On a Conjecture about Degree Deviation Measure of Graphs
aa r X i v : . [ m a t h . C O ] F e b On a Conjecture about Degree DeviationMeasure of Graphs
Ali Ghalavand ∗ and Ali Reza Ashrafi Department of Pure Mathematics, Faculty of Mathematical Sciences, University ofKashan, Kashan 87317–53153, I. R. Iran
February 24, 2020
Abstract
Let G be an n − vertex graph with m edges. The degree deviation measure of G isdefined as s ( G ) = P v ∈ V ( G ) | deg G ( v ) − mn | , where n and m are the number of verticesand edges of G , respectively. The aim of this paper is to prove the Conjecture 4.2of [J. A. de Oliveira, C. S. Oliveira, C. Justel and N. M. Maia de Abreu, Measuresof irregularity of graphs, Pesq. Oper. (3) (2013) 383–398]. The degree deviationmeasure of chemical graphs under some conditions on the cyclomatic number is alsocomputed. Keywords:
Irregularity, degree deviation measure, chemical graph.
Throughout this paper all graphs are assumed to be simple and undirected. If G is sucha graph, then the vertex and edge sets of G are denoted by V ( G ) and E ( G ), respectively.The degree of a vertex v is denoted by deg G ( v ) (or deg ( v ) for short), N [ v, G ] is the set ofall vertices adjacent to v and H ( n ) denotes the set of all connected n − vertex graphs.Let k and n be integer numbers such that 0 ≤ k ≤ n . A graph S ( n, k ) is a split graphif there is a partition of its vertex set into a clique of order k and a stable set of order n − k . A complete split graph, CS ( n, k ), is a split graph such that each vertex of theclique is adjacent to each vertex of the stable set [5].A graph in which all vertices have the same degree is said to be regular . Suppose α isa function from the set of all graphs into non-negative integers. If for each regular graph G , α ( G ) = 0, then the function α is called a measure of regularity .With the best of our knowledge the first irregularity measure was proposed by Albert-son [1]. He defined the irregularity of a graph G by irr ( G ) = P xy ∈ E ( G ) | deg ( x ) − deg ( y ) | and determined the maximum irregularity of various classes of graphs. As a consequence ∗ Corresponding author ([email protected])
1f his results, the irregularity of an n − vertex graph is less that n , and the bound is tight.Abdo et al. [3] was proposed a generalization of irregularity measure of Albertson whichis called the total irregularity . It is defined as irr t ( G ) = P u,v ∈ V ( G ) | deg ( u ) − deg ( v ) | .They obtained all graphs with maximal total irregularity and proved that among all treesof the same order the star has the maximal total irregularity.Following Nikiforov [6], the degree deviation measure of G is defined as s ( G ) = X v ∈ V ( G ) (cid:12)(cid:12)(cid:12)(cid:12) deg G ( v ) − mn (cid:12)(cid:12)(cid:12)(cid:12) , where n and m are the number of vertices and edges of G , respectively. Let G be agraph with maximum eigenvalue µ ( G ). The well-known result of Euler which states that P v ∈ V ( G ) deg G ( v ) = 2 | E ( G ) | implies that the degree deviation measure s ( G ) is a measureof irregularity. Nikiforov proved that s ( G )2 n √ m ≤ µ ( G ) − mn ≤ p s ( G )and these inequalities are tight up to a constant factor.de Oliveira et al. [5] investigated four distinct graph invariants used to measure theirregularity of a graph and proved the following theorem: Theorem 1.1.
Let k ∈ N and ≤ k ≤ n . If G = CS ( n, k ) is a complete split graph then s ( G ) = n k ( n − k )( n − − k ) . Besides, among all complete split graphs, the most irregularone by s ( G ) has to attend the following conditions on k : k = n | n n − | n − n − and n +13 | n − . They conjectured that
Conjecture 1.2.
Let H ( n ) be the set of all connected graphs G with n vertices. Then max G ∈ H ( n ) s ( G ) = s ( CS ( n, k )) , where k = n | n n − | n − n − and n +13 | n − . The aim of this paper is to prove Conjecture 1.2.
In this section, we will present a proof for Conjecture 1.2. To do this, we need somenotations as follows: V ↓ ( G ) = { v ∈ V ( G ) | deg G ( v ) ≤ mn } ,V ↑ ( G ) = { v ∈ V ( G ) | deg G ( v ) > mn } ,E ↓ ( G ) = { uv ∈ E ( G ) | { u, v } ⊆ V ↓ ( G ) } , ↑ ( G ) = {{ u, v } ⊆ V ↑ ( G ) | uv E ( G ) } . The well-known result of Euler which states that P v ∈ V ( G ) deg G ( v ) = 2 | E ( G ) | lead usto the following useful lemma: Lemma 2.1.
Let F ( n, m ) denote the family of all connected graphs with n vertices and m edges. Then,1. { G ∈ F ( n, m ) | V ↓ ( G ) = V ( G ) } = { G ∈ F ( n, m ) | G is mn − regular } , { G ∈ F ( n, m ) | V ↑ ( G ) = V ( G ) } = ∅ . Lemma 2.2.
Let G be a connected n − vertex irregular graph, n ≥ , and e = uv ∈ E ↓ ( G ) is not a cut edge of G . If G − = G − e then s ( G ) < s ( G − ) .Proof. Let V ↓ ( G ) = { v ∈ V ↓ ( G ) : deg G ( v ) > m − n } . By definition, s ( G − ) − s ( G ) = 2 m − n − deg G ( u ) + 1 + 2 m − n − deg G ( v ) + 1 − [ 2 mn − deg G ( u ) + 2 mn − deg G ( v )] + X w ∈ V ↓ ( G ) (cid:20) deg G ( w ) − m − n (cid:21) − X w ∈ V ↓ ( G ) \ ( V ↓ ( G ) ∪{ u,v } ) n + X v ∈ V ↑ ( G ) n ≥ − n h(cid:12)(cid:12)(cid:12) V ↓ ( G ) \ V ↓ ( G ) (cid:12)(cid:12)(cid:12) − (cid:12)(cid:12) V ↑ ( G ) (cid:12)(cid:12)i > , proving the lemma. Lemma 2.3.
Let G be a connected irregular n − vertex graph, n ≥ , and { w, z } ∈ E ↑ ( G ) .If G + = G + f , f = wz , then s ( G ) < s ( G + ) .Proof. Let V ↑ ( G ) = { v ∈ V ↑ ( G ) : deg G ( v ) ≤ m +2 n } . By definition, s ( G + ) − s ( G ) = deg G ( u ) + 1 − m + 2 n + deg G ( v ) + 1 − m + 2 n − [ deg G ( u ) − mn + deg G ( v ) − mn ] + X w ∈ V ↑ ( G ) (cid:20) m + 1 n − deg G ( w ) (cid:21) + X w ∈ V ↓ ( G ) n − X w ∈ V ↑ ( G ) \ ( V ↑ ( G ) ∪{ u,v } ) n ≥ − n h(cid:12)(cid:12)(cid:12) V ↑ ( G ) \ V ↑ ( G ) (cid:12)(cid:12)(cid:12) − (cid:12)(cid:12) V ↓ ( G ) (cid:12)(cid:12)i > , proving the lemma.Suppose G is a connected irregular graph with a cut edge e = uv ∈ E ↓ ( G ). It is clearthat there exists a vertex w ∈ V ↑ ( G ) such that at least one of the graphs G − uv + uw and G − uv + vw is connected. Without loss of generality, we assume that G − uv + uw is connected. Then the edge uw is called a connectedness factor of G − e with respect to V ↓ ( G ) and V ↑ ( G ). 3 emma 2.4. Let G be a connected irregular n − vertex graph, n ≥ , and e = uv ∈ E ↓ ( G ) is an cut edge of G . If w ∈ V ↑ ( G ) , G ∗ = G − uv + uw and uw is a connectedness factorof G − uv with respect to V ↓ ( G ) and V ↑ ( G ) , then s ( G ) < s ( G ∗ ) .Proof. By definition, s ( G ∗ ) − s ( G ) = 2 mn − ( deg G ( v ) −
1) + deg G ( w ) + 1 − mn − (cid:20) mn − deg G ( v ) + deg G ( w ) − mn (cid:21) = 2 , as desired. Lemma 2.5.
Suppose G is a connected irregular n − vertex graph such that V ↑ ( G ) = { u , . . . , u k } is a clique, V ↓ ( G ) = { v , . . . , v n − k } is a stable set and P n − ki =1 deg G ( v i ) < | V ↓ ( G ) | k . Let N [ G, v i ] = { u i , . . . , u i degG ( vi ) } , for ≤ i ≤ n − k .1. If | V ↓ ( G ) | > | V ↑ ( G ) | and G ‡ = G + { v i u j : 1 ≤ i ≤ n − k and u j ∈ V ↑ ( G ) \ N [ v, G ] } ,then s ( G ) < s ( G ‡ ) .2. If | V ↓ ( G ) | ≤ | V ↑ ( G ) | and G ⊺ = G − { v i u i j : 1 ≤ i ≤ n − k and ≤ j ≤ deg G ( v i ) } ,then s ( G ) ≤ s ( G ⊺ ) .Proof. Let α = |{ v i u j : 1 ≤ i ≤ n − k and u j ∈ V ↑ ( G ) \ N [ v, G ] }| and . For prove (1) bydefinition, s ( G ‡ ) − s ( G ) = k ( k −
1) + ( n − k ) k − m + αn | V ↑ ( G ) | + 2 m + αn | V ↓ ( G ) | − ( n − k ) k − (cid:20) k ( k −
1) + ( n − k ) k − α − mn | V ↑ ( G ) | + 2 mn | V ↓ ( G ) | − ( n − k ) k + α (cid:21) = αn | V ↓ ( G ) | − αn | V ↑ ( G ) | > , as desired. For Prove (2) by definition and similar with last proof, s ( G ⊺ ) − s ( G ) = αn | V ↑ ( G ) | − αn | V ↓ ( G ) | ≥
0, as desired.Let S ( n, k ) be a split graph such that all vertices in its stable set are of degree one.Define: H ( n,
1) := { G ∈ H ( n ) : f or one ≤ k ≤ n − , G ∼ = S ( n, k ) } ,CH ( n ) := { G ∈ H ( n ) : f or one ≤ k ≤ n − , G ∼ = CS ( n, k ) } . Lemma 2.6.
Let k ∈ N and < k < n − . Then max G ∈ H ( n, s ( G ) = s ( S ( n, k )) , where k = n | n ( n −
1) 3 | n − and n = 5 ( n + 1) n = 5 and | n − . Proof.
By definition s ( S ( n, k )) = n (3 k − k − n )( n − k ). For a given n , define g ( k ) = n (3 k − k − n )( n − k ). By a simple calculation the maximal value of g ( k ) is obtained by k = n . Since k is an integer, we have to determine ⌈ k ⌉ and ⌊ k ⌋ and then comparing g ( ⌈ k ⌉ ) and g ( ⌊ k ⌋ ) in all previous cases give the our result.4 emma 2.7. Let k ∈ N and < k < n − . Then max G ∈ H ( n, s ( G ) < max G ∈ CH ( n ) s ( G ) .Proof. Let max G ∈ CH ( n ) s ( G ) = λ and max G ∈ H ( n, s ( G ) = µ . By Theorem 1.1 and Lemma2.6, λ − µ = n | n
29 ( n +2) − n | n −
29 ( n +4)( n − n | n − , as desired.We are now ready to prove Conjecture 1.2. Theorem 2.8.
Let H ( n ) be the set of all connected graphs G with n vertices. Then, max G ∈ H ( n ) s ( G ) = s ( CS ( n, k )) , where k = n | n n − | n − n − and n +13 | n − . Proof.
Suppose G is not a regular graph. Then by repeated applications of Lemmas 2.2,2.3 and 2.4, we obtain a connected split graph H = S ( n, k ) such that s ( G ) ≤ s ( H ).Now by repeated applications of Lemmas 2.5 on graph H , we obtain a F = S ( n, k ) or F = CS ( n, k ) such that s ( G ) ≤ s ( H ) ≤ s ( F ) with equality if and only if G ∼ = S ( n, k ) or G ∼ = CS ( n, k ). The proof follows from Theorem 1.1 and Lemmas 2.6, 2.7. The aim of this section is to continue the interesting paper [2]. We will compute thedegree deviation measure of chemical graphs under some conditions on the cyclomaticnumber.Suppose n i = n i ( G ) is the number of vertices of degree i in a graph G . It can beeasily seen that P ∆( G ) i =1 n i = | V ( G ) | . If the graph G has exactly n vertices, m edges and k components, then c = m − n + k is called the cyclomatic number of G . A chemicalgraph is a graph with a maximum degree of 4. A connected chemical graphs with exactly n vertices and cyclomatic number c is called ( n, c )-chemical graph. Lemma 3.1. ( See [4] ) Let G be an ( n, c ) -chemical graph. Then n ( G ) = 2 − c + n + 2 n and n ( G ) = 2 c + n − − n − n . Lemma 3.2.
Let T be a chemical tree with n vertices. Then s ( T ) = 4( n − n + n − n [2 n ( T ) + 4 n ( T )] . Proof.
Since | E ( T ) | = n − s ( T ) = n ( T ) (cid:20) n − n − (cid:21) + n ( T ) (cid:20) − n − n (cid:21) + n ( T ) (cid:20) − n − n (cid:21) + n ( T ) (cid:20) − n − n (cid:21) n − n n ( T ) + 2 n n ( T ) + n + 2 n n ( T ) + 2 n + 2 n n ( T ) . We now apply Lemma 3.1 to deduce that s ( T ) = n − n + n − n [2 n ( T ) + 4 n ( T )], provingthe lemma. Corollary 3.3.
Let T be a chemical tree with n vertices. Then s ( T ) ≥ n − n , withequality if and only if T ∼ = P n . Lemma 3.4.
Let G be an ( n, -chemical graph. Then s ( G ) = 2 n ( G ) + 4 n ( G ) . Proof.
Since | E ( G ) | = n , s ( G ) = n ( G )[2 −
1] + n ( G )[2 −
2] + n ( G )[3 −
2] + n ( G )[4 − n ( G ) + n ( G ) + 2 n ( G ) , and by Lemma 3.1, s ( G ) = 2 n ( G ) + 4 n ( G ), as desired. Theorem 3.5.
Let G be an ( n, c ) -chemical graph such that c ≥ .1. If n > c − , then s ( G ) = 1 n h (2 n − c + 4) n ( G ) + (4 n − c + 4) n ( G ) i .
2. If n ≤ c − , then s ( G ) = n − c +1) n n ( G ) .Proof. By definition, | E ( G ) | = n + c − n > c −
2. Then, s ( G ) = n ( G ) (cid:20) n + c − n − (cid:21) + n ( G ) (cid:20) n + c − n − (cid:21) + n ( G ) (cid:20) − n + c − n (cid:21) + n ( G ) (cid:20) − n + c − n (cid:21) = n + 2 c − n n ( G ) + 2 c − n n ( G ) + n − c + 2 n n ( G ) + 2 n − c + 2 n n ( G ) , and by Lemma 3.1 s ( G ) = 1 n h (2 n − c + 4) n ( G ) + (4 n − c + 4) n ( G ) i . n ≤ c −
2. By the well-known result of Euler, 2 | E ( G ) | = P v ∈ V ( G ) deg G ( v ) ≤ P v ∈ V ( G ) n . Therefore, c = | E ( G ) | − n + 1 ≤ n + 1. Thus 2 c − ≤ n and s ( G ) = n ( G ) (cid:20) n + c − n − (cid:21) + n ( G ) (cid:20) n + c − n − (cid:21) + n ( G ) (cid:20) n + c − n − (cid:21) + n ( G ) (cid:20) − n + c − n (cid:21) = n + 2 c − n n ( G ) + 2 c − n n ( G ) + 2 c − − nn n ( G ) + 2 n − c + 2 n n ( G ) , and by Lemma 3.1, s ( G ) = n − c +1) n n ( G ).Hence the result. Acknowledgments.
The research of the second author was partially supported by theUniversity of Kashan under grant no 364988/109.6 eferences [1] M. O. Albertson, The irregularity of a graph,
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