Any H-free graph G is called H-saturated if the addition of any edge e / ∈ E(G) results in H as a subgraph of G. The minimum size of an H-saturated graph on n vertices is denoted by sat(n,H). The edge spectrum for the family of graphs with property P is the set of all sizes of graphs with property P . In this paper, we find the edge spectrum of K4-saturated graphs. We also show that if G is a K...

The paper deals with some spectral properties of (mostly infinite) quantum and combinatorial graphs. Quantum graphs have been intensively studied lately due to their numerous applications to mesoscopic physics, nanotechnology, optics, and other areas. A Schnol type theorem is proven that allows one to detect that a point λ belongs to the spectrum when a generalized eigenfunction with an subexpo...

In this work, the inverse eigenvalue problem is completely solved for a subfamily of clique-path graphs, in particular lollipop graphs and generalized barbell graphs. For matrix A with associated graph G, new technique utilizing strong spectral property introduced, allowing us to construct A′ whose obtained from G by appending clique while arbitrary list eigenvalues added spectrum. Consequently...

the first ($pi_1$) and the second $(pi_2$) multiplicative zagreb indices of a connected graph $g$, with vertex set $v(g)$ and edge set $e(g)$, are defined as $pi_1(g) = prod_{u in v(g)} {d_u}^2$ and $pi_2(g) = prod_{uv in e(g)} {d_u}d_{v}$, respectively, where ${d_u}$ denotes the degree of the vertex $u$. in this paper we present a simple approach to order these indices for connected graphs on ...

The spectrum of a network or graph G = (V,E) with adjacency matrix A, consists of the eigenvalues of the normalized Laplacian L = I − D−1/2AD−1/2. This set of eigenvalues encapsulates many aspects of the structure of the graph, including the extent to which the graph posses community structures at multiple scales. We study the problem of approximating the spectrum λ = (λ1, . . . , λ|V |), 0 ≤ λ...

This note complements results on cycle spectra of planar graphs by investigating the toroidal case. Let k ≥ 3 be an integer and G a cubic graph polyhedrally embedded torus with circumference at least . For = , it follows from Euler's formula that has some (facial) length in [ 6 ] 4 or 5, we show contains whose lies interval 12 ; for > 5 same holds 2 + is best possible all On other hand, any non...

Let $$G_S$$ be a graph with loops obtained from G of order n and at $$S \subseteq V(G).$$ In this paper, we establish neccesary sufficient condition on the bipartititeness connected spectrum $${\textrm{Spec}}(G_S)$$ $${\textrm{Spec}}(G_{V(G)\backslash S})$$ . We also prove that for every $$S\subseteq V(G),$$ $${\mathcal E}(G_S) \ge {\mathcal E}(G)$$ when is bipartite. Moreover, provide an ident...