Syddansk Universitet. The total irregularity of a graph. Abdo, H.; Brandt, S.; Dimitrov, D.

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1 Syddask Uiversitet The total irregularity of a graph Abdo, H.; Bradt, S.; Dimitrov, D. Published i: Discrete Mathematics & Theoretical Computer Sciece Publicatio date: 014 Documet versio Publisher's PDF, also kow as Versio of record Citatio for pulished versio (APA): Abdo, H., Bradt, S., & Dimitrov, D. (014). The total irregularity of a graph. Discrete Mathematics & Theoretical Computer Sciece, 16(1), Geeral rights Copyright ad moral rights for the publicatios made accessible i the public portal are retaied by the authors ad/or other copyright owers ad it is a coditio of accessig publicatios that users recogise ad abide by the legal requiremets associated with these rights. Users may dowload ad prit oe copy of ay publicatio from the public portal for the purpose of private study or research. You may ot further distribute the material or use it for ay profit-makig activity or commercial gai You may freely distribute the URL idetifyig the publicatio i the public portal? Take dow policy If you believe that this documet breaches copyright please cotact us providig details, ad we will remove access to the work immediately ad ivestigate your claim. Dowload date: 0. ja.. 019

2 Discrete Mathematics ad Theoretical Computer Sciece DMTCS vol. 16:1, -14, The total irregularity of a graph Hosam Abdo 1 Stepha Bradt Darko Dimitrov 1 1 Istitut für Iformatik, Freie Uiversität Berli, Germay Departmet of Mathematics ad Computer Sciece, Uiversity of Souther Demark, Odese, Demark received 1 st Oct. 01, revised 10 th Feb. 014, accepted 18 th Mar I this ote a ew measure of irregularity of a graph G is itroduced. It is amed the total irregularity of a graph ad is defied as irr t(g) = 1 u,v V (G) dg(u) dg(v), where dg(u) deotes the degree of a vertex u V (G). All graphs with maximal total irregularity are determied. It is also show that amog all trees of the same order the star has the maximal total irregularity. Keywords: the irregularity of a graph, the total irregularity of a graph, extremal graphs 1 Itroductio We cosider oly fiite udirected graphs without loops or multiple edges. For termiology ad otatio ot defied here we refer the reader to Body ad Murty s textbook [6]. I this ote we ivestigate irregularity measures of graphs. A graph is regular if all its vertices have the same degree, otherwise it is irregular. However, it is of iterest to measure how irregular it is. Several approaches have bee proposed that characterize how irregular a graph is. For a graph G, Albertso [4] defies the imbalace of a edge e = uv E(G) as d G (u) d G (v) ad the irregularity of G as irr(g) = uv E(G) d G (u) d G (v), (1) where d G (v) is the degree of v V (G). He preseted upper bouds o the irregularity of graphs, bipartite graphs, ad triagle-free graphs, as well as a sharp upper boud for trees. He showed that if a graph has maximal irregularity amog all graphs of order, the it is the joi of a clique ad a idepedet vertex set. Some claims about bipartite graphs give i [4] have bee formally proved i [16]. Related to the work of Albertso is the work of Hase ad Mélot [15], who characterized the graphs with vertices abdo@mi.fu-berli.de stepha.bradt@imada.sdu.dk darko@mi.fu-berli.de c -14 Discrete Mathematics ad Theoretical Computer Sciece (DMTCS), Nacy, Frace

3 0 Hosam Abdo, Stepha Bradt, Darko Dimitrov ad m edges with maximal irregularity. For more results o imbalace, the irregularity of a graph, ad other approaches, that measure how irregular a graph is, we refer the reader to [, 3, 5, 7 1, 17 0]. I the sequel we itroduce ad cosider a irregularity measure that is related to the irregularity measure (1). As well as (1), the ew measure also captures the irregularity oly by the differece of vertex degrees. For a graph G it is defied as irr t (G) = 1 (u,v) V (G) d G (u) d G (v). () For a ordered degree sequece of V (G) = {v 1, v,, v } with d(v 1 ) d(v ) d(v ), () ca be expressed i the form irr t (G) = i>j(d(v i ) d(v j )). (3) Because of the obvious coectio with the irregularity, we called irr t (G) the total irregularity of a graph. Note that for every graph the total irregularity is a eve umber sice the umber of vertices of odd degree is eve, ad it is completely determied by its degree sequece graphs with the same degree sequece have the same total irregularity, while this is ot always true with the irregularity of a graph (see Figure 1 for such a example). Sice the total irregularity of a graph depeds oly o its G 1 G Fig. 1: Two o-isomorphic graphs G 1 ad G with the same degree sequece 1, 1, 1, 1,, 3, 3. They have differet irregularities (irr(g 1) = 10 ad irr(g ) = 8), but the same total irregularity (irr t(g 1) = irr t(g ) = ). degree sequece, it ca be applied as irregularity measure whe the adjacecy iformatio of the vertices is ukow. Aother motivatio to itroduce irr t as a irregularity measure is the fact that the graphs with maximal irr, amog graphs of same order, are bidegreed graphs, more precisely the so-called clique-star graphs, graphs obtaied by joi of a clique ad isolated vertices [1, 4]. Also there are a graphs with large irr that have very small degree sets, the property that oe will ot expect from very irregular graphs. O the cotrary, as we show i this ote, the graphs with maximal irr t have large degree sets, ad some of them have eve the largest possible oes. Both measures are zero if ad oly if G is regular, ad irr t (G) is a upper boud of irr(g). Very recetly, these two irregularity measures were compared i [13], where it was show that for a coected graph G with vertices, irr t (G) irr(g)/4. Moreover, if G is a tree, the it was show that irr t (G) ( )irr(g). Sice the most irregular graphs with respect to irr are the joi of a complete graph ad a idepedet set, they have oly two differet vertex degrees. We will show below that the most irregular graphs with respect to irr t are graphs that have may differet vertex degrees, ad the graphs with the maximal umber of differet vertex degrees belog to the graphs with maximal total irregularity.

4 The total irregularity of a graph 03 Graphs with maximal total irregularity It is easy to see that there is o graph of order with differet vertex degrees. Askig for graphs with 1 differet degrees there are two possible sets of degrees: {1,,, 1} ad {0, 1,, } sice the graph caot cotai vertices of degree 0 ad 1 at the same time. A simple iductio proof verifies that for every there is a uique graph with the respective set of degrees up to isomorphism, see Figure. These graphs have bee called half-complete i [14]. The graph with a vertex of degree 1 is deoted H ad the graph with a vertex of degree 0 is its complemet H (this ca be see by mirrorig i Figure oe of the graphs H, H alog a horizotal lie). The graphs H ad H ca be obtaied by labellig the vertices v 1,, v ad addig all edges v i v j where i+j +1 (i+j > +1, respectively). Note that the two vertices of H with equal degrees have idices ad + 1. Fig. : The graphs H (with dashed edges) ad H (without dashed edges) for eve ad odd, respectively. Take the graph H with vertices labelled as above ad let S = {v i v j i + j = + 1}, the set of dashed edges i Figure. The H = H S. Now we ca formulate our mai result. Theorem.1 A graph G of order has maximal total irregularity, if ad oly if it is isomorphic to a graph H S, where S S. Observe that the set S of G = H S ca be determied oly from the degree sequece of G. This follows from the fact that the vertex labellig of G iherited from H has the property that d G (v 1 ) d G (v ) d G (v ). Therefore for j < i, i + j = + 1, we have v i v j S if ad oly if d G (v j ) = j 1. I particular, every two graphs G 1 = H S 1 ad G = H S with S 1 S S are ot isomorphic. Therefore, sice S = there are exactly o-isomorphic graphs of order with maximal total irregularity. Note that all these graphs have the property that they are subgraphs of H ad supergraphs of H, but the coverse does ot hold. E.g., the path o four vertices is a subgraph of H 4 ad a supergraph of H 4, but its total irregularity is 4, while the maximal total irregularity is 6. Proof: For a ordered degree sequece, cosider a edge v i v j, i > j. Addig v i v j to G or deletig v i v j from G oly chages the summads d(u) d(w) i (3) where {u, w} {v i, v j } = 1. Addig the edge v i v j to G icreases all summads d(v i ) d(v k ) with i > k ad i j by oe while the summads d(v k ) d(v i ) for k > i are decreased by oe (or icreased by oe, if d G (v k ) = d G (v i )). Similarly, all summads d(v j ) d(v k ) with j > k are icreased by oe, while the summads d(v k ) d(v i ) for k > i

5 04 Hosam Abdo, Stepha Bradt, Darko Dimitrov ad j i are decreased by oe (or icreased by oe, if d G (v k ) = d G (v j )). Therefore, irr t (G + v i v j ) irr t (G) + (i ) ( i) + (j 1) ( j 1) = irr t (G) + (i + j 1) which exceeds irr t (G) if i + j > + 1. Deletig a edge v i v j from G has the effect that all summads d(v k ) d(v i ) with k > i are icreased by oe while the summads d(v i ) d(v k ) with k < i, k j are decreased by oe (or icreased by oe, if d G (v k ) = d G (v i )). All summads d(v k ) d(v j ) with k > j, k i, are icreased by oe, while the summads d(v j ) d(v k ) with k < j, are decreased by oe (or icreased by oe, if d G (v k ) = d G (v j )). Therefore, irr t (G v i v j ) irr t (G) + ( i) (i ) + ( j 1) (j 1) (4) = irr t (G) + (i + j 1), with equality if ad oly if d G (v j ) > d G (v j 1 ) ad d G (v i ) > d G (v i 1 ) (d G (v i ) > d G (v i ) if j = i 1). Therefore, if G has maximal irregularity the G has all edges v i v j with i + j > + 1 ad o edge with i + j < + 1. Ay such graph ca be expressed i the form G = H S with S S. It is left to show that every such graph has the same total irregularity. Let G = H S where S S ad let v i v j S \ S. The v i ad v j have both their origial degree from H, but d(v i 1 ) ad d(v j 1 ) ca oly have decreased. Sice d H (v j ) > d H (v j 1 ) ad d H (v i ) > d H (v i 1 ) if j < i 1 ad d H (v i ) = d H (v j ) > d H (v j 1 ) if j = i 1, the iequality (4) holds with equality ad irr t (G v i v j ) = irr t (G). Corollary. For ay simple udirected graph G with vertices it holds that 1 1 ( ) odd, irr t (G) 1 1 (3 3 ) eve. Proof: The bouds follow with Theorem.1 from the total irregularity of H. The oly two vertices of H with equal degrees are v ad v +1. Sice, 1 = d(v 1) < < d(v ) = d(v +1 ) < < d(v ) = 1, we obtai irr t (H ) = 1 i 1 i ) d(v j )) = (d(v i ) d(v j )) i>j(d(v = = (i j) + 1 i 1 i= j=1 1 i 1 i= j=1 1 j + i=1 1 j=1 i= j=1 1 i + ( ) j i=1 i = + ( ) i= +1 ( ( ) i ) ( + ),

6 The total irregularity of a graph 05 which immediately gives irr t (H ) = 1 1 ( ) for odd, ad irr t (H ) = 1 1 (3 3 ) for eve. Propositio.3 If T is a tree with vertices, the irr t (T ) ( 1)( ). Moreover, equality holds if ad oly if T is a star. Proof: Let T be a tree that is ot a star, with u beig a vertex with maximal degree. Cosider a pedat vertex v that is a adjacet to a vertex w u. We remove the edge vw from T ad add the edge uv. After this replacemet, oly the degrees of u ad w are affected, amely, d(u) icreases by oe while d(w) decreases by oe. Thus, the summads d(u) d(w) ad d(w) d(u) i () both icrease by two. Each of the ( ) summads d(u) d(x) ad d(x) d(u) with x u, w icreases by oe while each of ( ) summads d(w) d(x) ad d(x) d(w) with x u, w icreases or decreases by oe. Thus, we have irr t (T wv + uv) irr t (T ) + + ( ) = irr t (T ) +. Therefore the largest total irregularity amog the trees of order is oly attaied by the star, whose total irregularity equals ( 1)( ). Refereces [1] H. Abdo, N. Cohe, ad D. Dimitrov. Graphs with maximal irregularity. Filomat, 014, i press. [] Y. Alavi, A. Boals, G. Chartrad, P. Erdős, ad O. R. Oellerma. k-path irregular graphs. Cogr. Numer., 65:01 10, [3] Y. Alavi, G. Chartrad, F. R. K. Chug, P. Erdős, R. L. Graham, ad O. R. Oellerma. Highly irregular graphs. J. Graph Theory, 11:35 49, [4] M. O. Albertso. The irregularity of a graph. Ars Comb., 46:19 5, [5] F. K. Bell. O the maximal idex of coected graphs. Liear Algebra Appl., 144: , [6] A. Body ad U. S. R. Murty. Graph Theory. Spriger Verlag, Berli, 008. [7] Y. Caro ad R. Yuster. Graphs with large variace. Ars Comb., 57:151 16, 000. [8] G. Chartrad, P. Erdős, ad O. R. Oellerma. How to defie a irregular graph. Coll. Math. J., 19:36 4, [9] G. Chartrad, K. S. Holbert, O. R. Oellerma, ad H. C. Swart. F -degrees i graphs. Ars Comb., 4: , [10] G. Chartrad, M. S. Jacobso, J. Lehel, O. Oellerma, S. Ruiz, ad F. Saba. Irregular etworks. Cogr. Numer., 64:197 10, [11] L. Collatz ad U. Siogowitz. Spektre edlicher Graphe. Abh. Math. Sem. Uiv. Hamburg, 1:63 77, 1957.

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