On the spectral radius of block graphs having all their blocks of the same size
OOn the spectral radius of block graphs having all their blocks of the same size
Cristian M. Conde ∗
1, 3 , Ezequiel Dratman † , and Luciano N.Grippo ‡ Instituto de Ciencias, Universidad Nacional de General Sarmiento, Consejo Nacional de Investigaciones Científicas y Tecnicas,Argentina Instituto Argentina de Matemática "Alberto Calderón" - ConsejoNacional de Investigaciones Científicas y Tecnicas, Argentina
Abstract
Let B ( n, q ) be the class of block graphs on n vertices having all its blocksof size q + 1 with q ≥ . In this article we prove that the maximum spectralradius ρ ( G ) , among all graphs G ∈ B ( n, q ) , is reached at a unique graph.We profit from this fact to present an tight upper bound for ρ ( G ) . We alsoprove that if G has at most three pairwise adjacent cut vertices then theminimum ρ ( G ) is attained at a unique graph. Likewise, we present a lowerbound for ρ ( G ) when G ∈ B ( n, q ) . The problem of finding those graphs that maximize or minimize the spectralradius of a connected graph on n vertices, within a given graph class H , haveattracted the attention of many researchers. Usually, this kind of problems aresolved by means of graphs transformations preserving the number of vertices, sothat the resulting graph also belongs to H , and having a monotone behavior re-spect to the spectral radius. We refer to the reader to [12] for more details aboutthis and other techniques. In [9], Lovász and Pelikán prove that the unique graphwith maximum spectral radius among the trees on n vertices is the star K ,n − and the unique graph with minimum spectral radius is the path P n . As far as ∗ [email protected] † [email protected] ‡ [email protected] a r X i v : . [ m a t h . SP ] J u l e know, this article is the first one within this research line. Since adding edgesto a graph increases the spectral radius (see Corollary 1), if H contains completegraphs and paths, then K n maximizes and P n minimizes ρ ( G ) among graphs in H ,meaning that this two graphs have the minimum and maximum spectral radiusamong graphs on n vertices when H is the class of all connected graphs. Con-sequently, several authors have considered the problem when H is a graph classnot containing either paths or complete graphs and defined by certain restrictionof classical graph parameters. Graphs with a given independence number [10, 6],graphs with a given clique number [13] and graphs with given connectivity andedge-connectivity [7]. It is worth mentioning that the foundation stone that givesplace to many late articles in connection with this problem is that of Brualdi andSolheid [3]. About our statement in connection with Lovász an Pelikánresult
For concepts and definitions used in this section we referred the reader to Sec-tion 2.In this article we consider the class B ( n, q ) of block graphs on n verticeshaving all their blocks on q + 1 vertices, for every q ≥ . For results relatedto the adjacency matrix of block graphs we refer to the reader to [2]. Trees areblock graphs with all their blocks on two vertices. In connection with the spectralradius on trees it was obtained the following result. Theorem 1.1. [9] If T is a tree on n vertices, then (cid:0) πn +1 (cid:1) = ρ ( P n ) ≤ ρ ( T ) ≤ ρ ( K ,n − ) = √ n − . In an attempt to generalize Theorem 1.1, we find the unique graph that reachesthe maximum spectral radius in B ( n, q ) and the unique graph that reaches theminimum spectral radius but in B ( n, q ) in the case in which the graph has atmost three pairwise adjacent cut vertices. We also present for the spectral radiusa lower bound and a tight upper bound. Theorem 1.2. If G ∈ B ( n, q ) , then ρ ( G ) ≤ ρ ( S ( n, q )) and S ( n, q ) is the uniquegraphs that maximizes the spectral radius. In addition, if G has at most threepairwise adjacent cut vertices then ρ ( P qb ) ≤ ρ ( G ) and P qb is the unique graph thatminimizes the spectral radius in the class B ( n, q ) , where b = n − q . We have strongly evidence obtained by the aid of Sage software that thehypothesis of having at most three pairwise adjacent cut vertices, in connectionwith the minimum of the spectral radius, can be dropped.
Organization of the article
This article is organized as follows. In Section 2 we present some definitions andpreliminary results. In Section 3 are presented two graph transformations having2 monotone behavior respect to the spectral radius. Section 4 is devoted to putall previous result together in order to prove our main result. In Section 5 a tightupper bound and a lower bound for the spectral radius are presented. Finally,Section 6 contains a short summary of our work and two conjectures are posted.
All graphs, mentioned in this article, are finite, have no loops and multiple edges.Let G be a graph. We use V ( G ) and E ( G ) to denote the set of vertices and theset of edges of G , respectively. A graph on one vertex is called trivial graph . Let v be a vertex of G , N G ( v ) (resp. N G [ v ] ) stands for the neighborhood of v (resp. N G ( v ) ∪ { v } ), if the context is clear the subscript G is omitted. We use d G ( v ) to denote the degree of v in G , or d ( v ) provided the context is clear. By G wedenote the complement graph of G . Given a set F of edges of G (resp. of G ), wedenote by G − F (resp. G + F ) the graph obtained from G by removing (resp.adding) all the edges in F . If F = { e } , we use G − e (resp. G + e ) for short.Let X ⊆ V ( G ) , we use G [ X ] to denote the graph induced by X . By G − X wedenote the graph G [ V ( G ) \ X ] . If X = { v } , we use G − v for short. Let G and H two graphs, we use G + H to denote the disjoint union between G and H ,and G + stands for the graph obtaining by adding an isolated vertex to G . Let A, B ⊆ V ( G ) we said that A is complete to (resp. anticomplete to ) B if everyvertex in A is adjacent (resp. nonadjacent) to every vertex of B . We denote by P n and K n to the path and the complete graph on n vertices.We denote by A ( G ) the adjacency matrix of G , and ρ ( G ) stands for thespectral radius of A ( G ) , we refer to ρ ( G ) as the spectral radius of G . If x is theprincipal eigenvector of A ( G ) , which is indexed by V ( G ) , we use x u to denotethe coordinate of x corresponding to the vertex u . We use P G ( x ) to denote thecharacteristic polynomial of A ( G ) ; i.e., P G ( x ) = det( xI n − A ( G )) . It is easy toprove that P K n ( x ) = ( x − n + 1)( x + 1) n − .A vertex v of a graph G is a cut vertex if G − v has a number of connectedcomponents greater than the number of connected components of G . Let H bea graph. A block of H , also known as -connected component , is a maximalconnected subgraph of H having no cut vertex. A block graph is a connectedgraph whose blocks are complete graphs. We use B ( n, q ) to denote the familyof block graphs on n vertices whose blocks have q + 1 vertices. Notice that if B ∈ B ( n, q ) and b is its number of blocks then b = n − q . Let G be a block graph,a leaf block is a block of G such that contains exactly one cut vertex of G . Weuse S ( n, q ) to denote the block graphs in B ( n, q ) having b blocks with only onecut vertex. By P qb we denote the block graph in B ( bq + 1 , q ) with at most two leafblocks when n − > q and no cut vertices when n − q , called ( q, b ) -path-block.3 xy h u vw y u vwx v g,h G H G · H Figure 1: The coalescence of graphs G and H at vertices g and h . This subsection is split into two parts. In the first one we present the resultsneeded to deal with the minimum spectral radius in B ( n, q ) , and in the second onewe developed the tools used to prove the result in connection with the maximumspectral radius in this class. Tools for finding the minimum
We will introduce a partial order on the class of graphs. We will use it to dealwith the graph transformations used to prove our main result. This techniquewas pioneered by Lovász and Pelikán [9].
Definition 1.
Let G and H be two graphs. We denote by G ≺ H , if P H ( x ) >P G ( x ) for all x ≥ ρ ( G ) . It is immediate that if G ≺ H then ρ ( H ) < ρ ( G ) . The spectrum radius isnondecreasing respect to the subgraph partial order.We repeatedly use the following Lemma to deal with the subgraph partialorder previously defined. Lemma 2.1. If H is a proper subgraph of G then ρ ( H ) < ρ ( G ) . The reader is referred to [1] for a proof of the above lemma. In particular,adding edges to a graph increases the spectral radius.
Corollary 1. If G is a graph such that uv / ∈ E ( G ) , then ρ ( G ) < ρ ( G + uv ) . The following technical lemma is a useful tool to develop graph transforma-tions.
Lemma 2.2. [8] If H is a spanning subgraph of the graph G then P G ( x ) ≤ P H ( x ) for all x ≥ ρ ( G ) . In addition, if G is connected then G ≺ H . Let G and H be two graphs. If g ∈ V ( G ) and h ∈ V ( H ) , the coalescence between G and H at g and h , denoted G · hg H , is the graph obtained from G and H , by identifying vertices g and h (see Fig. 1). We use G · H for short. Noticethat any block graph can be constructed by recursively using the coalescenceoperation between a block graph and a complete graph.In the 70s Schwenk published an article containing useful formulas for thecharacteristic polynomial of a graph [11]. The part corresponding to minimizing4he spectral radius of the main result of this research is based on the followingSchwenk’s formula, linking the characteristic polynomial of two graphs and thecoalescence between them. Lemma 2.3. [11] Let G and H be two graphs. If g ∈ V ( G ) , h ∈ V ( H ) , and F = G · H , then P F ( x ) = P G ( x ) P H − h ( x ) + P G − g ( x ) P H ( x ) − xP G − g ( x ) P H − h ( x ) . More details on Lemmas 2.2 and 2.3 can be found in [5].The following two technical lemmas will play an important role in order toprove the main result of this article.
Lemma 2.4.
Let H be a graph, let v, w ∈ V ( H ) such that H − w ≺ H − v andlet G be a connected graph. If G and G are the graphs obtained by means ofthe coalescence between G and H at u ∈ V ( G ) , and v or w respectively, then G ≺ G .Proof. By Lemma 2.3, the characteristic polynomial of G and G are P G ( x ) = P G − u ( x ) P H ( x ) + ( P G ( x ) − xP G − u ( x )) P H − v ( x ) and P G ( x ) = P G − u ( x ) P H ( x ) + ( P G ( x ) − xP G − u ( x )) P H − w ( x ) respectively, and thus P G ( x ) − P G ( x ) = ( P G ( x ) − xP G − u ( x ))( P H − w ( x ) − P H − v ( x )) . (1)By Lemmas 2.1 and 2.2, G ≺ ( G − u ) + . Therefore, since H − w ≺ H − v , byLemma 2.1 and (1) we have G ≺ G . Lemma 2.5.
Let H , H be two graphs such that either H = H or H ≺ H ,let v i ∈ V ( H i ) for each i = 1 , such that H − v ≺ H − v , and let G be aconnected graph. If G i is the graph obtained by means of the coalescence between G and H i at v ∈ V ( G ) and v i for each i = 1 , , then G ≺ G .Proof. By applying Lemma 2.3 as in Lemma 2.4 we obtain P G ( x ) − P G ( x ) = ( P G ( x ) − xP G − v ( x ))( P H − v ( x ) − P H − v ( x ))+ ( P H ( x ) − P H ( x )) P G − v ( x ) . (2)By Lemmas 2.1 and 2.2, G ≺ ( G − v ) + . Therefore, since either H = H or H ≺ H and H − v ≺ H − v , by (2) and Lemma 2.1 we conclude that G ≺ G . 5 v G vw u x y Figure 2: From left to right H , H and H . Tools for finding the maximum
In the following lemma we consider a set of vertices u , . . . , u (cid:96) of a graph G , where x i stands for x u i , the corresponding coordinate of u i in the principal eigenvector,for every ≤ i ≤ (cid:96) . Proposition 1. [10, 4] Let G be a connected graph and let u , . . . , u k , u k +1 , . . . , u (cid:96) vertices of G such that (cid:80) ki =1 x i ≤ (cid:80) (cid:96)i = k +1 x i , and let W ⊆ V ( G ) \ { u , . . . , u (cid:96) } .If { u , . . . , u k } is complete to W and { u k +1 , . . . , u (cid:96) } is anticomplete to W , then ρ ( G ) < ρ ( G − { wu i : w ∈ W and ≤ i ≤ k } + { wu i : w ∈ W and k + 1 ≤ i ≤ (cid:96) } ) . Proposition 1 has also shown to be of help to find the unique graph with max-imum spectral radius of block graphs with prescribed independence number [4].
To ease the reading of the next proposition we recommend to see Fig. 2.
Proposition 2.
Let G be a connected graph, and let u ∈ V ( G ) such that G − u is connected. Let H be the graph obtained from S ( k ( q −
1) + 1 , k ) by adding forall ≤ i ≤ k one pendant ( q, b i ) -path-block (possible empty, i.e., b i = 0 ) to eachleaf block, let v ∈ V ( H ) be the vertex of degree k ( q − , and let w ∈ V ( H ) be anyvertex in leaf block of H . If H is the graph obtained by the coalescence between G and H at u and v , H is the graph obtained by the coalescence between G and H at u and w , then H ≺ H .Proof. Observe that H − w is connected and H − v is a disconnected spanningsubgraph of H − w . Thus, by Lemma 2.2 H − w ≺ H − v . Therefore, the resultfollows from Lemma 2.4.The following proposition play a central role to prove the main result of thisarticle. We use G ( r, s ) to denote the graph obtained by means of the coalescencebetween G and a copy of K r at u ∈ V ( G ) and any vertex of the complete graph,and between G and K s at v ∈ V ( G ) and any vertex of the complete graph (seethe example depicted in Fig. 3). 6igure 3: From left to right G , G (4 , and G (3 , . Proposition 3.
Let G be a connected graph and let u, v ∈ V ( G ) . If r and s aretwo integers such that ≤ r ≤ s − , G − u ≺ G − v or G − u = G − v , then G ( r, s ) ≺ G ( r + 1 , s − .Proof. By applying Lemma 2.3 to G ( r, s ) at v we obtain P G ( r,s ) ( x ) = ( x + 1) s − [( x − s + 2) P G ( r,s ) − K s − ( x ) − ( s − P G ( r,s ) − K s ( x )] . (3)Applying again Lemma 2.3 to P G ( r,s ) − K s − ( x ) and P G ( r,s ) − K s ( x ) we obtain P G ( r,s ) ( x ) = ( x + 1) s + r − { ( x − s + 2)[( x − r + 2) P G ( x ) − ( r − P G − u ( x )] − ( s − x − r + 2) P G − v ( x ) − ( r − P G −{ u,v } ( x )] } . By symmetry P G ( r +1 ,s − ( x ) = ( x + 1) s + r − { ( x − s + 3)[( x − r + 1) P G ( x ) − rP G − u ( x )] − ( s − x − r + 1) P G − v ( x ) − rP G −{ u,v } ( x )] } . Hence P G ( r +1 ,s − ( x ) − P G ( r,s ) ( x ) = ( x + 1) s + r − { ( s − r − P G ( x ) + P G − u ( x )+ P G − v ( x ) + P G −{ u,v } ( x )] + ( x + 1)( P G − v ( x ) − P G − u ( x )) } . (4)Therefore, if ≤ r ≤ s − , by (4) and Lemma 2.1, G ( r, s ) ≺ G ( r + 1 , s − .A ( q, b ) -path-blocks in B ( n, q ) have blocks B , . . . , B b such that V ( B i ) ∩ V ( B i +1 ) = { v i } for every ≤ i ≤ b − and V ( B i ) ∩ V ( B j ) = ∅ whenever ≤ i < j ≤ n and | i − j | > . Let G be a graph and let v, w ∈ V ( G ) be two adjacent vertices. Weuse G [ q, k, (cid:96) ] to denote the graph obtained by adding a pendant ( q, (cid:96) ) -path-blockat v and a pendant ( q, k ) -path-block at w , where ≤ (cid:96) ≤ k , and G [ q, r, standsfor the graph obtained from G by adding just a ( q, r ) -path block at w . By “pen-dant at v ” we mean identifying a noncut vertex from one of the two leaf blockswith a noncut vertex v ∈ V ( B ) (see Fig. 4). Proposition 4.
Let G ∈ B ( n, q ) with at least one cut vertex. If (cid:96) and k are twopositive integers such that ≤ (cid:96) ≤ k and G [ q, k, (cid:96) ] has at most three adjacent cutvertices, then G [ q, k, (cid:96) ] ≺ G [ q, k + 1 , (cid:96) − .Proof. We use B , . . . , B (cid:96) and B (cid:48) , . . . , B (cid:48) k to denote the blocks of the ( q, (cid:96) ) -path-block, denoted P q(cid:96) , and the ( q, k ) -path-block, denoted P qk , respectively, v = v ,7 B B B (cid:48) vv v v (cid:48) B (cid:48) v v (cid:48) v (cid:48) v (cid:48) B (cid:48) B (cid:48) w B B B (cid:48) vv v v (cid:48) B (cid:48) v v (cid:48) v (cid:48) v (cid:48) B (cid:48) B (cid:48) wG G B (cid:48) v (cid:48) u Figure 4: From left to right G [3 , , and G [3 , , w = v (cid:48) , and v (cid:96) and v (cid:48) k stands for a fixed arbitrary noncut vertex of B (cid:96) and B (cid:48) k respectively. By G l [ q, k, (cid:96) ] (resp. G r [ q, k, (cid:96) ] ) we denote the graph G [ q, k, (cid:96) ] − v (cid:96) (resp. G [ q, k, (cid:96) ] − v (cid:48) k ). When both vertices are removed we use G lr [ q, k, (cid:96) ] . Byapplying Lemma 2.3 to G l [ q, k, (cid:96) − at v (cid:96) − is derived the following identity. P G [ q,k,(cid:96) ] ( x ) = ( x + 1) q − (( x − q + 1) P G [ q,k,(cid:96) − ( x ) − qP G l [ q,k,(cid:96) − ( x )) . (5)Analogously P G [ q,k,(cid:96) ] ( x ) = ( x + 1) q − (( x − q + 1) P G [ q,k − ,(cid:96) ] ( x ) − qP G r [ q,k − ,(cid:96) ] ( x )) . (6)By combining (5) and (6) we obtain P G [ q,k +1 ,(cid:96) − ( x ) − P G [ q,k,(cid:96) ] ( x ) = q ( x + 1) q − ( P G l [ q,k,(cid:96) − ( x ) − P G r [ q,k,(cid:96) − ( x ))) . (7)Again, by using properly Lemma 2.3, we derive the next identity P G l [ q,k,(cid:96) ] ( x ) = ( x + 1) q − { ( x − q + 1)[( x − q + 2) P G [ q,k − ,(cid:96) − ( x ) − ( q − P G l [ q,k − ,(cid:96) − ( x )] + q (( q − P G lr [ q,k − ,(cid:96) − ( x ) − ( x − q + 2) P G r [ q,k − ,(cid:96) − ( x )) } . Analogously, P G r [ q,k,(cid:96) ] ( x ) = ( x + 1) q − { ( x − q + 1)[( x − q + 2) P G [ q,k − ,(cid:96) − ( x ) − ( q − P G r [ q,k − ,(cid:96) − ( x )] + q (( q − P G lr [ q,k − ,(cid:96) − ( x ) − ( x − q + 2) P G l [ q,k − ,(cid:96) − ( x )) } . Hence P G l [ q,k,(cid:96) ] ( x ) − P G r [ q,k,(cid:96) ] ( x ) = ( x + 1) q − ( P G (cid:96) [ q,k − ,(cid:96) − ( x ) − P G r [ q,k − ,(cid:96) − ( x )) . (8)By applying (8) repeatedly we obtain for every ≤ j ≤ (cid:96)P G l [ q,k,(cid:96) ] ( x ) − P G r [ q,k,(cid:96) ] ( x ) = ( x + 1) j ( q − ( P G l [ q,k − j,(cid:96) − j ] ( x ) − P G r [ q,k − j,(cid:96) − j ] ( x )) . (9)8 (cid:48) v v (cid:48) B (cid:48) v (cid:48) w v v (cid:48) B (cid:48) wG − v G Figure 5: From left to right ( G − v )[3 , , and G r [3 , , .Replacing in (7) we obtain that for every ≤ j ≤ (cid:96) − P G [ q,k +1 ,(cid:96) − ( x ) − P G [ q,k,(cid:96) ] ( x ) = q ( x + 1) (2 j +1)( q − ( P G l [ q,k − j,(cid:96) − j − ( x ) − P G r [ q,k − j,(cid:96) − j − ( x )) . (10)In particular, if j = (cid:96) − P G [ q,k +1 ,(cid:96) − ( x ) − P G [ q,k,(cid:96) ] ( x ) = q ( x + 1) (2 (cid:96) − q − ( P ( G − v )[ q,k − (cid:96) +1 , ( x ) − P G r [ q,k − (cid:96) +1 , ( x )) . (11)Thus it suffices to prove that G r [ q, t, ≺ ( G − v )[ q, t, for all t ≥ (see Fig. 5).First observe that, since G one cut vertex, there exists a graph H ∈ B ( n, q ) suchthat G r [ q, t, is the coalescence between H and P qt +1 − v t at a noncut vertex x ∈ V ( H ) and a noncut vertex v ∈ V ( B ) of P qt +1 − v t respectively. Analogously, ( G − v )[ q, t, is the coalescence between H and P qt +1 − v at a noncut vertex x ∈ V ( H ) and a noncut vertex y ∈ V ( B ) \ { v } of P qt +1 − v respectively. Fromthis observation combined with Lemma 2.5 and Proposition 3 we conclude that G r [ q, t, ≺ ( G − v )[ q, t, . Let G ∈ B ( n, q ) and let B be a block of G . We say that B is a special blockof type one if B has at least two pendant path-blocks at v ∈ V ( B ) (see Fig. 2,the block whose vertex set is { v, x, y } is a special block of type one). We saythat B is a special of type two if B has a pendant path-block at v ∈ V ( B ) and apendant path-block at w ∈ V ( B ) with v (cid:54) = w (see Fig. 4, the block whose vertexset is { u, v, w } is a special block of type two). The below lemma, whose proof isomitted, will be used to prove our main result. Lemma 4.1. If G ∈ B ( n, q ) , then G either is a ( q, b ) -path-block, or has a specialblock of type one, or has a special block of type two. We obtain the graph maximizes the spectral radius within B ( n, q ) by applyingProposition 1. 9 heorem 4.1. If G ∈ B ( n, q ) , then ρ ( G ) ≤ ρ ( S ( n, q )) . Besides, S ( n, q ) is theunique graph maximizing ρ ( G ) .Proof. Suppose towards a contradiction that H ∈ B ( n, q ) maximizes the spectralradius within B ( n, q ) and it is not S ( n, q ) . Hence H has two leaf blocks B and B (cid:48) such that V ( B ) ∩ V ( B (cid:48) ) = ∅ . Assume that v and v (cid:48) are their correspondingcut vertices. Hence, x v ≥ x v (cid:48) or x v ≤ x v (cid:48) , say x v ≥ x v (cid:48) . Let F be the set of edges v (cid:48) u with u ∈ V ( B (cid:48) ) \ { v (cid:48) } . By Proposition 1, ρ ( H ) < ρ (( H − F ) ∪ { uv : u ∈ V ( B (cid:48) ) } ) , we reach a contradiction that arose from supposing that ρ ( H ) is the maximumamong all possible ρ ( G ) with G ∈ B ( n, q ) . Therefore, if G ∈ B ( n, q ) \ { S ( n, q ) } ,then ρ ( G ) < ρ ( S ( n, q )) .Now we are ready to put all pieces together in order to prove the main resultof the article. Proof of Theorem 1.2.
The upper bound follows from Theorem 4.1. Assume that G ∈ B ( n, q ) and it is not a path-block. By Lemma 4.1, G has either a specialblock of type one or a special block of type two. Hence, by Propositions 2 and 4,there exists a graph transformation onto G , involving the corresponding pendant-path-blocks, such that the resulting graph G (cid:48) satisfies ρ ( G (cid:48) ) < ρ ( G ) and G (cid:48) hasan special block less than G . Therefore, continuing with this procedure as longas G (cid:48) is a path-block, we conclude that ρ (cid:18) P q n − q (cid:19) < ρ ( G ) for all G ∈ B ( n, q ) . Theorem 5.1.
Let G ∈ B ( n, q ) and let n − > q . Then ρ ( G ) ≤ q − √ ( q − +4( n − , ρ ( G ) ≥ q + √ q for every ≤ q ≤ , and ρ ( G ) ≥ q + 4 + ( q − √ q + 3 √ , for every q ≥ . Besides, the upper bound holds if and only if G = S ( n, q ) .Proof. Since n − > q , the number of blocks b of G is at least two. By Theo-rem 1.2, S ( n, q ) is the graph in B ( n, q ) having maximum expectral radius. Let uscall B i for each i = 1 , . . . , b to the blocks of S ( n, q ) and v to its only cut vertex.By symmetry we can assume that all coordinates of the principal eigenvectorcorresponding to those vertices in B i \ { v } are equal to x for each ≤ i ≤ b and let y be the coordinate corresponding to v . By Perron-Frobenius theorem we10an assume that x and y are positive real numbers. If ρ is the spectral radius of S ( n, q ) , then ( q − x + y = ρxbq x = ρy. Hence, ρ − ( q − ρ − n + 1 = 0 and consequently ρ = q − √ ( q − +4( n − . ByTheorem 1.2 the equality holds if and only if G = S ( n, q ) .Assume now that q ≥ and b ≥ . By Lemma 2.1, we know that ρ ( P q ) ≤ ρ ( P qb ) ≤ ρ ( G ) for every graph G ∈ B ( n, q ) . By simple calculation, using Lemma 2.3,we obtain the characteristic polynomial of P q P P q ( x ) = ( x + 1) q − (cid:16) ( x − q )( x + 2) + 1 (cid:17)(cid:16) ( x − q ) (cid:16) ( x − q )( x + 2) + 1 (cid:17) − q (cid:17) . (12)Since ( x − q ) (cid:16) ( x − q )( x + 2) + 1 (cid:17) − q = − q when (cid:16) ( x − q )( x + 2) + 1 (cid:17) = 0 , wehave that ρ ( P q ) is the greatest root of f q ( x ) := ( x − q ) (cid:16) ( x − q )( x + 2) + 1 (cid:17) − q .Futhermore, since f q ( x ) is an increasing function on ( q, + ∞ ) and f q ( q ) < , wehave that ρ ( P q ) is the unique root of f q ( x ) on ( q, + ∞ ) .Using the following two facts f q (cid:18) q + √ q (cid:19) = q + 48 √ q + q ( q − ≤ and f q ( q + 1) = 4 − q ≥ , for every ≤ q ≤ , and f q ( q + 1) = 4 − q ≤ and f q ( q + √
2) = 4 + 3 √ ≥ , for every q ≥ , and taking into account that f (cid:48)(cid:48) q ( x ) > for every x ∈ ( q, + ∞ ) we conclude that ρ ( G ) ≥ q + √ q for every ≤ q ≤ , and ρ ( G ) ≥ q + 4 + ( q − √ q + 3 √ , for every q ≥ . We have found the unique graph maximizing the spectral radius among all graphsin B ( n, q ) . We have presented three graphs transformations to deal with the11inimum spectral radius of this class of block graphs, namely Propositions 2, 3and 4, but the last one has very strong hypothesis on the graph G . We donot know if they can be weakened. Nevertheless, we have collected very strongcomputational evidence that drives us to following conjecture. Conjecture 1. If G ∈ B ( n, q ) \ { P qb } , then P qb ≺ G . Consequently, if this statement were true the following weaker conjecturewould be also true.
Conjecture 2. If G ∈ B ( n, q ) \ { P qb } , then ρ ( P qb ) < ρ ( G ) . We believe that in for proving Conjecture 1 knew graph transformations needto be developed.Another interesting graph class, related to that considered in this paper, tostudy the problem of finding the maximum and minimum spectral radius is thatformed by those block graphs on n vertices having exactly b blocks not necessaryall of them with the same size. Acknowledgments
Cristian M. Conde acknowledges partial support from ANPCyT PICT 2017-2522.Ezequiel Dratman and Luciano N. Grippo acknowledge partial support from AN-PCyT PICT 2017-1315.
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