A class of inequalities relating degrees of adjacent nodes to the average degree in edge weighted uniform hypergraphs

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A CLASS OF INEQUALITIES RELATING DEGREES

OF ADJACENT NODES TO THE AVERAGE DEGREE

IN EDGE-WEIGHTED UNIFORM HYPERGRAPHS

P. D. JOHNSON JR. AND R. N. MOHAPATRA

Received 26 August 2004 and in revised form 28 September 2005

In 1986, Johnson and Perry proved a class of inequalities for uniform hypergraphs which included the following: for any such hypergraph, the geometric mean over the hyperedges of the geometric means of the degrees of the nodes on the hyperedge is no less than the average degree in the hypergraph, with equality only if the hypergraph is regular. Here, we prove a wider class of inequalities in a wider context, that of edge-weighted uniform hypergraphs.

1. Definitions and preliminary observations

Ak-uniform hypergraphis a pair (V,E)=Gin whichV, the set ofverticesornodesofG, is a set, andE, the set ofhyperedgesoredgesofG, is a multiset of multisets of elements of V, with eache∈Ehaving multiset cardinalityk. Whenk=2, such an object is called a graph, ormultigraph, and, in this case, whenEand every element ofEare sets, thenGis called asimple graph.

Throughout this paper,G=(V,E) will be a finitek-uniform hypergraph ofordern= |V|;|E|will be denoted by m. We require that every vertex appears on some edge. A weighting of the edges ofG is a functionwt:E→I, whereI is a real interval, usually [0,∞). For such a weighting, for any submultisetEofE,wt(E)=e∈Ewt(e). We will

refer to the pair (G,wt) as anedge-weighted hypernetwork, or, more simply, as aweighted network. Given (G,wt) andv∈V, thedegreeofvin (G,wt) is the sum of the weights of the edges to whichvbelongs, where each weight is counted according to the multiplicity ofvon the edge. More formally, forv∈V ande∈E, letδ(v,e) be the number of times thatvoccurs in the multisete. Then the degree ofvin (G,wt) is

d(v)= e∈E

δ(v,e)wt(e). (1.1)

Theaverage degreein (G,wt) is ¯d=(1/n)v∈Vd(v). Whendis a constant function,

(G,wt) isregular, with degreed=_{d. When}¯ _wt_≡_{1, the degree in the weighted network is} just the ordinary degree inG, denoted bydG.

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The k-uniformity of the hypergraph G implies a simple relation among ¯d,n, and wt(E):

¯ d= 1

n

v∈V

d(v)=1

n

v∈V

e∈E

δ(v,e)wt(e)

= 1

n

e∈E

v∈V

δ(v,e)

wt(e)=k

n

e∈E

wt(e)

=k

nwt(E).

(1.2)

Fore∈E, letu1(e),. . .,uk(e) be thekvertices one(repeated according to their

mul-tiplicities), in no particular order. Suppose thatFis a symmetric function ofkreal vari-ables. The inequality form announced in the title is∗(G,wt,F):

wt(E)−1

e∈E

wt(e)Fdu1(e),. . .,duk(e)

≥Fd,. . .¯ , ¯d. (1.3)

(Arrangements will be made so that all thek-tuples (d(u1(e)),. . .,d(uk(e))) and ( ¯d,. . .,

¯

d) are in the domain ofF.) Notice that∗(G,wt,F) holds with equality whenever (G,wt) is regular. We will be especially interested in families of weighted networks (G,wt) for which

∗(G,wt,F) holds, for some particularF, with equality only when (G,wt) is regular. Why the interest in inequalities of this form? One reason is that there are some quite memorable and useful inequalities of this form. For instance, in [2], it is shown that

∗(G,wt,F) holds strictly for all nonregular k-uniform G,wt≡1, and F(x1,. . .,xk)= k

i=1log(xi); manipulating this result, we obtain, withd=dG,

Πe∈E

Πk

j=1d

uj(e)

1/k1/|E|

≥_d,¯ _(1.4)

with equality only whenG=(G, 1) is regular. This says that the geometric mean, over the edges, of the geometric means of the degrees of the vertices on each edge is no less than the arithmetic average degree (with equality only whenGis regular), which seems, at first glance, to reverse the conventional wisdom about geometric means and arithmetic means. A moment’s thought shows that (1.4) is not very surprising, since vertices con-tribute as many factors to the product on the left-hand side of (1.4) as their degree, which gives rich vertices of high degree an undemocratic advantage in hoisting the result. Still, surprising or not, it is a pretty and inobvious inequality.Corollary 2.3is a generalization of it.

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such that∗(G,wt,F) holds for allGandwt. Before getting to that result, however, we make two easy remarks that bear on the general question.

(1) In definingk-uniform hypergraphs, we allowedEto be a multiset, that is, hyper-edges can be repeated. Not only is this customary, but it also seemed unavoidable, for full generality, in the main result in [2]. It is a happy consequence of the generalization to weighted networks that we can simplify to the cases whereEis merely a set. SupposeE is a multiset, and∗(G,wt,F) holds for somewt,F. LetGbe obtained by collapsing each multiple (repeated) hyperedge to a single hyperedge, and letwt be the weighting of the hyperedges ofGin which each hyperedge is weighted with the sum of thewtvalues on its copies inE. ThenGhas the same vertices asG, with the same degrees, and the fact that multiplication distributes over addition shows that∗(G,wt,F ) holds. The same observa-tion shows that going the other way, splitting a weighted edge into copies and distributing its weight over the copies, preserves the inequality.

(2) Suppose that 0 is allowed as an edge weight, and suppose that∗(G,wt,F) holds, for someF, and for allwt, or perhaps for allwtin some reasonable class, such as the class of weightings with respect to which every vertex has positive degree. (If, e.g.,F(x1,. . .,xk)

is defined only forx1,. . .,xk>0, and if 0 is allowed as an edge weight, then∗(G,wt,F)

would make sense only for weightings in that class.) Then we can conclude, if the class of weightingswtfor which∗(G,wt,F) holds permits, that∗(H,wt,F) holds for all suitable wt, and for allspanningsubhypergraphsHofG. A spanning subhypergraph ofGis one obtained by deleting edges of G, but so that each vertex ofG still lies on at least one of the edges remaining. Given such a spanning subhypergraph H, and a weightingwt of the edges ofH, extend the weighting to a weighting ofG simply by weighting each edge ofG which is not in H with zero. Then,∗(G,wt,F) clearly implies∗(H,wt,F), since the degrees of the vertices with respect to wtare the same, in GandH, and the terms on the left-hand side of the inequality corresponding to edges inG, but notH, are zero.

Observations (1) and (2) are trivial, but useful. For instance, to prove that∗(G,wt,F) holds, for some particularF, for allwt(or, allwtsuch that every degree is positive), and for all loopless multigraphs (k=2 and loops and isolates are not allowed)G, it suﬃces to verify the inequality for all simple complete graphs. To prove that the inequality holds for allwtand all multigraphsG, it suﬃces to show that it holds for allwtandGranging over the simple complete graphs with a loop added at each vertex.

2. Results and proofs

A real-valued function f isconvexon an intervalIif and only if

ftx+ (1−t)y≤t f(x) + (1−t)f(y) ∀x,y∈I,t∈[0, 1]. (2.1)

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Iff is convex onI,x1,. . .,xn∈I,λ1,. . .,λn>0, and _n

i=1λi=1, then

f

_n

i=1 λixi

≤

n

i=1

λifxi. (2.2)

If f is strictly convex onI, then equality holds in the inequality just above if and only if x1= ··· =xk. This inequality is known as Jensen’s inequality (see [1], pages 70–74). We

will be using it first withλ1= ··· =λn=1/n.

The following lemma is actually a generalization of most of the theorem to follow. We have relegated it to lemma status because it is quite unmemorable.

Lemma2.1. Suppose thatA=[ai j]is ann×mmatrix of real numbers with constant

col-umn sumc; that ϕis a real-valued function on a real intervalI such that f, defined by f(x)=xϕ(x), is convex onI; and that(x1,. . .,xm)is anm-tuple of real numbers such that

s=m

j=1xj>0anddi=mj=1ai jxj∈I for each i=1,. . .,n. Letd¯=(1/n)ni=1di. Then

s−1m j=1

n

i=1ϕ(di)ai jxj≥cϕ( ¯d), and if f is strictly convex, then equality holds only if

d1= ··· =dn.

Proof. First note that ¯d=(1/n)n_i₌1

m

j=1ai jxj=(1/n)

m

j=1( n

i=1ai j)xj=(c/n)s. With

this in mind, we have

s−1

m

j=1

n

i=1 ϕdi

ai jxj=s−1 n

i=1

_m

j=1 ai jxj

ϕdi

=s−1

n

i=1 diϕ

di

=s−1

n

i=1 fdi

≥s−1_{n f}_d_¯₌_s−1_n_dϕ_¯ _d_¯

=s−1_nc nsϕ

_¯

d=cϕd¯,

(2.3)

by the presumed convexity of f, and if f is strictly convex, then equality forces thedito

be equal.

Theorem2.2. Suppose thatϕis a real-valued function on a real intervalIsuch thatf(x)=

xϕ(x)is convex onI. For each integerk≥1, and eachk-uniform hypergraphG=(V,E), and each weightingwtof the edges ofGsuch thatwt(E)>0, and the weighted degrees of vertices are all inI, the inequality∗(G,wt,F)holds, withF defined byF(x1,. . .,xk)=ki=1ϕ(xi).

Further, if f is strictly convex, then equality implies that the weighted network(G,wt)is regular.

IfI⊆[0,∞),ϕis bounded on each compact subinterval ofIand∗(C4,wt,F)holds for all admissible edge weightings ofC4, the simple cycle of length4 (k=2), then f(x)=xϕ(x)is convex. Further, if equality implies that(C4,wt)is regular, then f is strictly convex.

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all equal tok. For a weightingwtsatisfying the hypothesis of the theorem, let (wt(e))e∈E

play the role of (x1,. . .,xm) in the lemma. Thens=wt(E), and forv∈V,

dv=

e∈E

δ(v,e)wt(e)=d(v). (2.4)

Further,e∈E

v∈Vϕ(d(v))δ(v,e)wt(e)=

e∈Ewt(e)F(u1(e),. . .,uk(e)). Thus, the

ine-quality concluded in the lemma is

wt(E)−1

e∈E

wt(e)Fu1(e),. . .,uk(e)

≥kϕd¯=Fd,. . .¯ , ¯d, (2.5)

and the conclusion in case of strict convexity follows, as well.

Now suppose thatI⊆[0,∞),ϕis “locally bounded,” as hypothesized, and∗(C4,wt,F) holds for all weightingswtsuch thatwt(E)>0 and the weighted degrees of the vertices ofC4 are all inI. Then, f is locally bounded. It is an elementary exercise to verify that, in this case, to show that f is convex, it suﬃces to show that f ismidpoint convex, that is, thatf((x+y)/2)≤(1/2)(f(x) + f(y)) for allx,y∈I. Suppose thatx,y∈I; we may as well suppose thatx=y. Weight the edges ofC4as depicted:

x/2 _d

d

x/2

y/2

d d

(2.6)

Then, I⊆[0,∞) and x=y imply that wt(E)=x+y >0, and the degrees of the ver-tices ofC4,x,y, and (x+y)/2 are all inI. The inequality∗(C4,wt,F) asserts that (1/(x+ y))[2(x/2)(ϕ(x) +ϕ((x+y)/2)) + 2(y/2)(ϕ(y) +ϕ((x+y)/2))]≥2ϕ((x+y)/2), which rearranges to f(x) +f(y)≥2f((x+y)/2), which verifies that f is midpoint convex, and therefore convex.

If f is convex but not strictly convex, then f(x)=ax+bfor allxin some subinterval ofI, for some constantsaandb. Choosingx and y,x=y, from that subinterval, and weighting the edges ofC4 as above, we have equality in∗(C4,wt,F), yet (C4,wt) is not

regular.

Corollary2.3. Suppose thatk≥1,G=(V,E)is ak-uniform hypergraph, andwt is a weighting of the edges ofGwith real weights so that the degree of each vertex ofGin(G,wt) is positive. Then

Πe∈EΠkj=1d

uj(e)wt(e)/k 1/wt(E)

≥_d,¯ _(2.7)

with equality only if(G,wt)is regular.

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Corollary2.4. Withk,G, andwtas inCorollary 2.3, suppose thatp >0. Then

1 wt(E)

e∈E

wt(e)

1 k

k

j=1 duj(e)

p 1/ p

≥d,¯ (2.8)

with equality only if(G,wt)is regular.

For the proof, apply the theorem withϕ(x)=xp_.

It is intriguing that the edge weights in the corollaries above are allowed to be nonpos-itive, as long as the degrees come out positive. It is worth noting thatkcan be 1 in these corollaries; in that case, each edge weight is a degree, and the inequalities turn out to be rather familiar, to inequality aficionados. The inequality inCorollary 2.4, for instance, whenk=1, is a familiar consequence of H¨older’s inequality.

Corollaries 2.3 and 2.4 are only the most obvious applications of Theorem 2.2. It is worth noting that the two are related: as p tends to zero from above, the left-hand side of the inequality in Corollary 2.4 tends to the left-hand side of the inequality in Corollary 2.3. Further, ifwt(e)≥0 for alle∈E, the left-hand side of the inequality in Corollary 2.4nonincreases to the left-hand side in Corollary 2.3, as p↓0, so the latter inequality is “better” than the former, in that case.

The following is a generalization ofCorollary 2.3. We have statedCorollary 2.3 sepa-rately for esthetic reasons.

Corollary2.5. Suppose thatϕtakes positive values on a real intervalI, and f defined by f(x)=xlogϕ(x)is convex onI. For each integerk≥1and eachk-uniform hypergraph G=(V,E), and each weightingwtof the edges ofGsuch thatwt(E)>0, and the weighted degrees of the vertices are all inI,

Πe∈E

Πk

i=1ϕ

dui(e)

wt(e)/k1/wt(E)

≥ϕd¯. (2.9)

Further, iff is strictly convex onI, then equality holds only if(G,wt)is regular. For the proof, applyTheorem 2.2withϕreplaced by logϕ.

Corollary2.6. Suppose thatk≥1,G is ak-uniform hypergraph,ϕandwt satisfy the hypotheses ofCorollary 2.5, andwt(e)≥0for eache∈E. LetF(x1,. . .,xk)=(Πki=1ϕ(xi))1/k.

Then∗(G,wt,F)holds, and if the function f inCorollary 2.5is strictly convex onI, then equality in∗(G,wt,F)holds if and only if(G,wt)is regular.

Proof. It is a well-known improvement of the arithmetic mean-geometric mean inequal-ity that ifλ1,. . .,λm≥0,mi=1λi=1, anda1,. . .,am≥0, then

m

i=1

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Applying this with the numberswt(e)/wt(E),e∈E, playing the roles of theλi, and the

numbers (Πkj=1ϕ(d(uj(e))))1/kplaying the roles ofai, we have

1 wt(E)

e∈E

wt(e)Πk j=1ϕ

duj(e) 1/k

≥Πe∈E

Πk j=1ϕ

duj(e)

wt(e)/k1/wt(E) .

(2.11)

Noting thatF( ¯d,. . ., ¯d)=(ϕ( ¯d)k₎1/k₌_{ϕ( ¯}_{d), the result now follows from}_{Corollary 2.5.}

With a few loose ends hanging,Theorem 2.2pretty well characterizes thoseϕsuch that

∗(G,wt,F) holds, for allk-uniform hypergraphsG, for admissiblewt(see the hypothesis ofTheorem 2.2), and forF defined byF(x1,. . .,xk)=

_k

i=1ϕ(xi). Theorem 2.7makes a

start on the same job, forFof the formF(x1,. . .,xk)=Πki=1ϕ(xi).

Theorem2.7. Letk≥1, letG=(V,E)be ak-uniform hypergraph, and letϕandwtsatisfy the hypotheses ofCorollary 2.5. Suppose thatFis defined byF(x1,. . .,xk)=Πki=1ϕ(xi). Then ∗(G,wt,F)holds, and if f(x)=xlogϕ(x)is strictly convex onI, then equality holds if and only if(G,wt)is regular.

Proof. Sinceg(x)=xk_{is convex and increasing on [0,}_∞_),

1 wt(E)

e∈E

wt(e)Πk_i₌1ϕ

dui(e)

= 1

wt(E)

e∈E

wt(e)Πki=1ϕ

dui(e) 1/kk

≥

1 wt(E)

e∈E

wt(e)Πk i=1ϕ

dui(e) 1/k

k

≥ϕd¯k=Fd,. . .¯ , ¯d,

(2.12)

byCorollary 2.6. The condition for equality also follows fromCorollary 2.6. Notice thatϕ(x)=xsatisfies the hypothesis of Theorems2.2and2.7, withI=(0,∞) in the case ofTheorem 2.7. The inequalities fromTheorem 2.7with thisϕseem espe-cially powerful, even though they are not, technically speaking, as “good” asCorollary 2.3. The reader is encouraged to try out a few examples, especially in the casek=2. For in-stance, the inequality fromTheorem 2.7withϕbeing the identity function, with the edge weighting ofC4 in the proof of Theorem 2.2, is equivalent to the venerable inequality x2₊_y2_≥_2xy.

3. Two problems

(1) Which other simple graphs can play the role thatC4plays inTheorem 2.2? Of course, we need not confine the question to simple graphs, but it seems a more attractive problem if we mercifully limit the question, at this point.

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whichGis a spanning subgraph will have that property. So solving this problem (taking, say,I=[0,∞)) will amount to finding the minimal graphs of each order, if any, that have that property. We wonder if all cycles have that property, and if any trees do.

(2) For 1≤s≤k, letA(k,s) denote the collection ofs-subsets of{1,. . .,k}and letPk,s

be defined byPk,s(x1,. . .,xk)=

B∈A(k,s)Πi∈Bxi;Pk,sis the “standard” symmetric

poly-nomial of degree s of k variables. Theorems 2.2 and 2.7 are about ∗(G,wt,F) being satisfied for allk-uniformG and all admissiblewt, withF of the form F(x1,. . .,xk)=

Pk,s(ϕ(x1),. . .,ϕ(xk)), fors=1 ands=k. The game we are playing is to ask for whichϕis ∗(G,wt,F) true for all (G,wt). Attacking this question for 1< s < klooks quite like a job. Here is a smaller, more approachable question: so far, fors=1 ands=k, ifI=[0,∞), the functionsϕ(x)=xp_,_{p >}_{0, have turned out to be “good;” will these functions be “good”}

for allsandk?

References

[1] G. H. Hardy, J. E. Littlewood, and G. P ´olya,Inequalities, Cambridge University Press,

Cam-bridge, 1934.

[2] P. D. Johnson Jr. and R. L. Perry,Inequalities relating degrees of adjacent vertices to the average

degree, European J. Combin.7(1986), no. 3, 237–241.

P. D. Johnson Jr.: Department of Mathematics and Statistics, College of Science and Mathematics, Auburn University, AL 36849-5307, USA

E-mail address:[email protected]

R. N. Mohapatra: Department of Mathematics, College of Arts and Sciences, University of Central Florida, Orlando, FL 32816-1364, USA