Mirko Lepovic

A Collective Property of Trees and Chemical Trees

The Wiener index (W) and the Hosoya polynomial (H) have been calculated for all trees (and thus for all chemical trees) with 20 and fewer vertices. Corroborating an earlier observation (Razinger, M. et al. J. Chem. Inf. Comput. Sci. 1985, 25, 23−27), we show that the ability of W to distinguish between nonisomorphic n-vertex trees (respectively, chemical trees) depends on n in an alternating manner:  it increases for even values of n and decreases for odd values of n. An analogous behavior is also found in the case of H.

No starlike trees are cospectral

A tree is said to be starlike if exactly one of its vertices has degree greater than two. We show that no two non-isomorphic starlike trees are cospectral.

The maximal exceptional graphs

A graph is said to be exceptional if it is connected, has least eigenvalue greater than or equal to -2, and is not a generalized line graph. Such graphs are known to be representable in the exceptional root system E8. We determine the maximal exceptional graphs by a computer search using the star complement technique, and then show how they can be found by theoretical considerations using a representation of E8 in R8. There are exactly 473 maximal exceptional graphs.

On integral graphs which belong to the class αKa∪βKb,b

Abstract Let G be a simple graph and let G denote its complement. We say that G is integral, if its spectrum consists of integral values. In this work we establish a characterization of integral graphs which belong to the class αK a ∪βK b,b , where mG denotes the m-fold union of the graph G.

On bipartite graphs with small number of laplacian eigenvalues greater than two and three

All connected bipartite graphs with exactly two Laplacian eigenvalues greater than two are determined. Besides, all connected bipartite graphs with exactly one Laplacian eigenvalue greater than three are determined.

On formal products and angle matrices of a graph

Abstract Let G be a simple graph of order n and let P G ∗ (λ)=|λI−A ∗ | denote the Seidel characteristic polynomial, where A ∗ is the Seidel adjacency matrix of G. Let P ∗ (G) be the collection of Seidel characteristic polynomials P G i ∗ (λ) of vertex deleted subgraphs G⧹i (i=1,2,…,n) . If G and H are two switching equivalent graphs, using the Seidel formal product and the Seidel angle matrices, we prove that P ∗ (G)= P ∗ (H) . Further, let PG(λ)=|λI−A| be the characteristic polynomial of the graph G, where A is the adjacency matrix of G. Let S be any subset of the vertex set V(G) and let GS be the graph obtained from the graph G by adding a new vertex x which is adjacent exactly to the vertices from S. In particular, if G is a regular graph of degree r, we prove that P G S (λ)= (−1) n+1 λ+r+1 (λ− r )P G S ( λ )+ (λ+r+1−|S|) 2 λ+r+1 P G ( λ ) , where G S denotes the complement of G S , r =(n−1)−r and λ =−λ−1 . Using the last relation we prove that the polynomial reconstruction conjecture is true for all graphs GS for which G is regular.

On formal products and spectra of graphs

For any non-singular matrix M we denote by M the matrix formed by the algebraic cofactors of order (n − 1) so that {M}T = |M| M−1. Let G be an arbitrary simple graph of order n and let A = [Aij] = {λI − A}, where A is the adjacency matrix of G. Besides, let X, Y be any two subsets of the vertex set V(G) and define 〈X, Y〉 = ∑i∈X ∑j∈Y Aij. The expression 〈X, Y〉 is called the formal product of the sets X and Y associated with the graph G. For any S ⊆ V(G), denote by GS the graph obtained from the graph G by adding a new vertex x which is adjacent exactly to the vertices from S, which is called the overgraph of G. Further, for any adjacency matrix A of G, let Ak = [aij〈k〉]. If S ⊆ V(G) then S(t) = ∑k = 0∞ cktk is called the formal generating function associated with GS, where ck = ∑i∈S ∑j∈S aij(k) (k = 0, 1, 2,…). In this paper, using the formal product and the formal generating functions, some results about cospectral graphs are proved. In particular, for any two overgraphs GS1 and GS2 of G of order (n + 1) we give necessary and sufficient conditions under which GS1 and GS2 are cospectral.

On strongly asymmetric graphs

Abstract Let G be an arbitrary simple graph of order n . G is called strongly asymmetric if all induced overgraphs of G of order ( n + 1) are nonisomorphic. In this paper we give some properties of such graphs and prove that the class L c of all connected strongly asymmetric graphs is infinite.

Graphs with small numbers of independent edges

The graphs with exactly one, two or three independent edges are determined.

Some kinds of energies of graphs

Abstract Torgasev (1986) described all finite connected graphs whose energy (i.e. the sum of all positive eigenvalues including also their multiplicities), does not exceed 3. In this paper, we introduce definitions of some other kinds of energies and we prove some properties of them.