with respect to the vertices u and v. Denote by a (or b) the number of white (black) vertices that are closer to the vertex v (or u) than to the vertex u (or v). Since Fk is a bipartite graph, σ (u ) − σ (v ) = a − b = (3k + 2 ) − (3k + 1 ) = 1 and, therefore, Fk is a STI graph.  For example, graphs G2 , H3 , and F1 of these families have the following vertex transmissions: σ (v1 ) = 10, σ (u ) = σ (v2 ) = 11, and σ (w ) = 12 in G2 ; σ (v3 ) = σ (x ) = 32, σ (v2 ) = σ (y ) = 33, σ (v1 ) = 34, and σ (u ) = 35 in H3 ; σ (u ) = 15 and σ (v ) = 14 in F1 . The considered examples show that STI graphs may have the increasing diameter and the large cyclomatic number. The next two infinite families consist of 3-connected STI graphs with growing diameter. Consider two graphs in Fig. 5. Graph Tk is constructed from the prism with 4(k + 1 ) vertices by attaching a new vertex x. Graph Tk has order 4k + 5 and diameter k + 2. Graph Mk is obtained from the Möbius ladder like graph Tk , and it has 4k + 3 vertices and diameter k + 1. It is known that the prism and the Möbius ladder are vertex transitive graphs. Denote by σ P and σ L the transmissions of a vertex in the prism with 4k + 4 vertices and the Möbius ladder with 4k + 2 vertices, respectively. Then σP = 2k2 + 6k + 4 and σL = 2k2 + 4k + 1. Proposition 8. Tk and Mk are STI graphs for all k ≥ 1. Proof. Since graphs Tk and Mk have symmetries, it is sufficient to check vertex transmissions for several edges. For vertices of graph Tk , we have σ (vi ) = σP + i, σ (ui ) = σP + i + 1, and σ (w1 ) = σP + 1, σ (w2 ) = σP + 2, σ (w3 ) = σP + k + 1, σ (w4 ) = σP + k + 2, i = 1, 2, . . . , k. Since vertices w2 and x have common neighbors, σ (x ) = σ (w2 ). Then σ (vi+1 ) − σ (vi ) = σ (ui+1 ) − σ (ui ) = σ (ui ) − σ (vi ) = σ (w2 ) − σ (w1 ) = σ (w4 ) − σ (w3 ) = 1. For the other edges, σ (w3 ) − σ (vk ) = σ (w4 ) − σ (uk ) = σ (w2 ) − σ (v1 ) = σ (u1 ) − σ (w1 ) = 1, i = 1, 2, . . . , k. Therefore, Tk is a STI graph. Consider graph Mk . It easy to see that its vertex transmissions can be calculated in the same way as for graph Mk . Namely, σ (vi ) = σL + i, σ (ui ) = σL + i + 1, and σ (w1 ) = σL + 1, σ (w2 ) = σ (x ) = σL + 2, i = 1, 2, . . . , k. Then Mk is also a STI graph. 

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Fig. 6. Torus graph C6 Ck and STI graph Sk .

Fig. 7. STI graph Hk , 2 ≤ k ≤ 6.

For instance, the initial graphs of these families have the following transmissions: σ (v1 ) = σ (w1 ) = 13, σ (u1 ) = σ (w2 ) = σ (w3 ) = σ (x ) = 14, σ (w4 ) = 15 in T1 , and σ (v1 ) = σ (w1 ) = 8, σ (u1 ) = σ (w2 ) = σ (x ) = 9 in M1 . Consider the torus graph C6 Ck that is the Cartesian product of two cycles, where k ≥ 2 is even (see Fig. 6). We assume that C2 ∼ =P2 . Let graph Sk be obtained from the torus graph by attaching a new vertex x as shown in Fig. 6. Proposition 9. Sk is a STI graph for all k ≥ 2. Proof. The torus graph is vertex and edge transitive for k ≥ 2. It is easy to see that the shortest paths between black vertices in graph Sk are the same as in the torus graph. Because of symmetry of Sk , it is sufficient to consider edges (vi , vi+1 ), i = 1, 2, . . . , k/2, edges (ui , vi ), i = 1, 2, . . . , k, and edge (v1 , x). We have σ (ui ) − σ (vi ) = d (ui , x ) − d (vi , x ) = 1, σ (vi ) − σ (vi+1 ) = d (vi , x ) − d (vi+1 , x ) = −1. By construction of Sk , vertex x is closer to three cycles Ck than vertex v1 while vertex v1 is closer to the other three cycles Ck . Then σ (v1 ) − σ (x ) = d (v1 , x ) = 1. Therefore, Sk is a STI graph.  The next family contains STI graphs Hk of order 6k + 1 (see Fig. 7). Parameter k ≥ 2 counts the number of the centered hexagons in these graphs. The vertices of neighboring hexagons are shown in different colors. We conjecture that graphs Hk form an infinite family of STI graphs. The initial graphs of the family are checked by computer for 2 ≤ k ≤ 6. Considerations of examples of STI graphs lead to the following conjecture (more than 1300 graphs have been examined). Conjecture. The girth of a STI graph is equal to 4.

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4. Topological indices of STI graphs 1. The Wiener index, W(G), of a graph G is defined as



W (G ) =

d (u, v ).

{u,v}⊂V (G )

It was introduced as structural descriptor for characterization of acyclic structures [25]. Details on the chemical applications and mathematical properties of the Wiener index can be found in numerous books and reviews (see selected books [5,14,19,20,23,24] and articles [10,12,21,22]). The parity of the Wiener index of bipartite graphs was studied in [4]. The following statement immediately follows from [4]. Proposition 10. The Wiener index of a STI graph is even. Indeed, distance d(u, v) of a bipartite graph is even if vertices u and v belong to the same bipartite part and d(u, v) is odd when u and v belong to the distinct bipartite parts. Since the cardinalities of bipartite parts of a STI graph G have distinct parity, the number of all odd distances of G is always even. The minimal STI graphs with the same Wiener index have 11 vertices. 2. The Szeged index, Sz(G), is a generalization of the Wiener index for cyclic graphs introduced in [15]:



Sz(G ) =

nu nv .

(u,v )∈E (G )

Basic properties of the Szeged index can be found in [18]. It was noted that some properties of indices W and Sz are similar while the others are very different. All STI graphs with the same number of vertices and edges have the same Szeged index, i.e., graphs of this class can not be distinguished by the index. Proposition 11. Let G be a STI graph with n vertices and m edges. Then

Sz(G ) =

1 m ( n2 − 1 ). 4

Proof. The proof follows from the fact that the Szeged index can be calculated via transmission irregularities in a bipartite graph G [11]:

Sz(G ) =



1 2 n m − 4





[ σ ( u ) − σ ( v ) ]2 .

(u,v )∈E (G )

Then for a STI graph G, Sz(G ) = (n2 m − m )/4.



This proposition also follows from the fact that nu = (n + 1 )/2 and nv = (n − 1 )/2 for an edge (u, v) of an n-vertex STI graph. Recall that (n2 − 1 )/4 is the maximal number of edges in STI graphs. Since (n2 − 1 ) ≡ 0 (mod 8 ) for odd n, the Szeged and Wiener indices have the same parity. Corollary 3. The Szeged index of a STI graph is divisible by 2m. The similar property is not valid for the Wiener index. For example, the last graph in Fig. 2 has 8 edges and W = 38. 3. The Degree distance, DD(G), of a graph G is a modification of the Wiener index which appeared in [9,16]:



DD(G ) =

deg(v )σ (v ).

v∈V (G )

Proposition 12. The Degree distance and the number of edges of a STI graph have the same parity. Proof. Let G be a STI graph with m edges. Then

m=



(u,v )∈E (G )

=





[ σ ( u ) − σ ( v ) ]2 =

v∈V (G )

deg(v )σ (v )2 − 2



σ ( v )σ ( u )

(u,v )∈E (G )

deg(v )σ (v )2 − 4(. . . ).

v∈V (G )

The quantities deg(v )σ (v )2 and deg(v )σ (v ) have the same parity.



If m ≡ 0 (mod 4 ), then the Degree distance is also divisible by 4. The minimal STI graphs with the same Degree distance have 11 vertices.

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Acknowledgments This work was supported by the Laboratory of Topology and Dynamics, Novosibirsk State University. (contract no. 14.Y26.31.0025 with the Ministry of Education and Science of the Russian Federation). References [1] Y. Alizadeh, V. Andova, S. Klavžar, R. Škrekovski, Wiener dimension: fundamental properties and (5,0)-nanotubical fullerenes, MATCH Commun. Math. Comput. Chem. 72 (2014) 279–294. [2] Y. Alizadeh, S. Klavžar, Complexity of topological indices: the case of connective eccentric index, MATCH Commun. Math. Comput. Chem. 76 (2016) 659–667. [3] Y. Alizadeh, S. Klavžar, On graphs whose Wiener complexity equals their order and on wiener index of asymmetric graphs, Appl. Math. Comput. 328 (2018) 113–118. [4] D. Bonchev, I. Gutman, O.E. Polansky, Parity of the distance numbers and Wiener numbers of bipartite graphs, MATCH Commun. Math. Comput. Chem. 22 (1987) 209–214. [5] M. Dehmer, F. Emmert-Streib (Eds.), Quantitative Graph Theory: Mathematical Foundations and Applications, Discrete Mathematics and Its Applications, Chapman and Hall/CRC, 2014. [6] A.A. Dobrynin, Infinite family of 2-connected transmission irregular graphs, Appl. Math. Comput. 340 (2019) 1–4. [7] A.A. Dobrynin, Infinite family of transmission irregular trees of even order, Discrete Math. 342 (2019) 74–77. [8] A.A. Dobrynin, Infinite family of 3-connected cubic transmission irregular graphs, Discrete Appl. Math. 257 (2019) 151–157. [9] A.A. Dobrynin, A.A. Kochetova, Degree distance of a graph: a degree analogue of the Wiener index, J. Chem. Inf. Comput. Sci. 34 (1994) 1082–1086. [10] A.A. Dobrynin, R. Entringer, I. Gutman, Wiener index of trees: theory and applications, Acta Appl. Math. 66 (2001) 200–249. [11] A.A. Dobrynin, I. Gutman, On a graph invariant related to the sum of all distances in a graph, Publ. l’Inst. Math. 56 (1994) 18–22. [12] A.A. Dobrynin, I. Gutman, S. Klavžar, P. Žigert, Wiener index of hexagonal systems, Acta Appl. Math. 72 (3) (2002) 247–294. [13] R.C. Entringer, D.E. Jackson, D.A. Snyder, Distance in graphs, Czechoslovak Math. J. 26 (1976) 283–296. [14] I. Gutman, O.E. Polansky, Mathematical Concepts in Organic Chemistry, Springer–Verlag, Berlin, 1986. [15] I. Gutman, A formula for the Wiener number of trees and its extension to graphs containing cycles, Graph Theory Notes N. Y. 27 (1994) 9–15. [16] I. Gutman, Selected properties of the Schultz molecular topological index, J. Chem. Inf. Comput. Sci. 34 (1994) 1087–1089. [17] I. Gutman, Stepwise irregular graphs, Appl. Math. Comput. 325 (2018) 234–238. [18] I. Gutman, A.A. Dobrynin, The Szeged index — a success story, Graph Theory Notes N. Y. 34 (1998) 37–44. [19] I. Gutman, B. Furtula (Eds.), Distance in Molecular Graphs — Theory. Mathematical Chemistry Monographs, 12, Univ. Kragujevac, Kragujevac, Serbia, 2012. [20] I. Gutman, B. Furtula (Eds.), Distance in Molecular Graphs — Applications, Mathematical Chemistry Monographs, 13, Univ. Kragujevac, Kragujevac, Serbia, 2012. [21] M. Knor, R. Škrekovski, A. Tepeh, Mathematical aspects of Wiener index, Ars Math. Contemp. 11 (2) (2016) 327–352. [22] S. Nikolic´ , N. Trinajstic´ , Z. Mihalic´ , The Wiener index: developments and applications, Croat. Chem. Acta 68 (1995) 105–129. [23] R. Todeschini, V. Consonni, Handbook of Molecular Descriptors, Wiley–VCH, Weinheim, 20 0 0. [24] N. Trinajstic´ , Chemical Graph Theory, 2nd ed., CRC Press, Boca Raton, FL, 1983. 1992 [25] H. Wiener, Structural determination of paraffin boiling points, J. Am. Chem. Soc. 69 (1947) 17–20.