Edge-connectivity matrices and their spectra
aa r X i v : . [ m a t h . C O ] F e b Edge-connectivity matrices and their spectra
Tobias Hofmann, Uwe Schwerdtfeger [email protected]@gmail.com
Abstract.
The edge-connectivity matrix of a weighted graph is thematrix whose off-diagonal v - w entry is the weight of a minimum edgecut separating vertices v and w . Its computation is a classical topic ofcombinatorial optimization since at least the seminal work of Gomoryand Hu. In this article, we investigate spectral properties of these ma-trices. In particular, we provide tight bounds on the smallest eigenvalueand the energy. Moreover, we study the eigenvector structure and showin which cases eigenvectors can be easily obtained from matrix entries.These results in turn rely on a new characterization of those nonnega-tive matrices that can actually occur as edge-connectivity matrices. Keywords. graph spectra, graph energy, minimum cuts, edge-disjointpaths, path matrices
MSC Subject classification.
For an undirected graph G = ( V, E ) with nonnegative edge weights its edge-connectivity matrix is the V × V matrix C ( G ) whose off-diagonal v - w entry denotesthe minimum weight of an edge set whose removal disconnects the vertices v and w .The diagonal entries are defined as zero. In the seminal article [5], Gomory andHu show that there exists a weighted tree T = ( V, F ) on the same vertex set as G ,but not necessarily with F ⊆ E , such that C ( T ) = C ( G ). Even stronger, for eachpair of vertices v and w the two sides of a minimum cut that separates v and w in T also induce a minimum cut that separates v and w in G . An auxiliary re-sult of Gomory and Hu, which is particularly important for our investigation, is acharacterization of those matrices that can occur as edge-connectivity matrices of1 3 4 612Figure 1: A graph G for which the matrix P ( G ) + D ( G ) is not positive semidefiniteweighted graphs. This description is in terms of a special triangle inequality, whichis stated in Theorem 2.1. In this article, we provide another characterization anddemonstrate how it can be utilized to gain further insights about the spectrum.Our considerations about the spectrum of the edge-connectivity matrix are inspiredby recent articles that investigate the spectrum of the vertex-connectivity matrix or path matrix P ( G ) of an unweighted graph G = ( V, E ). This is the V × V matrixwhose off-diagonal v - w entry is the maximum number of independent v - w pathswith only zeros on the diagonal. Shikare and coauthors [14] raised the conjecturethat the energy of P ( G ), that is the sum of the absolute values of the eigenvalues,is at most 2( n − . Ilić and Bašić [9] claim to prove this bound by employingthe approach of Koolen and Moulton [10]. However, following their argumentscarefully, they only obtain an upper bound of ( n − / +( n − / , which is strictlylarger than ( n − for all n ∈ N . Nevertheless, our numerical investigationshave not yet revealed any counterexamples to the stated bound. An analogousresult for the edge-connectivity matrix is presented in Item (i) of Theorem 1.1.In another article [13], Patekar and Shikare claim that P ( G ) + D ( G ) is positivesemidefinite, where D ( G ) denotes the diagonal matrix of vertex degrees. Thiswould immediately imply the conjecture about the energy. However, the claimthat P ( G ) + D ( G ) is positive semidefinite is indeed false. A counter example isgiven by the graph G in Figure 1 for which P ( G ) + D ( G ) = . = det − − = 4 det " − − = − . Another attempt to prove the energy bound was to show the weaker assumptionthat the smallest eigenvalue of P ( G ) is at least − ( n − − ( n + 1).It appears rather natural to consider also the related problem where the numberof independent paths is replaced by the number of edge-disjoint paths or, what isthe same, edge cuts. The preceding equivalence is the edge version of Menger’stheorem [12], which, for instance, is presented by Diestel [3, Section 3]. It turns outthat the aforementioned assertions are both true for the edge-connectivity matrixand even stronger statements hold. Theorem 1.1.
For a weighted graph G on vertex set V let C = ( c vw ) be itsedge-connectivity matrix. Denote for v ∈ V by m ( v ) := max { c vw : w ∈ V \ { v }} the maximum off-diagonal entry in row v and by M := max { c vw : v, w ∈ V } themaximum entry of C . Then the following statements hold.(i) The matrix C + diag( m ( v ) : v ∈ V ) is positive semidefinite.(ii) The smallest eigenvalue of C is − M .(iii) The energy of C is at most 2( n − M with equality if and only if G isuniformly M -edge-connected.Note that Item (i) comprises the result that C ( G )+ D ( G ) is positive semidefinite forany graph G because all v, w ∈ V ( G ) satisfy c vw ≤ min { deg( v ) , deg( w ) } and thusalso m ( v ) ≤ deg( v ) is true for all v ∈ V ( G ). Proofs for the items of Theorem 1.1are given in the subsequent sections in a more general setting.Basic properties of the edge-connectivity matrix are also investigated by Akbariand coauthors [1]. They determine the spectrum of bicyclic graphs and consider thestructure of uniformly k -edge-connected graphs. These are graphs in which k ∈ N is the maximum number of edge-disjoint paths between any two vertices. Analo-gously, a graph is called uniformly k -connected if k ∈ N is the maximum numberof independent paths between any two vertices. More about the structure of thesegraph classes can be found in Göring, Hofmann, and Streicher [7]. Also notethat there are results on the largest eigenvalue ̺ of vertex-connectivity matrices.Shikare and coauthors [14] show that ( n − ≤ ̺ ≤ ( n − for a graph on n ver-tices by using Perron-Frobenius arguments, which are presented comprehensivelyby Horn and Johnson [8, Chapter 8]. Both bounds are tight. They are attained3or trees or complete graphs, respectively. Moreover, the same arguments andobtained bounds hold analogously for edge-connectivity matrices. Outline.
In Section 2, we present an alternative characterization of edge-connec-tivity matrices. We use these results in Section 3 to prove spectral bounds andeigenvector properties. Finally, Section 4 provides an alternative interpretation ofour results for distance matrices whose entries satisfy an ultrametric.We conclude this introduction with further notation that is particularly importantfor our investigation. All vectors and matrices in this article are indexed by somefinite set V or V × V , respectively, and we write n = | V | for short. For a subset of X ⊆ V we write X for the vector x with x v = 1 if v ∈ X and x v = 0 otherwise.By J X we denote the V × V matrix for which ( J X ) vw = 1 if ( v, w ) ∈ X × X and 0 otherwise. In other words, J X = X ⊤ X . By we mean the all onesvector, for J V we occasionally write J for short and I is the identity matrix.Furthermore, we denote V × V diagonal matrices whose diagonal entries are givenby a sequence ( x v ) v ∈ V by diag( x v : v ∈ V ). The vectors of the standard basis of R V are denoted by e v for v ∈ V . For a symmetric matrix C with eigenvalues λ , . . . , λ n its energy E ( C ) is defined as E ( C ) = P ni =1 | λ i | . An overview about methods anddifferent variants of the energy concept is given by Li, Shi, and Gutman in themonograph [11]. By an equitable partition of a matrix C we mean a partition of C into submatrices C = C C k C k C kk , where each block C ij has constant row sums equal to q ij . The matrix Q = ( q ij ) iscalled an equitable quotient matrix of C and all eigenvalues of Q are also eigenvaluesof C . You and coauthors [16] provide details about the structure of such matrices.Lastly, for graph theoretical terminology we refer to Diestel [3]. To begin, we recall the following characterization that is due to Gomory and Hu [5].
Theorem 2.1.
A nonnegative real symmetric V × V matrix C with only zeros onits diagonal occurs as an edge-connectivity matrix of a weighted graph if and onlyif it satisfies the Gomory-Hu triangle inequality c xz ≥ min { c xy , c yz } for all x, y, z ∈ V with x = y = z = x . 4ur characterization is in terms of the following property. Definition 2.2.
For a real symmetric V × V matrix C = ( c vw ) and a real number ℓ we denote by S ℓ ( C ) = { ( v, w ) ∈ V × V : c vw ≥ ℓ } the superlevel set of C . We call C terraced if for each ℓ ∈ R there is a set T ( ℓ ) ofpairwise disjoint subsets of V such that S ℓ ( C ) = [ X ∈T ( ℓ ) X × X Both the Gomory-Hu triangle inequality as well as the terraced structure makeperfect sense without any assumptions on nonnegativity. So we formulate thenext theorem for arbitrary symmetric matrices.
Theorem 2.3.
For a real symmetric V × V matrix C the following statementsare equivalent.(i) The matrix C is terraced.(ii) The matrix C satisfies the Gomory-Hu triangle inequality for all x, y, z ∈ V .In particular, c vv ≥ c vw for all v, w ∈ V . Proof.
We show first that (i) implies (ii). So let C be terraced and define T ( ℓ )as in Definition 2.2. We choose x, y, z ∈ V without loss of generality such thatmin { c xy , c yz } = c yz =: ℓ , in other words c xy ≥ c yz ≥ ℓ . Then there is a set X in T ( ℓ ) with ( x, y ) ∈ X × X , which means that x, y ∈ X . Likewise, there is aset X ′ with y, z ∈ X ′ . Because z ∈ X ∩ X ′ , we obtain that X = X ′ , as the setsin T ( ℓ ) are defined to be pairwise disjoint. We observe that x, y, z ∈ X and inparticular ( x, z ) ∈ X × X which implies c xz ≥ ℓ , as desired.We show that (ii) implies (i) by induction on n := | V | . For n = 1 there is nothingto show. So let n > L := min { c vw : v, w ∈ V } . Suppose that thecolumn with index z contains an entry equal to L and define two sets X and Y by X = { x ∈ V : x = z and c xz = L } and Y = V \ X. Observe that X = ∅ because c zz ≥ c vz for v ∈ V . Furthermore, we have Y = ∅ because z ∈ Y . For x ∈ X and y ∈ Y it follows that c xz ≤ c yz = c zy andequality can only hold if Y = { z } and c zz = L . Applying the Gomory-Hu triangleinequality twice, we obtain that c xy ≥ min { c xz , c zy } = c xz ≥ min { c xy , c yz } = c xy . y = z , then c xz < c yz , asotherwise x / ∈ X . Thus for the preceding inequality to hold, the last minimummust be equal to c xy . If y = z , the last equality follows from c zz ≥ c xz , which istrue by the assumption in (ii). In both cases, we obtain that c xy = c xz = L for anarbitrary pair ( x, y ) ∈ X × Y .So we observe that all entries of C restricted to X × Y and Y × X are equal to L .Consequently, for ℓ > L the superlevel set S ℓ ( C ) is contained in ( X × X ) ∪ ( Y × Y ).The restrictions C | X × X and C | Y × Y of C to X × X and Y × Y , respectively, satisfythe triangle inequality and, by induction, are terraced. Because S L ( C ) = V and S ℓ ( C ) = S ℓ (cid:16) C | X × X (cid:17) ∪ S ℓ (cid:16) C | Y × Y (cid:17) for ℓ > L, we conclude that C is of the desired form.In the specific case of an edge-connectivity matrix C of a graph G on n vertices,we know by the results of Gomory and Hu [5] that such a matrix can have atmost n − n − With the structural results of the previous section at hand, we proceed with spec-tral investigations, which enable us to resolve the claims from Theorem 1.1.
Theorem 3.1.
A real nonnegative terraced matrix is positive semidefinite.
Proof.
Let 0 ≤ ℓ < · · · < ℓ k be the distinct values of entries of a real nonnegativeterraced matrix C . With the notation of Definition 2.2, we can write C as C = ℓ J V + k X i =1 X X ∈T ( ℓ i ) ( ℓ i − ℓ i − ) J X . This is a nonnegative linear combination of positive semidefinite matrices whichis why C is positive semidefinite as well. Proof of Item (i) of Theorem 1.1.
Let C = ( c vw ) be the edge-connectivity matrixof a weighted graph G . Then C satisfies the Gomory-Hu triangle inequality byTheorem 2.1. Furthermore, the matrix C + diag( m ( v ) : v ∈ V ) by definitionsatisfies c vv ≥ c vw for all v, w ∈ V , because m ( v ) denotes the maximum entryin row v ∈ V . So Theorem 2.3 tells us that C + diag( m ( v ) : v ∈ V ) is terraced.Moreover, it is clearly real and nonnegative and thus positive semidefinite, byTheorem 3.1. 6ome eigenvalues and eigenvectors of terraced matrices can be read off from rowmaxima directly. In particular, we easily obtain the smallest eigenvalue in case ofa nonnegative terraced matrix. Theorem 3.2.
Let C = ( c vw ) be a real symmetric matrix whose off-diagonalentries satisfy the Gomory-Hu triangle inequality and let x, y ∈ V with x = y .Then the value c xy is maximal among the off-diagonal elements in its row andcolumn and c xx = c yy if and only if e x − e y is an eigenvector of C with correspondingeigenvalue c xx − c xy . Proof.
First, let x, y ∈ V with x = y satisfy c xx = c yy and let c xy = c yx be amaximum off-diagonal entry in its row and column. As C satisfies the Gomory-Hutriangle inequality, we obtain for each z ∈ V with z = x and z = y that c xz ≥ min { c xy , c yz } = c yz ≥ min { c yx , c xz } = c xz . This implies that c xz = c yz . As a consequence, (cid:16) C ( e x − e y ) (cid:17) z = z = x and z = y,c xx − c xy if z = x,c yx − c yy = c xy − c xx if z = y and thus e x − e y is an eigenvector for c xx − c xy .Conversely, let e x − e y be an eigenvector of C with corresponding eigenvalue c xx − c xy for some x, y ∈ V . Then the eigenequation ( C ( e x − e y )) z = c zx − c zy = 0 for z ∈ V with z = x and z = y implies that c zx = c zy . Because C is symmetric itfollows that c xz = c zy = min { c xz , c zy } ≤ c xy by the Gomory-Hu triangle inequality.Hence, c xy is the maximum off-diagonal entry in its row and column. Furthermore,as c xx − c xy is an eigenvalue of C with eigenvector e x − e y , the eigenequation( C ( e x − e y )) y = c yx − c yy = c xy − c xx implies that c xx = c yy , as desired. Proof of Item (ii) of Theorem 1.1.
Let C = ( c vw ) be the edge-connectivity matrixof a weighted graph. Then C is nonnegative with only zeros on its diagonal andits off-diagonal entries satisfy the Gomory-Hu triangle inequality by Theorem 2.1.Elements x, y ∈ V ( G ) with c xy = M := max { c vw : v, w ∈ V } satisfy the conditionsof Theorem 3.2 and so c xx − c xy = − M is an eigenvalue of C . Moreover, byTheorem 2.3, M I + C is a nonnegative terraced matrix all of whose eigenvalues arenonnegative according to Theorem 3.1. So − M is indeed the smallest eigenvalueof C .The energy bound (iii) of Theorem 1.1 follows directly from the following moregeneral statement. 7 heorem 3.3. Let C be a real nonnegative symmetric matrix whose off-diagonalentries satisfy the Gomory-Hu triangle inequality with n rows and columns andonly zeros on its diagonal. Then the energy of C is at most 2( n − M , where M := max { c vw : v, w ∈ V } . Furthermore, M ( J − I ) is the only such matrix thatattains this bound. Proof.
Because the trace of C is zero, the energy of C is twice the sum of theabsolute values of the negative eigenvalues. This sum is at most ( n −
1) times theabsolute value of the smallest eigenvalue which by Theorem 3.2 is − M .It is an easy computation that the matrix M ( J − I ) attains the bound. Further-more, it is a consequence of the Perron-Frobenius theory that any other matrixwhose entries are bounded from above by M has a largest eigenvalue strictly lessthan ( n − M . So M ( J − I ) is indeed the only matrix which attains the bound.In terms of graph theory, the preceding energy bound is attained if and only if weare given a uniformly M -edge-connected graph. In the remainder of this section,we refine our results on the eigenvector structure of edge-connectivity matrices andachieve a lower bound for the energy. Theorem 3.4.
Let C = ( c xy ) be a symmetric V × V matrix whose off-diagonalentries satisfy the Gomory-Hu triangle inequality and denote the maximum off-diagonal entry in row v ∈ V by m ( v ) := max { c vw : w ∈ V \ { v }} . Then thefollowing statements hold.(i) The matrix C induces an equivalence relation on V by x ∼ y : ⇔ x = y or m ( x ) = m ( y ) = c xy . (ii) If x ∼ x and y ∼ y , then c x y = c x y or, equivalently, an entry c xy depends only on the equivalence classes of x and y .(iii) Let X , . . . , X k be the equivalence classes with respect to the relation ∼ andassume further that the diagonal entries satisfy c xx = c yy whenever x ∼ y .Then the restrictions C | X i × X j of C to X i × X j induce an equitable partitionof C with equitable quotient matrix Q = ( q ij ) i,j =1 ,...,k , where q ij = c xy | X j | if i = j, where x ∈ X i and y ∈ X j ,c xx + m ( x )( | X j | −
1) if i = j, where x ∈ X j . C is nonnegative and has only zeros on itsdiagonal. Denote further by m ( X i ) the common value m ( x ) for x ∈ X i .Then E ( C ) = E ( Q ) + trace( Q ) ≥ k X i =1 ( | X i | − m ( X i ) . This inequality is an equality if and only if the equitable quotient matrix Q has no negative eigenvalues. Proof.
We begin with Item (i). The relation ∼ is reflexive by definition. It issymmetric because C is. As for transitivity assume that x ∼ y and y ∼ z . Hence, m ( x ) = c xy = m ( y ) = c yz = m ( z ). Thus we obtain by applying the Gomory-Hu triangle inequality that c xy = min { c xy , c yz } ≤ c xz ≤ m ( x ) = c xy and hence c xz = c xy = m ( x ) = m ( z ), as desired.For Item (ii), let x ∼ x and y ∼ y . This means that c x x is the maximum entryin rows x and x . Likewise, c y y is the maximum entry in rows y and y . Byapplying the Gomory-Hu triangle inequality and the symmetry of C repeatedly,we obtain that c x y ≥ min { c x y , c y y } = c x y ≥ min { c x x , c x y } = c x y ≥ min { c x y , c y y } = c x y ≥ min { c x x , c x y } = c x y . We observe that all inequalities in this chain are indeed equalities and consequently c x y = c x y , as desired.For Item (iii), note that Item (ii) provides us with the fact that the submatrices C | X i × X j are constant for i, j ∈ V with i = j . This implies in particular that theirrow sums are constant. Similarly, for i ∈ V all off-diagonal entries of C | X i × X i havethe same value by the definition of the relation ∼ and thus all diagonal entrieshave the same value by assumption. The stated formula for the entries q ij is adirect consequence.We finally turn to Item (iv). Let X = { x , . . . , x s } be an equivalence class withrespect to the relation ∼ . Then we obtain s − e x − e x i for i = 2 , . . . , s with a corresponding eigenvalue − m ( X ) ≤
0. Thiscontributes − ( s − m ( X ) to the sum of the negative eigenvalues. Therefore,summation over all classes, contributes k X i =1 (cid:16) ( | X i | − m ( X i ) (cid:17) = trace( Q )to E ( C ). From the partition of V into equivalence classes X , . . . , X k , we obtainin total | V | − k nonpositive eigenvalues, when counting multiplicities. The corre-sponding linearly independent eigenvectors are of the form e x − e y where x ∼ y .9he remaining k eigenvectors in an eigenbasis can be chosen orthogonal on theaforementioned vectors. This means, they can be chosen constant on the classes X i or, equivalently, of the form Z = k X i =1 z i X i for appropriate z i ∈ R . The corresponding eigenequation CZ = λZ is equivalent to Qz = λz , where z = ( z , . . . , z k ) ⊤ . This shows that the remaining k eigenvalues of C , in particularthe positive ones, are among the eigenvalues of Q . This provides us with therelation E ( C ) = E ( Q ) + trace( Q ) which in turn implies the bound to be shown,as clearly E ( Q ) ≥ trace( Q ) with equality if and only if all eigenvalues of Q arenonnegative. Corollary 3.5.
The equitable quotient matrix Q from the previous theorem issimilar to the symmetric matrix Q ′ = ( q ′ ij ) with q ′ ij = c xy ( | X j || X i | ) / if i = j, where x ∈ X i and y ∈ X j ,c xx + m ( x )( | X j | −
1) if i = j, where x ∈ X j . If Q ′ is positive semidefinite, the inequality in the previous theorem is an equality. Proof.
We have Q ′ = W QW − with W = diag( | X i | / : i = 1 , . . . , k ) . Example.
The edge-connectivity matrix of the graph G in Figure 1 is C ( G ) = . The partition V = X ∪ X with X = { , , , } and X = { , } yields theequitable quotient matrix Q = "
12 612 3 . From X we obtain three eigenvalues − e − e , e − e , e − e and X provides us with an eigenvalue − e − e . A lower bound on the energy is therefore2 k X i =1 ( | X i | − m ( X i ) = 2((4 − − . Q has anothernegative eigenvalue, as its determinant is −
36. However, the achieved value is quiteclose to the actual energy of 15 + 3 √ ≈ . The structure of distance matrices of graphs is an active research topic. See Aouch-iche and Hansen [2] for an overview or Stevanović and Indulal [15] for results onthe distance energy. Most of the studies in that field involve shortest paths dis-tances, whereas our results shed light on the spectral properties of ultrametric dis-tance matrices. To see that, consider a graph G and its edge-connectivity matrix C = ( c vw ) and define for two vertices v, w ∈ V ( G ) the distance d ( v, w ) := ( c vw ) − .Then the strong triangle inequality d ( x, z ) ≤ max { d ( x, y ) , d ( y, z ) } holds for all x, y, z ∈ V ( G ). See also Gurvich [6] for how this metric is some kind of resistancedistance. Many of the results of the previous section have analogues in this setting.Indeed, if the entries of a matrix D arise from an ultrametric, then − D satisfiesthe Gomory-Hu triangle inequality and all of our previous results that do not re-quire a nonnegativity assumption hold for − D as well. We state those results forcompleteness, occasionally with a slightly different wording. Theorem 4.1.
Let d : V × V → R be an ultrametric and denote the correspondingdistance matrix by D = ( d ( x, y )). This in particular requires d ( x, x ) = 0 for all x ∈ V . Furthermore, denote by r ( x ) = min { d ( x, y ) : y ∈ V \ { x }} the distance of x to a nearest point. Then the following statements hold:(i) The points x, y ∈ V with x = y are mutually nearest points (this means that d ( x, y ) ≤ d ( x, z ) and d ( x, y ) ≤ d ( z, y ) for each z ∈ V \ { x, y } ), if and only if e x − e y is an eigenvector of D with corresponding eigenvalue − d ( x, y ).(ii) The matrix D induces an equivalence relation on V by x ∼ y : ⇔ x = y or r ( x ) = r ( y ) = d ( x, y ) . (iii) If x ∼ x and y ∼ y , then d ( x , y ) = d ( x , y ) or, equivalently, d ( x, y )depends only on the equivalence classes of x and y , respectively.(iv) Let X , . . . , X k be the equivalence classes with respect to the relation ∼ anddenote by r ( X i ) the common value r ( x ) of x ∈ X i . Then the restrictions D | X i × X j of D to X i × X j induce an equitable partition of D with equitablequotient matrix Q = ( q ij ) i,j =1 ,...,k , where q ij = d ( x, y ) | X j | if i = j, where x ∈ X i and y ∈ X j ,r ( X j )( | X j | −
1) if i = j. D and its equitable quotient matrix Q are related by E ( D ) = E ( Q ) + trace( Q ) ≥ k X i =1 ( | X i | − r ( X i ) . We also have a lower bound on the smallest eigenvalue of an ultrametric distancematrix, which essentially relies on the following result of Zhan [17, Therorem 1]for general symmetric interval matrices.
Theorem 4.2.
Let X be a real symmetric V × V matrix with entries in aninterval [ m, M ]. Denote by λ n ( X ) the smallest eigenvalue of X and consider theproblem to minimize λ n ( X ) such that mJ ≤ X ≤ M J, (P1)where the inequalities are meant componentwise. Then the only optimal solutionof problem (P1) up to simultaneous permutations of rows and columns is X = X ∗ := " mJ V × V M J V × V M J V × V mJ V × V , where V ∪ V = V is a bipartition of V with || V | − | V || ≤
1. The optimal valueis λ n ( X ∗ ) = n ( m − M ) / n is even, (cid:16) nm − q m + ( n − M (cid:17) / n is odd. Theorem 4.3.
Let d : V × V → R be an ultrametric and let D = ( d ( x, y )) be thecorresponding distance matrix. Denote m := min { d ( x, y ) : x, y ∈ V with x = y } , M := max { d ( x, y ) : x, y ∈ V with x = y } , and n := | V | . Then the smallesteigenvalue λ n ( D ) of D satisfies λ n ( X ∗ ) ≥ n ( m − M ) / − m if n is even, (cid:16) nm − q m + ( n − M (cid:17) / − m if n is odd.This bound is attained if and only if there is a bipartition V = V ∪ V of V with || V | − | V || ≤ x, y ) with x = y is d ( x, y ) = m if ( x, y ) ∈ V × V or ( x, y ) ∈ V × V ,M if ( x, y ) ∈ V × V or ( x, y ) ∈ V × V . Equivalently, the stated bound is attained if and only if D = X ∗ − mI , where X ∗ is defined as in Theorem 4.2. 12 roof. We recall first that if d is an ultrametric, then d ( x, x ) = 0 for all x ∈ V .Consequently, D has only zeros on its diagonal. Thus the optimal value of theproblem to minimize λ n ( X ) such that m ( J − I ) ≤ X ≤ M ( J − I ) (P2)is certainly a lower bound for the smallest eigenvalue of an ultrametric matrix.Now, if X is an optimal solution of problem (P2), then X + mI is a feasible solutionof problem (P1) and thus opt(P2) + m ≥ opt(P1). Conversely, by Theorem 4.2, X ∗ + mI is an optimal solution of problem (P1) and X ∗ is feasible for problem (P2).Consequently, opt(P1) − m ≥ opt(P2), as desired. The stated bound on thesmallest eigenvalue follows directly from Theorem 4.2 and the fact that the uniqueoptimal solution to (P2) is an ultrametric matrix. Motivated by the recent interest in the spectra of vertex-connectivity matrices, wefound even stronger structural properties for the edge-connectivity matrix. Notethat both problems, the issue of whether the energy of a vertex-connectivity matrixof a graph on n vertices is bounded from above by 2( n − as well as the questionfor a tight lower bound on its smallest eigenvalue, are still open.Furthermore, we obtained that for edge-connectivity matrices the upper bound( n − on the largest eigenvalue as well as the lower bound − ( n −
1) on thesmallest eigenvalue are attained simultaneously by the matrix that arises from thecomplete graph. This provides us with a tight upper bound ( n − n −
2) on the spread of such matrices, which is defined as the largest distance between any twoeigenvalues of a matrix. This resolves a special case of an intriguing open problemstated by Zhan [17, Problem 2] for which Fallat and Xing [4] formulated a detailedconjecture. There, the question for the spread is stated for general symmetricinterval matrices. Note, however, that our result on the spread is bound to thevery rich structure of edge-connectivity matrices. In different settings, one mightnot expect to find a matrix that attains an upper bound on the largest eigenvalueand a lower bound on the smallest eigenvalue simultaneously.
Acknowledgments
We greatly thank Dragan Stevanović for drawing our attention to the questionsabout vertex-connectivity matrices. This inspired our work on edge-connectivitymatrices. Our research was partially funded by the Deutsche Forschungsgemein-schaft (DFG, German Research Foundation) – Project-ID 416228727 – SFB 1410.13 eferences [1] Saieed Akbari, Seyran Azizi, Modjtaba Ghorbani, and Xueliang Li. On edge-path eigenvalues of graphs.
Linear and Multilinear Algebra , pages 1–11, 2020.[2] Mustapha Aouchiche and Pierre Hansen. Distance spectra of graphs: A sur-vey.
Linear Algebra and its Applications , 458:301–386, 2014.[3] Reinhard Diestel.
Graph Theory . Springer, 2017.[4] Shaun M. Fallat and YongJun Xing. On the spread of certain normal matrices.
Linear and Multilinear Algebra , 60(11-12):1391–1407, 2012.[5] Ralph E. Gomory and Tien Chung Hu. Multi-terminal network flows.
SIAMJournal , 9(4):551–570, 1961.[6] Vladimir Gurvich. Metric and ultrametric spaces of resistances.
DiscreteApplied Mathematics , 158(14):1496–1505, 2010.[7] Frank Göring, Tobias Hofmann, and Manuel Streicher. Uniformly connectedgraphs.
Preprint , 2021.[8] Roger A. Horn and Charles R. Johnson.
Matrix Analysis . Cambridge Univer-sity Press, 1990.[9] Aleksandar Ilić and Milan Bašić. Path matrix and path energy of graphs.
Applied Mathematics and Computation , 355:537–541, 2019.[10] Jack H. Koolen and Vincent Moulton. Maximal energy graphs.
Advances inApplied Mathematics , 26(1):47–52, 2001.[11] Xueliang Li, Yongtang Shi, and Ivan Gutman.
Graph Energy . Springer, 2012.[12] Karl Menger. Zur allgemeinen Kurventheorie.
Fundamenta Mathematicae ,10(1):96–115, 1927.[13] Prashant P. Patekar and Maruti M. Shikare. On the path cospectral graphsand path signless Laplacian matrix of graphs.
Journal of Mathematical andComputational Science , 10(4):922–935, 2020.[14] Maruti M. Shikare, Prashant P. Malavadkar, Shridhar C. Patekar, and IvanGutman. On path eigenvalues and path energy of graphs.
MATCH Commu-nications in Mathematical and in Computer Chemistry , 79:387–398, 2018.[15] Dragan Stevanović and Gopalapillai Indulal. The distance spectrum and en-ergy of the compositions of regular graphs.
Applied Mathematics Letters ,22(7):1136–1140, 2009. 1416] Lihua You, Man Yang, Wasin So, and Weige Xi. On the spectrum of an equi-table quotient matrix and its application.
Linear Algebra and its Applications ,577:21–40, 2019.[17] Xingzhi Zhan. Extremal eigenvalues of real symmetric matrices with entriesin an interval.