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In this paper, we study the prime order Cayley graph assigned to the group ℤn for different values of n. We specify some of the graph theoretical properties such as chromatic and perfect matching numbers. Furthermore, we determine the adjacency matrices and eigenvalues of the prime order Cayley graph associated with groups ℤn and 𝒟2n.

In this article, we consider the relationships between walks in a signed graph G˙\dot{G} and its eigenvalues, with a particular focus on the largest absolute value of its eigenvalues ρ(G˙)\rho \left(\dot{G}), known as the spectral radius. Among other results, we derive a sequence of lower bounds for ρ(G˙)\rho \left(\dot{G}) expressed in terms of walks or closed walks. We also prove that ρ(G˙)\rho \left(\dot{G}) attains the spectral radius of the underlying graph GG if and only if G˙\dot{G} is switching equivalent to GG or its negation. It is proved that the length kk of the shortest negative cycle in G˙\dot{G} and the number of such cycles are determined by the spectrum of G˙\dot{G} and the spectrum of GG. Finally, a relation between kk and characteristic polynomials of G˙\dot{G} and GG is established.

For a simple graph with vertex set {v1,v2,…,vn}\left\{{v}_{1},{v}_{2},\ldots ,{v}_{n}\right\} and degree sequence dvii=1,2,…,n{d}_{{v}_{i}}\hspace{0.33em}i=1,2,\ldots ,n, the inverse sum indeg matrix (ISI matrix) AISI(G)=(aij){A}_{{\rm{ISI}}}\left(G)=\left({a}_{ij}) of GG is a square matrix of order n,n, where aij=dvidvjdvi+dvj,{a}_{ij}=\frac{{d}_{{v}_{i}}{d}_{{v}_{j}}}{{d}_{{v}_{i}}+{d}_{{v}_{j}}}, if vi{v}_{i} is adjacent to vj{v}_{j} and 0, otherwise. The multiset of eigenvalues τ1≥τ2≥⋯≥τn{\tau }_{1}\ge {\tau }_{2}\hspace{0.33em}\ge \cdots \ge {\tau }_{n} of AISI(G){A}_{{\rm{ISI}}}\left(G) is known as the ISI spectrum of GG. The ISI energy of GG is the sum ∑i=1n∣τi∣\mathop{\sum }\limits_{i=1}^{n}| {\tau }_{i}| of the absolute ISI eigenvalues of G.G. In this article, we give some properties of the ISI eigenvalues of graphs. Also, we obtain the bounds of the ISI eigenvalues and characterize the extremal graphs. Furthermore, we construct pairs of ISI equienergetic graphs for each n≥9n\ge 9.

We consider signed graphs with just 2 or 3 distinct eigenvalues, in particular (i) those with at least one simple eigenvalue, and (ii) those with vertex-deleted subgraphs which themselves have at most 3 distinct eigenvalues. We also construct new examples using weighing matrices and symmetric 3-class association schemes.

In a graph G, if di is the degree of a vertex vi, the geometric-arithmetic matrix GA(G) is a square matrix whose [Formula: see text]-th entry is [Formula: see text] whenever vertices i and j are adjacent and 0 otherwise. The set of all eigenvalues of GA(G) including multiplicities is known as the geometric-arithmetic spectrum of G. The difference between the largest and the smallest geometric-arithmetic eigenvalue is called the geometric-arithmetic spread [Formula: see text] of G. In this article, we investigate some properties of [Formula: see text] We obtain lower and upper bounds of [Formula: see text] and show the existence of graphs for which equality holds. Further, [Formula: see text] is computed for various graph operations.

We investigate signed graphs with just 2 or 3 distinct eigenvalues, mostly in the context of vertex-deleted subgraphs, the join of two signed graphs or association schemes. [ABSTRACT FROM AUTHOR]