Necessary conditions for an undirected graph G to contain a graph H as induced subgraph involving the smallest ordinary or the largest normalized Laplacian eigenvalue of G are presented.

Abstract If ${\mathbf v} \in {\mathbb R}^{V(X)}$ is an eigenvector for eigenvalue $\lambda $ of a graph X and $\alpha automorphism , then ({\mathbf v})$ also . Thus, it rather exceptional vertex-transitive to have multiplicity one. We study cubic graphs with nontrivial simple eigenvalue, discover remarkable connections arc-transitivity, regular maps, number theory.

We give a necessary and sufficient condition for a graph to be bipartite in terms of an eigenvector corresponding to the largest eigenvalue of the adjacency matrix of the graph.

Abstract We offer a new method for proving that the maxima eigenvalue of normalized graph Laplacian with n vertices is at least $$\frac{n+1}{n-1}$$ n + 1 - provided not complete and equality attained if only com...

the energy of a graph is equal to the sum of the absolute values of its eigenvalues. two graphs of the same order are said to be equienergetic if their energies are equal. we point out the following two open problems for equienergetic graphs. (1) although it is known that there are numerous pairs of equienergetic, non-cospectral trees, it is not known how to systematically construct any such pa...

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In this paper, we investigate how the smallest signless Laplacian eigenvalue of a graph behaves when the graph is perturbed by deleting a vertex, subdividing edges or moving edges.

for a simple connected graph $g$ with $n$-vertices having laplacian eigenvalues‎ ‎$mu_1$‎, ‎$mu_2$‎, ‎$dots$‎, ‎$mu_{n-1}$‎, ‎$mu_n=0$‎, ‎and signless laplacian eigenvalues $q_1‎, ‎q_2,dots‎, ‎q_n$‎, ‎the laplacian-energy-like invariant($lel$) and the incidence energy ($ie$) of a graph $g$ are respectively defined as $lel(g)=sum_{i=1}^{n-1}sqrt{mu_i}$ and $ie(g)=sum_{i=1}^{n}sqrt{q_i}$‎. ‎in th...

For a graph $G$, we associate family of real symmetric matrices, $\mathcal{S}(G)$, where for any $M \in \mathcal{S}(G)$, the location nonzero off-diagonal entries $M$ is governed by adjacency structure $G$. The ordered multiplicity Inverse Eigenvalue Problem Graph (IEPG) concerned with finding all attainable lists eigenvalue multiplicities matrices in $\mathcal{S}(G)$. connected graphs order si...