1. ON UNICYCLIC REFLEXIVE GRAPHS

2. The spectrum of a simple graph (non-oriented, without loops and multiple edges) is the spectrum of its adjacency matrix, along with the usual assumption The Interlacing theorem: Let be the eigenvalues of a graph G and eigenvalues of its induced subgraph H. Then the inequalities,, hold. Reflexive graphs are graphs having

3. Graph G is a maximal reflexive graph inside a given class of graphs C if G is reflexive and any extension G+v that belongs to C has Theorem (Smith): For a simple graph G (resp. ) if and only if each component of G is an induced subgraph (resp. proper induced subgraph) of one of the graphs of Fig. 1, all of which have

4. Figure 1. Connected graphs that have their largest eigenvalue (the index) equal to 2 are known as Smith graphs

5. Theorem (Schwenk): Given a graph G, let ( ) denote the set of all cycles containing a vertex and an edge of G, respectively. Then (i) (ii) where Adj(v) denotes the set of neighbors of v, while G – V(C) is the graph obtained from G by removing the vertices belonging to the cycle C.

6. Corollary 1. Let G be a graph obtained by joining a vertex of a graph to a vertex of a graph by an edge. Let ( ) be the subgraph of ( ) obtained by deleting the vertex ( ) from (resp. ). Then Corollary 2. Let G be a graph with a pendant edge, being of degree 1. Then where ( ) is the graph obtained from G (resp. ) by deleting the vertex (resp. )

7. Theorem RS Let G be a graph with a cut vertex u. i.If at least two components of G-u are supergraphs of Smith graphs, and if at least one of them is a proper supergraph, then ii.If at least two components of G-u are Smith graphs, and the rest are subgraphs of Smith graphs, then iii.If at most one component of G-u is a Smith graph, and the rest are proper subgraphs of Smith graphs, then

8. Maximum number of loaded vertices of the cycle in unicyclic reflexive graph

9. Theorem 1. The cycle of unicyclic reflexive graph of length greater than 8 cannot have more than 7 loaded vertices. Theorem 2. The cycle of unicyclic reflexive graph of length greater than 10 cannot have more than 6 loaded vertices.

10. The length of the cycle with six loaded vertices Theorem 3. Maximal length of the cycle of unicyclic reflexive graph with 6 loaded vertices is l = 12.

12. The length of the cycle with five loaded vertices Theorem 4. 1.Maximal length of the cycle of unicyclic reflexive graph with 5 loaded vertices, if these vertices are not consecutive, is l = 14. 2. Maximal length of the cycle of unicyclic reflexive graph with 5 consecutive loaded vertices is l = 16.

13. The length of the cycle with four loaded vertices Theorem 5. 1. Maximal length of the cycle of unicyclic reflexive graph with 4 loaded vertices, if there are no consecutive loaded vertices on the cycle, is l = 16. 2. Maximal length of the cycle of unicyclic reflexive graph with 4 loaded vertices, if there are are two (but not three, and not four) consecutive loaded vertices on the cycle is l = 21. 3. Maximal length of the cycle of unicyclic reflexive graph with 4 loaded vertices, if there are are three (but not four) consecutive loaded vertices on the cycle is l = 38. 4. The length of the cycle of unicyclic reflexive graph with 4 consecutive loaded vertices, has no upper bound.

14. Let m, p, n, q be the lengths of the paths (a; b), (c; d), (a; d), (b; c), respectively The length of the cycle of graph G is l = m + p + q + n PG(2) = mpqn-4mpn-4mnq-4pqm- 4pqn+12mn+12mq+12pq+12np+16nq+16mp- 32m - 32n - 32p - 32q

15. m = 1: PG(2) = -3pqn + 8pn + 12nq + 8pq - 20n - 20q - 16p - 32 n mnqPG(2)λ 2 ≤2λ2>2λ2>2l 1.1224p-64p≤16p≥1721 2.1236p-60p≤10p≥1116 3.1248p-56p≤7p≥814 4.12510p-52p≤5p≥613 5.12612p-48p≤4p≥513 6.12714p-44p≤3p≥413 7.12816p-40p≤2p≥313 8.12918p-36p≤2p≥314 9.121020p-32p≤1p≥214 10.121122p-28p≤1p≥215 11.121224p-24p≤1p≥216 12.1335p-44p≤8p≥915 13.1344p-28p≤7p≥815 14.1353p-12p≤4p≥513 15.1362p+4/p≥1/ 16.137p+20/p≥1/ 17.13836/p≥1/

16. m = n = 1: PG(2) = 5pq - 8p - 8q – 52 p≥35: PG(2)= 167q - 332 > 0 p≥11 and q≥3: PG(2)= 47q - 140 > 0 pPG(2)λ 2 ≤2λ2>2λ2>2l 1.22q-68q≤34q≥3538 2.37q-76q≤10q≥1115 3.412q-84q≤7q≥813 4.517q-92q≤5q≥612 5.622q-100q≤4q≥512 6.727q-108q≤4q≥513 7.832q-116q≤3q≥413 8.937q-124q≤3q≥414 9.1042q-132q≤3q≥415

17. m=n=p=1: PG(2) = -3q - 60 < 0

18. The length of the cycle with three loaded vertices Theorem 6. Let G be the unicyclic reflexive graph with exactly three loaded vertices of the cycle, and let m, n and k be the lengths of the paths between its loaded vertices, p= min (m,n,k). 1. If p≥3 then the maximal length of the cycle is 18. 2. If p=2: 2.1. m=n=2, the length of the cycle is not bounded. 2.2. m=2, n≥3, k≥3, maximal length of the cycle is 23. 3. If p=1: 3.1. m=n=1, or m=1, n=2, the length of the cycle is not bounded. 3.2 m=1, n≥3, k≥3, maximal length of the cycle is 40.

19. Let m, n, k be the lengths of the paths (a; b), (b; c), (c; a), respectively The length of the cycle of graph G is l = m + n + k p = min (m,n,k) PG(2) = -mnk + 4mn + 4mk + 4nk - 12m - 12n - 12k

20. p=1: p=3: p=2: p=4: mnkl 13k≤367≤l≤40 14k≤118≤l≤16 15k≤79≤l≤13 16k≤610≤l≤13 mnkl 23k≤188≤l≤23 24k≤109≤l≤16 25k≤710≤l≤14 26k≤611≤l≤14 mnkl 33k≤129≤l≤18 34k≤910≤l≤16 35k≤711≤l≤15 36k≤612≤l≤15 mnkl 44k≤812≤l≤16 45k≤713≤l≤16 46k≤614≤l≤16

21. p=5: 1.m=n=k=5, l=15 2.m=n=5, k=6, l=16 3.m=5, n=k=6, l=17 p=6: m=n=k=6, l=18