Graphs of Maximal Energy with Fixed Maximal Degree
aa r X i v : . [ m a t h . C O ] F e b Graphs of Maximal Energy with Fixed MaximalDegree
Octavio ArizmendiCentro de Investigaci´on en Matem´aticasGuanajauto, Mexico. Email: [email protected] Fernandez HidalgoUniversidad Nacional Aut´onoma de MexicoMexico City, Mexico. Email: [email protected] (accepted in MATCH)
Abstract
We give a bound for the graph energy with given maximal degree in terms of thesecond and fourth moments of a graph. In the case in which the graph is d -regularwe obtain the bound that is given in Van Dam, E. et al. (2014). through elementarymethods. In this paper we study the energy of a graph as defined by Gutman [3]. For a graph G on n vertexes with adjacency eigenvalues λ , . . . , λ n , the energy of the graph G is given by sum of the absolute value of its adjacency eigenvalues, n X i =1 | λ i | . Several results on bounds for the energy of a graph have been considered in thetheory. e.g. [6,8,12]. A tool to give general bounds is given by the so-called spectralmoments of a graph [4, 5, 7, 9, 10, 13]. The k -th spectral moment of a graph is givenby M k ( G ) = n X i =1 λ ki . he usefulness of these quantities comes from the fact that, when k is an integer, M k has a combinatorial interpretation: M k ( G ) is the trace of the k ’th power of theadjacency matrix of G , and consequently, is given by the number of closed walks oflength i in G .Here we present a simple but effective method to bound the energy of a graphin terms of its spectral moments which we describe in Lemma 3.1.A direct application of this method provides upper and lower bounds for theenergy of a graph with n vertexes, and maximum degree ∆, in terms of the secondand fourth moments.To state this result, we shall use the notation A = M ∆ , B = M ∆ , C = ∆ n. (1.1)Our main theorem is as follows. Theorem 1.1.
Let G be a connected graph with at least vertexes, E ( G ) ≤ − B + B √− A + B √− B + C − C ( A + √− A + B √− B + C ) A − B + C (1.2) with equality if and only if G is a complete graph K n , a strongly regular graph with λ = µ or the incidence graph of a symmetric − ( v, k, λ ) design. For regular graphs one can see that (1.2) is decreasing in A (see Lemma 3.6),and thus the above theorem subsumes the main theorem of Van Dam et al. [11]. Theorem 1.2 ( [11]) . For a regular graph G one has E ( G ) ≤ n d + ( d − d ) √ d − d − d + 1 with equality if and only if G is the incidence graph of a symmetric − ( v, d, design. We must mention that the above result was derived originally by very differentmethods from the ones in this paper and we do not know if the methods from [11],may be applied to show our main theorem.Apart from the introduction, the paper has two more sections. Section 2 givesthe basic preliminaries needed for this paper and the proofs of the main results aregiven in Section 3.
Preliminaries on graphs
Throughout this paper, we shall work exclusively with simple finite graphs. Weshall denote the number of vertexes of the graph G by n and the number of edgesby m . We will denote the vertexes of a graph G using v , . . . , v n . We say vertex v i is a neighbour of v j if v i and v j are adjacent (in other words if ( v i , v j ) is an edge of G ). We define the degree of a vertex v i as its number of neighbours and we denotethis quantity by d i . The maximum degree over all vertexes of the graph will bedenoted by ∆, while the minimum degree will be denoted by δ . If ∆ = δ = d wesay that G is a d -regular graph.Given a graph G , the adjacency matrix of G is the n × n symmetric matrix A such that A ij is 1 if i and j are adjacent and A i,j is 0 otherwise. Being a symmetricmatrix, A has n real eigenvalues, counted with multiplicity, which are called theadjacency eigenvalues of the graph G , we shall sometimes simply refer to them asthe eigenvalues of G . The spectral moments of the graph G are the quantities M k = T r ( A k ). Since A isnormal then T r ( A k ) = n P i =1 λ ki . Let us note that the 0-th, first and second momentsare given by M = n, M = 0 , and M = 0 , let us denote the number of 4-cycles in G , by Q and use Z for the Zagreb indexgiven by Z = n P i =1 d i , then M = 2 Z − m + 8 Q. Hence, with the notation of the introdution, we have, A = 2 Z − m + 8 Q ∆ , B = 2 m ∆ , C = ∆ n. (2.1) Finally, we describe the graphs which satisfy the equality in Theorems 1.1 and 1.2.We say that a graph is strongly connected with parameters λ, µ if it is regularand it satisfies: the following properties
Every two adjacent vertexes have λ common neighbours. • Every two non adjacent vertexes have µ common neighbours.The spectrum of a strongly connected graph with n vertexes and common degree d is: • d with multiplicity 1. • [( λ − µ )+ p ( λ − µ ) + 4( d − µ )] with multiplicity [( v − − k +( v − λ − µ ) √ ( λ − µ ) +4( d − µ ) ] , • [( λ − µ ) − p ( λ − µ ) + 4( d − µ )] with multiplicity [( v − k +( v − λ − µ ) √ ( λ − µ ) +4( d − µ ) ] . We refer to a symmetric 2 − ( v, k, λ ) design as a set X with v elements, alongwith a family F consisting of v subsets of X (called blocks) such that each blockhas k elements and for any two points x and y in X there are exactly λ blocks thatcontain both.We define the incidence graph of a symmetric 2 − ( v, k, λ ) design as the graphwith vertexes F ∪ X such that there is an edge between a block B ∈ F and a point x ∈ X if and only if x ∈ B .The spectrum of the incidence graph of a symmetric 2 − ( v, k, λ ) design is: • ± k , each with multiplicity 1. • ±√ k − λ , each with multiplicity v − The main idea comes from the following observation. Let P ( x ) = a m x m + · · · + a be a polynomial such that P ( x ) ≥ | x | for all x ∈ [ − ρ, ρ ], then E ( G ) = n X i =1 | λ i | ≤ n X i =1 P ( λ i ) = m X i =0 a m M m . Here we emphasize that in order to find useful bounds for E ( G ) we only need totake into consideration x ∈ [ − ρ, ρ ] and not in all R . Since obtaining ρ directly fromcombinatorial properties of the graph is, in general hard we use instead ∆, obtainingthe following lemma, which includes analogous lower bounds for E ( G ). emma 3.1. Let G be a graph with maximum degree ∆ .1) For any polynomial P ( x ) = a m x m + · · · + a a polynomial such that P ( x ) ≥ | x | on x ∈ [ − ∆ , ∆] we have E ( G ) ≤ m X i =0 a m M m . (3.1)
2) For any polynomial Q ( x ) = a m x m + · · · + a be a polynomial Q ( x ) ≤ | x | , on x ∈ [ − ∆ , ∆] we have E ( G ) ≥ m X i =0 a m M m . (3.2) Moreover, equality in (3.1) (resp. (3.2) ) occurs if and only if P ( λ i ) = | λ i | (resp. Q ( λ i ) = | λ i | ) for all i . The strength of the previous lemma comes from the freedom of choosing theabove polynomials, and using the information of the class of graphs studied is veryhelpful as we will see in the next sections.
Let us consider the case where P is a polynomial of degree 4. Since the absolutevalue is an even function considering x or x in the polynomial will produce one ofthe sides to tilt which will worsen our approximation. Thus we assume that P iseven. Then we are led consider a polynomial of the form P ( x ) = ax + bx + c. For this polynomial P ( x ) = ax + bx + c we have T r ( P ( A )) = n X i =1 P ( λ i ) = aM + bM + cM = 2 aZ + 8 aQ + 2( b − a ) m + cn. (3.3)Moreover, we see that T r ( P ( A )) = aM + bM + cM = a (2( n X i =0 d i ) − m + 8 Q ) + bm + cn (3.4)= a Q + ( b − m + ( c + ad ) n = a Q + (cid:18) c + ad + bd − d (cid:19) n. We note for further reference that when G is d -regular we have 2 m = dn and Z = nd which gives us n X i =1 P ( λ i ) = 2 and + 8 aQ + ( b − a ) nd + cn. (3.5)Now our goal is to choose a, b and c which minimize the left-hand side of (3.4)restricted to the condition that P ( x ) ≥ x for ∆ ≥ x ≥
0. For this we seek forpolynomials that are tangent to f ( x ) = x . emma 3.2. Given < r < there is a unique polynomial P r = ax + bx + c suchthat P r ( r ) = r, P r (1) = 1 and P r ( x ) ≥ x for x ∈ [0 , .Proof. We shall call the desired polynomial P r = ax + bx + c . Since P r must betangent to the absolute value function at r the following equations must be satisfied P r ( r ) = ar + br + c = r,P r (1) = a + b + c = 1 ,P ′ r ( r ) = 4 ar + 2 br = 1 . The solution to this system of equations is unique and gives us a = − r ( r + 1) , b = 3 r + 2 r + 12 r ( r + 1) , c = r (2 r + 1)2 r ( r + 1) . (3.6)First we show that P r ( x ) > x for all 0 < x < x = r . Consider for this,the function Q ( x ) = P ( x ) − x . Its second derivative is given by 12 ax − b , whichis decreasing and thus Q ( x ) is convex on (0 , r ) and concave on ( r , r = q r +2 r +16 is the unique solution to Q ′′ ( x ) = 0. Since r ∈ (0 , r < r < , r ] the function is convex it reaches its minimum at thepoint such that Q ′ ( x ) = 0. This point is r and we have Q ( r ) = 0. In the interval[ r ,
1] the function is concave and therefore reaches its minimum at one of r and 1.We have Q ( r ) > Q ( r ) = 0 and we have Q (1) = 0. So indeed P r is as desired.Thus from Lemma 3.1 we obtain the following bound. Theorem 3.3.
Let r ∈ (0 , . For G a graph with maximum degree ∆ the energyof G is bounded by E ( G ) ≤ − r ( r + 1) Z − m + 8 Q ∆ + ( 3 r + 2 r + 12 r ( r + 1) ) 2 m ∆ + n ∆ r (2 r + 1)2 r ( r + 1) Proof.
Let P ( x ) be as in Lemma 2.2. Then P r ( x ) ≥ | x | for x ∈ [ − , P r, ∆ ( x ) = ∆ P r ( x/R ), be the dilation by ∆ of the polynomial P r . Then P r, ∆ ( x ) ≥ | x | for x ∈ [ − ∆ , ∆]. By part 1) of Lemma 2.1 we conclude.For the following we recall the notation A = 2 Z − m + 8 Q ∆ , B = 2 m ∆ , C = ∆ n e want to minimize for r . For this, we claim that A ≤ B ≤ C and that0 ≤ B − AC − B ≤
1. To show that B ≤ C we simply notice 2 m ≤ ∆ n ∆ n , and to prove A ≤ B we see it is equivalent to M ≤ m ∆ . Now, notice that for every orientededge u, v there can be at most ∆ closed walks of length 4 starting with u, v .By elementary calculus one sees that subject to the restriction A ≤ B and B ≤ C . The minimum value of the function − r ( r +1) A + ( r +2 r +12 r ( r +1) ) B + r (2 r +1)2 r ( r +1) C for r ∈ (0 ,
1) is reached at √ B − A √ C − B , giving the value − B + B √ B − A √ C − B − C ( A + √ B − A √ C − B ) A − B + C . (3.7)Thus we arrive to the following theorem.
Theorem 3.4.
For any graph G one has E ( G ) ≤ − B + B √− A + B √− B + C − C ( A + √− A + B √− B + C ) A − B + C with equality if and only if the spectrum of the graph is contained in {± √ B − A √ C − B , ± ∆ } Proof.
The inequality follows from the considerations above. In order to have equal-ity, the polynomial P √ B − A √ C − B , ∆ ( x ) and the function Abs ( x ) := | x | should coincide forall eigenvalues of G . This is only possible for the set {± √ B − A √ C − B , ± ∆ } . The following propositon characterizes the graphs for which equality is reachedand thus completes the main theorem of this paper.
Proposition 3.5.
The only connected graphs with at least vertexes for whichequality is attained are the complete graphs K n , strongly regular graphs with λ = µ and incidence graphs of symmetric − ( v, k, λ ) designs.Proof. Equality occurs if and only if the spectra of G is contained in {± √ B − A √ C − B , ± ∆ } .If ∆ is not an eigenvalue of G then − ∆ is not an eigenvalue either, so G hasexactly two eigenvalues. This implies G is complete, however complete graphs have∆ as an eigenvalue. We conclude ∆ is an eigenvalue, meaning G is regular.If − ∆ is an eigenvalue then G is bipartite and therefore has symmetric spectra.If G has exactly 2 eigenvalues then G is complete and bipartite (So G = K ).If G has exactly 4 eigenvalues then G is a regular bipartite graph with exactly 4eigenvalues and must therefore be the incidence graph of a symmetric 2 − ( v, k, λ )design, as shown in [1].f − ∆ is not an eigenvalue then G must have exactly 3 eigenvalues or exactly 2eigenvalues. The only connected regular graphs with exactly 3 eigenvalues are thestrongly regular graphs, as is show in [2]. Moreover λ = µ is required so that oneeigenvalue is the negative of the other. Finally, if G has only two eigenvalues itmust be a complete graph.It should be clear that equality is indeed achieved in all these cases, as a poly-nomial P r, ∆ ( x ) can be found such that P ( x ) = | x | for all eigenvalues x of G .To minimize (3.7) we notice the following. Lemma 3.6.
The expression given in (3.7) is decreasing in A .Proof. Note that the formula − r ( r +1) A + ( r +2 r +12 r ( r +1) ) B + r (2 r +1)2 r ( r +1) C is decreasing in A for every value of r ∈ (0 , r ∈ (0 ,
1) and thus should also be decreasing on A .Thus, from (2.1), one should expect that having quadrangles should decreasethe energy. d -regular graphs Finally, we may recover the result in [11]. Indeed, for the case of regular graphs wehave A = 2 nd + nd + 8 Qd , B = n, C = nd. Lemma 3.6 shows that, for n and d are fixed, (3.7) is decreasing in Q and then Q = 0gives a bound for all regular graphs. In this case the minimum value simplifies andis reached at d √ d − Corollary 3.7.
For a regular graph G one has E ( G ) ≤ n d + ( d − d ) √ d − d − d + 1 with equality if and only if G is the incidence graph of a symmetric − ( v, d, design. cknowledgment O. Arizmendi was supported by a CONACYT Grant No. 222668 and by the Eu-ropean Union’s Horizon 2020 research and innovation programme under the MarieSk lodowska-Curie grant agreement 734922, during the writing of this paper.
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