Metric Characteristics. Matrix Representations of Graphs.

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1 Graph Theory Metrc Characterstcs. Matrx Representatons of Graphs. Lecturer: PhD, Assocate Professor Zarpova Elvra Rnatovna, Department of Appled Probablty and Informatcs, RUDN Unversty Translated by: Krll Bolshakov, 2 nd year student

2 Recommended lterature. Зарипова Э.Р., Кокотчикова М.Г. Лекции по дискретной математике: Теория графов. Учебное пособие. М., изд-во: РУДН, 203, 62 с. 2. Харари Ф. «Теория графов», М.: КомКнига, с. 3. Судоплатов С.В., Овчинникова Е.В. «Элементы дискретной математики». Учебник. М.: Инфра-М; Новосибирск: НГТУ, с. 4. Шапорев С.Д. «Дискретная математика. Курс лекций и практических занятий». СПб.: БХВ-Петербург, с.: ил. 5. Сайт кафедры прикладной информатики и теории вероятностей РУДН (информационный ресурс). Режим доступа: свободный. 6. Учебный портал кафедры прикладной информатики и теории вероятностей РУДН (информационный ресурс) Режим доступа: для зарегистрированных пользователей. 7. Учебный портал РУДН, раздел «Теория конечных графов» 2

3 Metrc characterstcs Let s take the connected, unweghted, undrected graph G ( E, ),,,. k Let d(, ) be the length (amount of edges) of the shortest path between and, and we ll say that d(, ) f and are located n dfferent connected components. These lengths wll satsfy the followng metrc axoms: ) d(, ) 0, 2) d(, ) 0, 3) d(, ) d(, ), 4) d(, ) d(, ) d(, ). k k 3

4 Metrc characterstcs The eccentrcty of a fxed vertex s the value e( ) max d(, ). The dameter of graph G ( E, ) s the value d( G) max e( ). ertex s perpheral f e( ) d( G). The radus of graph G ( E, ) s the value r( G) mn e( ). ertex s central f e( ) r( G). The set of all the central vertces of a graph s called the center of the graph. 4

5 Unorented ncdence matrces Let G ( E, ) be an undrected graph wth n vertces ( n ) and m edges ( E m ),.e., e e e,,..., E,,..., 2 n 2 m. The followng matrx s called the ncdence matrx of graph G ( E, ) : A e e... e 2 A a a [ ],, n,, m 2,... n m 5

6 where a, Unorented ncdence matrces and, n,, m. Property : a 0, f edge e sn't ncdent wth ;, f edge e s ncdent wth ; 2, f edge e s a loop at vertex. n m a 2, a a,, ( ). Every column has exactly two s, except for those that represent loops. Columns that represent loops only have one 2. Property 2: In cases, when the graph can be cut nto 2 or more dsconnected components, the ncdence matrx wll have a dagonally-segmented structure on condton that the vertces of the frst component are numbered frst, the vertces of the second component - second, etc. (.e. n ascendng order). 6

7 Unorented ncdence matrces e 5 e 2 4 e 3 e 2 e e 4 Example. Create an ncdence matrx for the undrected graph. Note that the graph s dsconnected and the vertces and edges are numbered sequentally 7

8 A dagonally-segmented ncdence matrx for Example Snce edge e connects vertces and, we place a nto the frst 2 column, frst row and another nto the frst column, second row. Due to e beng a loop at vertex, we place a 2 nto the ffth column, fourth 5 4 row, and so on: e 2 e 2 3 e 4 e e 6 e 5 A A e e e e e e

9 Adacency matrces: undrected graphs Let G ( E, ) be an undrected graph wth n vertces ( n,,..., ). The followng matrx B s called the adacency 2 n matrx of graph G ( E, ) : b, B... 2 B b b {, [ ],,, n 2,... n n, amount of edges, smultaneously ncdent wth and,,, n.} Property. Adacency matrces of undrected graphs are always symmetrc about the man dagonal. 9

10 Adacency matrces: undrected graphs e 5 e 2 4 e 3 e 2 e e 4 Example 2. Create an adacency matrx for the undrected graph. Note that the graph s dsconnected and the vertces and edges are numbered sequentally 0

11 Adacency matrx for Example 2 e 2 e 2 3 e 4 e e 6 e 5 B B ertces and are connected by one edge, so we place a nto the 2 frst row, second column and nto the second row, frst column. Note, that even though there are loops n the graph, they are denoted by a n the adacency matrx, because there s only one loop. The matrx s symmetrc about the man dagonal.

12 Orented ncdence matrces Let G E, be a dgraph wth n vertces ( n ) and m arcs E ):, e e e,,..., E,,..., 2 n 2 m. The matrx A s called the ncdence matrx of dgraph G E, : ( m A e e... e 2 A a a [ ],, n,, m 2,... n m, where a, 0, f arc e sn't ncdent wth ;, f arc e s ncdent from vertex (.e. t comes out of vertex );, f the arc s ncdent to vertex (.e. t goes nto ); 2, f arc e s a loop at vertex,, n,, m. 2

13 Orented ncdence matrces e 2 5 e 2 e e 4 Example 3. Create an ncdence matrx for the dgraph. Note that the graph s dsconnected and the vertces and edges are numbered sequentally 3

14 Incdence matrx for Example 3 e A e e e e e e A e Snce arc e s drected from vertex to, we 2 place a nto the frst row, frst column and a - nto the second row, frst column. The loop at vertex 4 gves us a 2 n the fourth row, fourth column and so on. 4

15 Adacency matrx: dgraphs Let G E, be a dgraph wth n vertces ( n,,,..., ). Matrx B s called the 2 n adacency matrx of dgraph G E, : B... 2 B b b [ ],,, n 2,... n where b s the amount of arcs, drected from vertex, vertex,,, n. n, to 5

16 Adacency matrx: dgraphs e 2 5 e 2 e e 4 Example 4. Create an adacency matrx for the dgraph. Note that the graph s dsconnected and the vertces and edges are numbered sequentally 6

17 Adacency matrx for Example 4 e 2 e 2 3 e 4 4 e 3 5 B B Note that graph G E, s dsconnected, and ths s reflected n the dagonally-segmented structure of the ncdence and adacency matrces. 7

18 Adacency lsts for weakly connected graphs The adacency lst of vertex s the set of vertces whch are adacent to sad vertex. u( ) :(, ) E (for undrected graphs) and u( ) :, E (for drected graphs). 8

19 Adacency lsts for weakly connected graphs Example 5. Create an adacency lst for the gven dgraph. 9

20 Adacency lst for Example u( ) {, }, 3 4 u( ) { }, 2 3 u( ) { }, 3 u( ) { }

21 Theorem on adacency matrx powers n Matrx B gves the amount of drected walks of length n between any two vertces of a dgraph. 2

22 Theorem on adacency matrx powers Proof. (usng mathematcal nducton) ) Let s take graph G E,. Let m. We ll ntroduce the followng terms: b amount of arcs, connectng vertex wth vertex, k, k b amount of arcs, connectng vertex wth vertex, k, k (2) b amount of dfferent drected walks of length 2, (.e. walks consstng of 2 arcs) from vertex to vertex whch pass through vertex, k, m. k Then m k b b b (2), k k,,. The theorem s obvous for 2 B. 22

23 Theorem on adacency matrx powers n 2) Let the theorem be true for matrx. We ll show that t s also n n true for matrx B B B. ( n ) If b amount of drected walks of length ( n ) from to, k k, B b amount of arcs from to, then k, k b ( n) b amount of all drected walks from to, whch pass, k k, through. k Then m ( n) ( n) b b b s the amount of all drected walks of length, k k,, k ( n) n drected from to, and, Comment : If there exsts an, b s an element of matrx n B. n l n l : B 0, then there are no cycles n the graph. Comment 2: Ths theorem s also true for undrected graphs. 23

24 Pseudographs A pseudograph s a graph that allows loops and multple parallel edges. 3 2 Example 6. 24

25 Multgraphs A multgraph s a graph whch allows parallel edges but has no loops. 3 2 Example 7. 25

26 Smple undrected graphs An undrected graph s smple f t doesn t contan any loops or multple parallel edges. 3 2 Example 8. 26

27 Planar and flat graphs A graph s flat, f, when drawn on a flat surface, all of ts edge ntersectons are also ts vertces. A graph s planar f t s somorphc wth a flat graph. G 5 G Example 9. Planar and flat graphs 27

28 Planar and flat graphs G 3 5 G Example 0. Planar and flat graphs 28

29 Exercse: s ths graph flat? Planar? G Example. Explan your reasonng when dong ths exercse 29

30 Planar and flat graphs G 6 5 G Example. Planar and flat graphs 30

31 Weght matrces For graph G ( E, ), where n, the weght matrx W s defned as follows:... n W,,, n and... w, n mnmum weght of the edge from vertex to vertex, w 0, f,,, f edge (, ) doesn't exst. 3

32 Next Lecture: «Kruskal s Algorthm» 32

Non-Split Restrained Dominating Set of an Interval Graph Using an Algorithm

Non-Split Restrained Dominating Set of an Interval Graph Using an Algorithm Internatonal Journal of Advancements n Research & Technology, Volume, Issue, July- ISS - on-splt Restraned Domnatng Set of an Interval Graph Usng an Algorthm ABSTRACT Dr.A.Sudhakaraah *, E. Gnana Deepka,