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The rank of adjacency matrix plays an important role in construction of linear codes from a directed strongly regular graph using different techniques, namely, code orthogonality, adjacency matrix determinant and adjacency matrix spectrum. The problem of computing the dimensions of such codes is an intriguing one. Several conjectures to determine the rank of adjacency matrix of a DSRG Γ over a finite field, keep researchers working in this area. To address the same to an extent, we have considered the problem of finding the rank over a finite field of the adjacency matrix of a DSRG Γ (v , k , t , λ , μ) with μ = 1 , including some mixed Moore graphs and corresponding codes arising from them, in this paper. [ABSTRACT FROM AUTHOR]

Let G be a finite simple graph with λ as an eigenvalue (i.e. an eigenvalue of the adjacency matrix of G), and let H be a star complement for λ in G. Motivated by a controllability condition, we say that H is a strong star complement for λ if G and H have no eigenvalue in common. We explore this concept in the context of line graphs, exceptional graphs, strongly regular graphs and graphs with a prescribed star complement. [ABSTRACT FROM AUTHOR]

Let A(G) be the adjacency matrix of a simple connected undirected graph G. A graph G of order n is said to be non-singular (respectively singular) if A(G) is non-singular (respectively singular). The spectrum of a graph G is the set of all its eigenvalues denoted by spec(G). The antireciprocal (respectively reciprocal) eigenvalue property for a graph G can be defined as "Let G be a non-singular graph G if the negative reciprocal (respectively positive reciprocal) of each eigenvalue is likewise an eigenvalue of G, then G has anti-reciprocal (respectively reciprocal) eigenvalue property." Furthermore, a graph G is said to have strong anti-reciprocal eigenvalue property (resp. strong reciprocal eigenvalue property) if the eigenvalues and their negative (resp. positive) reciprocals are of same multiplicities. In this article, graphs satisfying anti-reciprocal eigenvalue (or property (-R)) and strong anti-reciprocal eigenvalue property (or property (-SR)) are discussed. [ABSTRACT FROM AUTHOR]

Let G be a graph, and let $ A(G) $ A (G) be the adjacency matrix of G. The computation of permanent of $ A(G) $ A (G) is #p-complete. Computing permanent of $ A(G) $ A (G) is of great interest in quantum chemistry, statistical physics, among other disciplines. In this paper, we characterize the ordering of permanents of adjacency matrices of all graphs obtained from regular complete bipartite graph $ K_{p, p} $ K p , p by deleting six edges. As an application, we show that all graphs with a perfect matching obtained from $ K_{p, p} $ K p , p with six edges deleted are determined by their permanental polynomials. [ABSTRACT FROM AUTHOR]

In this article, we discuss the spectral properties of the general extended adjacency matrix for chain graphs. In particular, we discuss the eigenvalues of the general extended adjacency matrix of the chain graphs and obtain its general extended adjacency inertia. We obtain bounds for the largest and the smallest general extended adjacency eigenvalues and characterize the extremal graphs. We also obtain a lower bound for the spread of the general extended adjacency matrix. We characterize chain graphs with all the general extended adjacency eigenvalues being simple and chain graphs that are non-singular under the general extended adjacency matrix. Further, we determine the explicit formula for the determinant and the trace of the square of the general extended adjacency matrix of chain graphs. Finally, we discuss the energy of the general extended adjacency matrix and obtain some bounds for it. We characterize the extremal chain graphs attaining these bounds. [ABSTRACT FROM AUTHOR]

Let G be a simple connected graph and A (G) be its adjacency matrix. The terms singularity, eigenvalues, and characteristic polynomial of G mean those of A (G). A nonsingular graph G is said to have the reciprocal eigenvalue property if the reciprocal of each eigenvalue of G is also an eigenvalue. A graph G (possibly singular) is said to have the weak reciprocal eigenvalue property if the reciprocal of each nonzero eigenvalue of it is also an eigenvalue. In Barik et al. (2022) [3] , the authors proved that there is no nontrivial tree with the weak reciprocal eigenvalue property and posed the following question: "Does there exist a nontrivial graph with the weak reciprocal eigenvalue property?" Suppose that G is singular and the characteristic polynomial of G is x n − k (x k + a 1 x k − 1 + ⋯ + a k). Assume that A (G) has rank k , so that a k ≠ 0. Can we ever have | a k | = 1 ? The answer turns out to be negative. As an application, we settle the question posed in Barik et al. (2022) [3]. Another similar application is also mentioned. It is natural to wonder, "Does there exist a nontrivial, simple, connected weighted graph with the weak reciprocal eigenvalue property?" We provide a class of such graphs. Furthermore, we extend our results to weighted graphs. [ABSTRACT FROM AUTHOR]

We consider linear ternary codes generated by A + β I , where A is the adjacency matrix of a strongly regular signed graph, I is the identity matrix and β ∈ F 3. Our results include theoretical examinations on dimension and distance, and the interplay between the codes. We also compute many structural parameters of codes for some infinite families of strongly regular signed graphs and establish more than 60 particular codes of comparatively small dimension. It occurs that some known linear ternary codes are covered by this approach. [ABSTRACT FROM AUTHOR]