On bicyclic reflexive graphs

Transcription

1 Discrete Mathematics 308 (2008) On bicycic refexive graphs Zoran Raosavjević, Bojana Mihaiović, Marija Rašajski Schoo of Eectrica Engineering, Buevar Kraja Aeksanra 73, Beogra, Serbia Receive 27 September 2006; accepte 11 Juy 2007 Avaiabe onine 21 Augsust 2007 Abstract A simpe graph is sai to be refexive if the secon argest eigenvaue of a (0, 1)-ajacency matrix oes not excee 2. We use graph moifications invoving Smith trees to construct four casses of maxima bicycic refexive graphs Esevier B.V. A rights reserve. MSC: 05C50 Keywors: Graph theory; Secon argest eigenvaue; Refexive graph; Bicycic graph; Cactus 1. Introuction For a simpe graph G (a non-oriente graph without oops or mutipe eges), having a matrix A as its (0, 1)- ajacency matrix, we efine P G (λ) = et(λi A) to be its characteristic poynomia an enote it simpy by P(λ) if it is cear which graph it is reate to. The roots of P G (λ) are the eigenvaues of G, making up the spectrum of G an, since they a are rea numbers, we can assume they are in non-increasing orer: λ 1 (G) λ 2 (G) λ n (G). The argest eigenvaue λ 1 (G) is aso cae the inex of G. In a connecte graph λ 1 > λ 2 hos, whie in the case of a isconnecte graph we can have λ 1 = λ 2 if these are equa inices of two components. The reation between the spectrum of a graph an the spectra of its inuce subgraphs is estabishe by the interacing theorem: Let λ 1 λ 2 λ n be the eigenvaues of a graph G an μ 1 μ 2 μ m eigenvaues of its inuce subgraph H. Then the inequaities λ n m+i μ i λ i (i = 1,...,m)ho. Thus, e.g. if m = n 1, λ 1 μ 1 λ 2 μ 2....Aso λ 1 > μ 1 if G is connecte. Graphs having λ 2 2 are cae refexive graphs (an aso hyperboic graphs if λ 2 2 λ 1 ). They correspon to certain sets of vectors in the Lorentz space R p,1 an have some appications to the construction an cassification of refexion groups [6]. Thus far, refexive trees have been stuie in [3,5] an bicycic refexive graphs with a brige between the cyces in [11] (see aso [7]). Recenty, various casses of muticycic refexive cacti have been investigate in [4,8 10]. In this paper we continue with the investigations initiate by the artice [11] an extene in the meantime by consieration of some other casses of refexive graphs. A cactus, oratreeike graph, is a graph in which any two cyces have at most one common vertex, i.e. are ege isjoint. E-mai aress: zorangraf@etf.bg.ac.yu (Z. Raosavjević) X/$ - see front matter 2007 Esevier B.V. A rights reserve. oi: /j.isc

2 716 Z. Raosavjević et a. / Discrete Mathematics 308 (2008) Since the property λ 2 2isahereitary one (any inuce subgraph of a refexive graph is refexive itsef), it is natura to present such graphs aways through sets of maxima graphs. Aso, since the spectrum can be extene using an arbitrary number of components, when ooking for refexive graphs we aways assume them to be connecte. In orer to provie the starting base an the toos for getting the resuts of this paper, in Section 2 we give some important genera an auxiiary facts an some parts of recent resuts concerning maxima refexive cacti. The remaining sections are evote to the aim of this artice, i.e. the construction of various casses of bicycic refexive graphs. At some stages the work has been supporte by using the expert system GRAPH [2]. The terminoogy of the theory of graph spectra in this paper foows [1]. 2. Some former, genera an auxiiary resuts Graphs whose inex equas 2 are known as Smith graphs. Lemma 1 (Smith [13], see aso Cvetković [1, p. 79]). λ 1 (G) 2 (resp. λ 1 (G) < 2) if an ony if each component of graph G is a subgraph (resp. proper subgraph) of one of the graphs of Fig. 1, a of which have inex equa to 2. Let us emphasize the simpe, but important fact, that any connecte graph is either an inuce subgraph or an inuce supergraph of some Smith graphs. If we form a tree T by ientifying vertices u 1 an u 2 (u 1 = u 2 = u) of two (roote) trees T 1 an T 2, respectivey (the coaescence T 1 T 2 of T 1 an T 2 ), we may say that T can be spit at its vertex u into T 1 an T 2 (Fig. 2(a)). Of course, spitting at a given vertex is not etermine uniquey if its egree is greater than 2. If we spit a tree T at a its vertices u in a possibe ways, an in each case attach the parts at vertices of spitting u 1 an u 2 to some vertices v 1 an v 2 of a graph G (i.e. ean the parts on G by ientifying u 1 with v 1 an u 2 with v 2, an vice versa), we sha say that in the obtaine famiy of graphs the tree T pours between v 1 an v 2 (Fig. 2(b)). Of course, this proceure incues attachment of the intact tree T, at each vertex, to v 1 an v 2. Pouring of Smith trees pays an important roe in escribing of maxima refexive cacti [8,10]. The foowing formuae give usefu interreations between the characteristic poynomia of a graph an its subgraphs. Lemma 2 (Schwenk [12]). If G 1 an G 2 are two roote graphs with roots u 1 an u 2, then the characteristic poynomia of their coaescence G 1 G 2 is P G1 G 2 (λ) = P G1 (λ) P G2 u 2 (λ) + P G1 u 1 (λ) P G2 (λ) λp G1 u 1 (λ) P G2 u 2 (λ) n n-1 Cn 1 2 n Wn Fig. 1. T 1 T u u T 1 T T 1 G T v v 2 T u Fig. 2.

3 Z. Raosavjević et a. / Discrete Mathematics 308 (2008) k G1 k eges, k>=0 G2 G3 m G4 k G5 m k n m,n>=3, k>=0 Fig. 3. Lemma 3 (Schwenk [12]). Given a graph G, et C(v) (C(uv)) enote the set of a cyces containing a vertex v an an ege uv of it G, respectivey. Then (i) P G (λ) = λp G v (λ) u Aj(v) P G v u(λ) 2 C C(v) P G V(C)(λ), (ii) P G (λ) = P G uv (λ) P G v u (λ) 2 C C(uv) P G V(C)(λ), where Aj(v) enotes the set of neighbours of v, whie G V(C)is the graph obtaine from G by removing the vertices beonging to the cyce C. These reations have the foowing obvious consequences (see, e.g. [1, p. 59]). Coroary 1. Let G be a graph obtaine by joining a vertex v 1 of a graph G 1 to a vertex v 2 of a graph G 2 by an ege. Let G 1 (G 2 ) be the subgraph of G 1(G 2 ) obtaine by eeting the vertex v 1 (v 2 ) from G 1 (resp. G 2 ). Then P G (λ) = P G1 (λ)p G2 (λ) P G 1 (λ)p G 2 (λ). Coroary 2. Let G be a graph with a penant ege v 1 v 2, v 1 being of egree 1. Then P G (λ) = λp G1 (λ) P G2 (λ), where G 1 (G 2 ) is the graph obtaine from G (resp. G 1 ) by eeting the vertex v 1 (resp. v 2 ). A ist of vaues of P G (2) for some sma graphs has prove to be very usefu in searching for maxima refexive graphs. Lemma 4 (Raosavjević an Rašajski [9,10], Raosavjević an Simić [11]). Let G 1,...,G 5 be the graphs ispaye in Fig. 3. Then (i) P G1 (2) = k + 2; (ii) P G2 (2) = 4; (iii) P G3 (2) = km + k + + m + 2; (iv) P G4 (2) = 4(1 k); (v) P G5 (2) = (3k + 2)mn. First inuce supergraphs of Smith graphs have the foowing property. Lemma 5 (Raosavjević an Simić [11]). Let G be a graph obtaine by extening any of Smith graphs by a vertex of arbitrary positive egree. Then P G (2)<0(i.e. λ 2 (G) < 2 < λ 1 (G)). A ot of refexive graphs can be etecte by the next genera theorem.

4 718 Z. Raosavjević et a. / Discrete Mathematics 308 (2008) Q 1 Q 2 T 1 T 2 c 1 c 1 c 1 c 1 c 4 c 4 Fig. 4. c 1 c 4 (>3) Fig. 5. Theorem 1 (Raosavjević an Simić [11]). Let G be a graph with cut-vertex u. (i) If at east two components of G u are inuce supergraphs of Smith graphs, an if at east one of them is a proper supergraph, then λ 2 (G) > 2. (ii) If at east two components of G u are Smith graphs, an the rest are inuce subgraphs of Smith graphs, then λ 2 (G) = 2. (iii) If at most one component of G u is a Smith graph, an the rest are proper inuce subgraphs of Smith graphs, then λ 2 (G) < 2. This theorem can be appie to a wie cass of graphs with a cut-vertex: among others it covers competey a bicycic graphs whose cyces are joine by a path of ength greater than 1. But if it happens that G u has one proper inuce supergraph an the rest of proper inuce subgraphs of Smith graphs, Theorem 1 is not appicabe an such a situation is interesting for further investigation. If a cyces of a cactus have a unique common vertex, they are sai to form a bune. Since now a refexive cactus can have an infinite number of cyces, searching for maxima refexive cacti in this case is much harer than otherwise. That is why in former investigations bunes have been omitte. The first cass of maxima refexive cacti to be foun was that of bicycic graphs with a brige between its cyces. Theorem 1 oes not appy here except ceary in cases when there is a cyce in which a vertices but one are of egree 2. The resut incues an exceptiona case of a tricycic cactus, which appeare naturay by repacing Smith trees with cyces [11]. The next resut concerns the maximum number of cyces. Theorem 2 (Raosavjević an Rašajski [10]). A treeike refexive graph to which Theorem 1 cannot be appie an whose cyces o not form a bune has at most five cyces. The ony such graphs with five cyces, which are a maxima, are the four famiies of graphs in Fig. 4 (a cyces attache at the cut-vertices (the c-vertices) are of arbitrary engths). Starting from these graphs, it has been possibe to etermine a maxima refexive cacti with four cyces uner the same two conitions non-appicabiity of Theorem 1 an no bune [10, partia resuts in 9]. These maxima graphs now contain cyces with ony one vertex of egree >2(c-vertex), as we as those with some aitiona vertices with

5 Z. Raosavjević et a. / Discrete Mathematics 308 (2008) >2. The cyces of the first kin wi be cae free cyces; otherwise we sha say that the corresponing vertex is oae (by a penant ege, a tree, etc.). Now, using these resuts, one can aso recognize four characteristic casses of tricycic refexive cacti to be subjecte to further investigation [4], ispaye in Fig. 5. Accoring to Theorem 2, the first three aow the aition of more cyces, whie the ast one oes not (i.e. aways remains tricycic). Thus far, there are some partia resuts for some of these casses [4,8]. 3. Repacement of free cyces Former resuts on maxima refexive cacti with four an three cyces areay incue some cases of repacing free cyces by Smith trees (see [10, Propositions 1 3]), as we as more interesting situations in which such a substitution has been generaize into pouring of Smith trees between two vertices at which free cyces are attache [8,10]. Let us now consier the maxima refexive cactus of Fig. 6(a), which is the coaescence of a cyce C of ength n an a cactus G. Appying Lemma 2, we fin that P(2) = P C (2) P G v (2) + P C v (2) P G (2) 2P C v (2) P G v (2). If P(2) = 0, it foows from Lemma 4(i) that n(p G (2) 2P G v (2)) = 0. (1) If C is now repace by a Smith tree S (Fig. 6(b)), then we obtain P(2) = P S v (2)(P G (2) 2P G v (2)) = 0. (2) Extensions of a maxima refexive cactus may have P(2)>0 or may have P(2) = 0 (e.g. if λ 2 > 2, but λ 3 = 2). Suppose that an extension of the graph G by a penant ege gives a graph G 1 for which P G1 (2) 2P G1 v(2)>0; this means that P(2)>0 in (1), impying P(2)>0 in (2). Aso, if we exten S to S + by aing one new (non-isoate) vertex an assume P G (2), P G v (2) = 0, then appying Lemma 5 we see that P(2) = P S+ (2) P G v (2)>0, which means that any graph in Fig. 6(b) is maxima, too. If P G (2) = P G v (2) = 0 (which simpy means that the conition λ 2 = 2 has been attaine before the graph has become maxima, i.e. that λ 2 = 2 is being preserve through some steps of the extension), then such cases have to be verifie iniviuay. Some of them have areay been escribe, enabing an immeiate concusion that those graphs are aso maxima [10, Propositions 1 3]. If a graph of Fig. 6(a) is a maxima refexive cactus such that P G (2) 2P G v (2)<0 (i.e. λ 2 < 2), then aso P(2)<0 in (2). Since now attaching a new penant ege to G prouces P G1 (2) 2P G v (2)>0, the same hos when the cyce is repace by a Smith tree, which means that the graphs of Fig. 6(b) cannot be extene at vertices of G. It is cear, however, that there is no guarantee that an extension of S wi give λ 2 > 2. Therefore, if in a maxima refexive cactus with λ 2 < 2 a free cyce is repace by a Smith tree, the new cactus has λ 2 < 2 an nee not necessariy be maxima. These concusions ea to the foowing theorem. Theorem 3. Suppose that a graph of the form shown in Fig. 6(a) is a maxima refexive cactus for which P(2) = 0 an P G (2)<0 an for any extension G 1 forme by attaching to G a penant ege P G1 (2) 2P G1 v(2)>0 hos. If the free cyce C is repace by an arbitrary Smith tree, then the resuting graph is again a maxima refexive cactus. C v G S v G Fig. 6.

6 720 Z. Raosavjević et a. / Discrete Mathematics 308 (2008) S 1 S 2 c1 c 4 S 1 S 2 Fig. 7. S 2 S 1 S 1 S 2 S 3 c1 S 3 m Fig. 8. This can be the base for the construction of various casses of bicycic refexive cacti. Possibiities given by the graphs Q 1 an Q 2 of Fig. 4 have areay been consiere in [4,8 10]. These graphs attain λ 2 = 2 areay at the stage of ony two cyces joine by the path of ength 2. Their tricycic inuce subgraph shown in Fig. 5(a) aows pouring of a pair of Smith trees between an c 4 [4,8] (Fig. 7), an it can be verifie that the repacement of any of the two free cyces (incuing both) by Smith trees oes not change λ 2 = 2. On the other han, if we remove, e.g. an appy Theorem 1 to c 4, we see that none of these Smith trees can be extene, an these graphs are maxima. The same hos for the graphs H 1 H 48 efine in [9,10] whose cyces at are not free. Unike Q 1, an Q 2, the graphs T 1 an T 2 amit the appication of Theorem 3, but since they aso aow pouring of Smith trees [8,10], they wi be consiere in the next section. The resuting graphs of [9] are characterize by the fact that they possess at east one cyce with attachments, but since they aso have free cyces, by appying Theorem 3 they can give rise to a ot of maxima bicycic refexive cacti. Theorem 3 can aso be appie to graphs of Fig. 5() (which cannot be extene by cyces an have at east one free cyce). A further search for maxima tricycic refexive cacti wi at the same time yie corresponing bicycic graphs. 4. Pouring of tripes of Smith trees The cass of refexive cacti with four cyces base on the graphs T 1 an T 2 in Fig. 4, an generate by the pouring of Smith trees between vertices an, has been ientifie in [10]; a these graphs are maxima, except for one characteristic case (spitting of W n into two anaogous parts) which requires attachment at some vertices of the free cyce. The cass of tricycic refexive cacti constructe by pouring of pairs of Smith trees between the same vertices has been escribe in [8]; most of these graphs are maxima, whie two characteristic exceptions become maxima ony by attachment at some vertices of the free cyce in the same way as before. We wi examine now the generaization of these cases the pouring of tripes of Smith trees. Consier the graph B of Fig. 8 an et tripes of Smith trees pour between an (coaescences S i S i, i = 1, 2, 3, are Smith trees). Let us introuce the abes P Si (2) = p i, P S i (2) = p i ; P Si v(2) = Σ i, P S i v(2) = Σ i (i = 1, 2, 3), v Aj S i v Aj S i

7 Z. Raosavjević et a. / Discrete Mathematics 308 (2008) S 1 S 2 S 1 S 2 S 1 S S 3 S 3 S 3 S 1 S 2 S 1 S 2 S 1 S 2 1 (1,2,3>=0) Fig. 9. where Aj c i enotes the set of vertices ajacent to c i (i = 2, 3). Appying Lemma 3(i) to c 1 we fin that P B (2) = m[p 1 p 2 p 3 (2p 1 p 2 p 3 Σ 1 p 2 p 3 p 1 Σ 2 p 3 p 1 p 2 Σ 3 ) + p 1 p 2 p 3 (2p 1p 2 p 3 Σ 1 p 2 p 3 p 1 Σ 2 p 3 p 1 p 2 Σ 3 ) + 2p 1 p 2 p 3 p 1 p 2 p 3 ] = m[p 2 p 3 p 2 p 3 (2p 1p 1 Σ 1p 1 Σ 1 p 1) + p 1 p 3 p 1 p 3,(2p 2 p 2 Σ 2p 2 Σ 2 p 2) + p 1 p 2 p 1 p 2 (2p 3p 3 Σ 3p 3 Σ 3 p 3)]. (3) On the other han, appying the same Lemma to the spitting vertex of a Smith tree, we get 2p i p i Σ ip i Σ i p i = 0 (i = 1, 2, 3), (4) impying P B (2) = 0 in (3). If some of the three coaescences S i S i are proper inuce subgraphs of a Smith tree, the corresponing expression in (4) is positive, impying P B (2)<0, whie in the case of a proper inuce supergraph, because of Lemma 5, P B (2)>0. Thus, we have refexive graphs, which are maxima in the sense that none of them can be extene at any vertex of S i or S i (i = 1, 2, 3). Now, a question arises whether it is possibe to a something at c 1 or some of other vertices of the cyce. If we attach a penant ege to c 1, by appying Coroary 2 we see that the property P(2) = 0 wi be preserve if an ony if P E (2) = 0, where E is the component of B c 1 ifferent from the path of ength m 2. If at east two compete Smith trees are attache at, for exampe, Theorem 1(iii) gives P E (2)<0. If two intact Smith trees S 1 an S 2 are attache at the opposite vertices of the brige, whie the thir one pours, the situation is anaogous to that consiere in [8]. Using Coroary 1 we see that P E (2) = p 1 p 2 (Σ 3 Σ 3 p 3p 3 ) an since for a Smith graphs Σ 3 Σ 3 p 3p 3 < 0 except in the case p 3 = p 3 = Σ 3 = Σ 3 = 4 (spitting of W n into two anaogous parts), we come to the case which can be extene as far as the graph of Fig. 9(a). In an anaogous way one can make sure that the same hos for the remaining two exceptions of Fig. 9(b, c). If an intact Smith tree S 1 is attache at an the remaining two pour, the corresponing expression becomes P E (2) = p 1 (2Σ 2 Σ 3 p 2 p 3 Σ 2p 3 Σ 2 p 3 p 2Σ 3 p 2 Σ 3 + p 2p 3 p 2 p 3 ). (5) If the two pouring Smith trees are spit in such a way that S 2 an S 3 are K 2 (penant eges), we have p i = 2, Σ i = 1, p i = 2 3 Σ i, (i = 2, 3) an P E(2) = 0 in (5) (the exceptiona case of Fig. 9()). Otherwise P E (2)<0 (for a other ways of

8 722 Z. Raosavjević et a. / Discrete Mathematics 308 (2008) S 1 S 1 S 1 S 2 S 2 S 2 S 3 S 3 S 3 Fig. 10. spitting p i 4 3 Σ i an the proof is ientica to that in [8]). The remaining two exceptions (Fig. 9(e, f)) can be obtaine in an anaogous way. Suppose now that none of the six parts of the three Smith trees is empty. The appication of Coroary 1 gives P E (2) = (2p 1 p 2 p 3 Σ 1 p 2 p 3 p 1 Σ 2 p 3 p 1 p 2 Σ 3 )(2p 1 p 2 p 3 Σ 1 p 2 p 3 p 1 Σ 2 p 3 p 1 p 2 Σ 3 ) p 1 p 2 p 3 p 1 p 2 p 3, (6) an now we can etermine whether P E (2)<0orP E (2) = 0. A simpe inspection of the possibiities for spitting Smith trees an corresponing vaues of p i,p i, Σ i, Σ i, supporte by the appication of Lemma 4, shows that a their spittings can be cassifie into six casses. If W n is spit into two anaogous parts, p 1 = p 1 = Σ 1 = Σ 1 = 4. A other spittings aways give one simpe path, say S i. It turns out that Σ i = αp i, Σ i = (2 α)p i, where α = 2 1, 2 3, 4 3, 4 5, 5 6 epening on whether the mentione path is of ength 1, 2, 3, 4, 5, respectivey. Numerica examination of (6) has shown that a the exceptiona cases (P E (2) = 0), can be escribe by referring to those iustrate in Fig. 9. Thus, we can now formuate the concusion. Theorem 4. Let a bicycic graph G consist of a cyce of arbitrary ength an a triange, et them have the common vertex c 1 an et tripes of Smith trees pour between the remaining two vertices an (Fig. 8). Then G is maxima refexive graph, with the foowing exceptions: (1) A compete Smith tree is attache at an another at, whie the thir (pouring) tree is W n, spit as shown in Fig. 9(a c), in which case these three (famiies of) graphs are maxima refexive graphs. (2) A compete Smith tree S 1 is attache at, an each of the remaining two Smith trees is spit into K 2 an S i (i = 1, 2), as shown in Fig. 9( f), in which case these three (famiies of) graphs are maxima refexive graphs. (3) For one of the two coaescences of three parts of three pouring Smith trees, say S 1,S 2,S 3, there exist corresponing parts S 1, S 2, S 3 such that S i an S i (i = 1, 2, 3) have the same vaues p i an Σ i (i.e. beong to the same one of the formery escribe six casses) which, of course, incues the possibiity S i = S i for some i, an such that their anaogous coaescence consists of a compete Smith tree an two aitiona penant eges at c 1 (as in Fig. 9( f)), in which case the three exceptiona (famiies of) graphs are forme in the same way as in former cases. The graphs of Fig. 10 iustrate the escription of case (3) of this theorem. 5. A case of two free cyces The casses of bicycic refexive graphs iscusse above o not amit two free cyces. Therefore, et us now consier a pair of free cyces (of arbitrary engths m an n) with a common vertex c to which we attach trees. Theorem 1 says that we can attach to c infinitey many trees without vioating the property λ 2 2 an in view of the mutitue of possibe

9 Z. Raosavjević et a. / Discrete Mathematics 308 (2008) c m n ( >=0) S S 1 ( >=0) (*) (*) (*) i j k (i,j,k)=(3,4,19), (3,5,1 1), (3,6,8), (*) (3,7,7), (4,4,9), (4,5,6), (*) (5,5,5) i j (i,j)= (17,6), (9,7), (6,8), (*) (5,9), (4,10), (*) (3,13), (2,24) i (i,j)= (2,14), (3,5), (*) j (4,4) Fig. 11. cases we imit ourseves to a narrower cass. Thus, suppose that c is of egree exacty 5 an that we are seeking ony those maxima refexive graphs which cannot be etecte by Theorem 1. Let us enote by the vertex ajacent to c not beonging to the cyces. If we attach to a Smith tree S an an aitiona penant ege, the appication of Coroaries 1 an 2 gives λ 2 = 2. Since a trees are comparabe with Smith trees, this famiy covers a cases with a penant ege at. Next we examine various shapes of trees attache to, starting with Coroary 1 (appie to the ege c) an appying Lemma 4. The resuting set of maxima refexive graphs is ispaye in Fig. 11. The graphs S 1 are inuce subgraphs of Smith trees obtaine by removing a penant ege, which means that the ege c augments them to Smith trees. The cases for which maxima graphs have λ 2 < 2 are marke by asterisk. Theorem 5. A bicycic graph having two free cyces with a common vertex of egree 5, to which Theorem 1 cannot be appie, is refexive if an ony if it is an inuce subgraph of some graph in Fig. 11.

10 724 Z. Raosavjević et a. / Discrete Mathematics 308 (2008) θ-graphs If two cyces of a bicycic graph have a common path, we sha say that they form a θ-graph an the same name wi be use for any bicycic graph with such cyces. If we appy Lemma 3 to the θ-graph of Fig. 12 an make use of Lemma 4, we get P θ (2) = km 2(k + km + m). (7) This expression gives bouns for a refexive θ-graph. Assuming k m we fin that for k = 1, 2 the parameters an m are not imite, whie for k = 3, = 3, 4, 5, 6 an k = = 4 the parameter m can be arbitrary. Otherwise k,, m are boune an k 6. Any of these particuar cases can be investigate by attaching trees to its vertices in orer to fin the corresponing set of maxima refexive graphs. Aso, some initia extensions of the starting graph can cause unimite engths to become boune. For exampe, if k = = 4 an if a penant ege is attache to a vertex on the thir path, then we have m 10. Let us examine here ony one of the bounary cases k = = m = 6, giving P θ (2) = 0 in (7) (Fig. 13(a)). If we remove the three c-vertices, the remaining pair of Smith trees shows that an extension of the starting graph is possibe at none of their vertices. Thus, it remains to test the extension at c-vertices. Theorem 6. An extension of a θ-graph with k = = m = 6 is refexive if an ony if it is an inuce subgraph of some of the four graphs of Fig. 13(b). k m Fig. 12. c 1 (>=0) Fig. 13.

11 Z. Raosavjević et a. / Discrete Mathematics 308 (2008) In some of these resuting graphs one can recognize again pairs of Smith trees (as inicate in the rawings) to make sure that they are maxima. Acknowegements The work on this artice, incuing the reate former resuts of these authors, has been faciitate by the programming package GRAPH [2]. The authors are thankfu to the Serbian Ministry of Science an Environment Protection for the financia support. References [1] D.M. Cvetković, M. Doob, H. Sachs, Spectra of Graphs Theory an Appication, Deutscher Verag er Wissenschaften, Acaemic Press, Berin, New York, 1980 (secon eition 1982; thir eition, Johann Ambrosius Barth Verag, Heieberg, Leipzig, 1995). [2] D. Cvetković, L. Kraus, S. Simić, Discussing graph theory with a computer, Impementation of agorithms. Univ. Beogra Pub. Eektrotehn. Fak. Ser. Mat. Fiz. 716 (734) (1981) [3] G. Maxwe, Hyperboic trees, J. Agebra 54 (1978) [4] B. Mihaiović, Z. Raosavjević, On a cass of tricycic refexive cactuses, Univ. Beogra Pub. Eektrotehn. Fak. Ser. Mat. 16 (2005) [5] A. Neumaier, The secon argest eigenvaue of a tree, Linear Agebra App. 46 (1982) [6] A. Neumaier, J.J. Seie, Discrete hyperboic geometry, Combinatorica 3 (1983) [7] M. Petrović, Z. Raosavjević, Spectray constraine graphs, Facuty of Science, Kragujevac, Serbia, [8] Z. Raosavjević, B. Mihaiović, M. Rašajski, Decomposition of Smith graphs in maxima refexive cacti, Discrete Math. (2007), oi: /j.isc [9] Z. Raosavjević, M. Rašajski, A cass of refexive cactuses with four cyces, Univ. Beogra Pub. Eektrotehn. Fak. Ser. Mat. 14 (2003) [10] Z. Raosavjević, M. Rašajski, Muticycic treeike refexive graphs, Discrete Math. 296 (1) (2005) [11] Z. Raosavjević, S. Simić, Which bicycic graphs are refexive?, Univ. Beogra Pub. Eektroteh. Fak. Ser. Mat. 7 (1996) [12] A.J. Schwenk, Computing the characteristic poynomia of a graph, in: R. Bari, F. Harary (Es.), Graphs an Combinatorics, Lecture Notes in Mathematics, vo. 406, Springer, Berin, Heieberg, New York, 1974, pp [13] J.H. Smith, Some properties of the spectrum of a graph, in: R. Guy, H. Hanani, N. Sauer, J. Schonheim (Es.), Combinatoria Structures an Their Appications, Goron an Breach, Science, New York, Lonon, Paris, 1970, pp