Non-isomorphic Graphs Research Articles

Construction of graphs with distinct eigenvalues

Let G (resp. Gn) be the set of connected graphs (resp. with n vertices) whose eigenvalues are mutually distinct, and G∗ (resp. Gn∗) the set of connected graphs (resp. with n vertices) whose eigenvalues are mutually distinct and main. Two graphs G and H are said to be cospectral if they share the same adjacency spectrum. In this paper, we give a new method to construct infinite families of graphs in G and G∗. Concretely, given a graph G in Gn or Gn∗, the infinite families of G or G∗ are constructed from G, and furthermore the spectra of such graphs are also characterized by the spectrum of G. By the way, we use this method to construct some infinite families of non-isomorphic cospectral graphs, especially, including the graphs in G and G∗.

Constructing Matching Equivalent Graphs

Two nonisomorphic graphs G and H are said to be matching equivalent if and only if G and H have the same matching polynomials. In this paper, some families matching equivalent graphs are constructed. In particular, a new method to construct cospectral forests is given.

Graph Isomorphism, Color Refinement, and Compactness

Color refinement is a classical technique used to show that two given graphs G and H are non-isomorphic; it is very efficient, although it does not succeed on all graphs. We call a graph G amenable to color refinement if the color refinement procedure succeeds in distinguishing G from any non-isomorphic graph H. Babai et al. (SIAM J Comput 9(3):628–635, 1980) have shown that random graphs are amenable with high probability. We determine the exact range of applicability of color refinement by showing that amenable graphs are recognizable in time $${O((n+m)\log n)}$$ , where n and m denote the number of vertices and the number of edges in the input graph. We use our characterization of amenable graphs to analyze the approach to Graph Isomorphism based on the notion of compact graphs. A graph is called compact if the polytope of its fractional automorphisms is integral. Tinhofer (Discrete Appl Math 30(2–3):253–264, 1991) noted that isomorphism testing for compact graphs can be done quite efficiently by linear programming. However, the problem of characterizing compact graphs and recognizing them in polynomial time remains an open question. Our results in this direction are summarized below:

Which Q-cospectral graphs have same degree sequences

Let Cn denote the cycle with n vertices, and let τ(G) be the number of triangles of G. If G is a graph with n vertices, then μ1(G)≥μ2(G)≥⋯≥μn(G) denote the signless Laplacian spectrum of G. Two graphs are said to be Q-cospectral if they have the same signless Laplacian spectrum. A graph G is said to be Q−DS if there is no other non-isomorphic graph H such that G and H are Q-cospectral. In this paper, we prove that “Let G be a graph with n≥12 vertices and 1≤μn(G)≤μ2(G)≤5<n<μ1(G) or let G be a graph with n≥10 vertices and 0<μn(G)≤μ2(G)≤4<n<μ1(G). If G and H are Q-cospectral, then G and H share the same degree sequences and τ(G)=τ(H)”. As a consequence of our results, we show that all multi-wheel graphs K1∨(Cq1∪Cq2∪⋯∪Cqt) are Q−DS.

Decoherence-enhanced performance of quantum walks applied to graph isomorphism testing

Computational advantages gained by quantum algorithms rely largely on the coherence of quantum devices and are generally compromised by decoherence. As an exception, we present a quantum algorithm for graph isomorphism testing whose performance is optimal when operating in the partially coherent regime, as opposed to the extremes of fully coherent or classical regimes. The algorithm builds on continuous-time quantum stochastic walks (QSWs) on graphs and the algorithmic performance is quantified by the distinguishing power (DIP) between non-isomorphic graphs. The QSW explores the entire graph and acquires information about the underlying structure, which is extracted by monitoring stochastic jumps across an auxiliary edge. The resulting counting statistics of stochastic jumps is used to identify the spectrum of the dynamical generator of the QSW, serving as a novel graph invariant, based on which non-isomorphic graphs are distinguished. We provide specific examples of non-isomorphic graphs that are only distinguishable by QSWs in the presence of decoherence.

A large family of cospectral Cayley graphs over dihedral groups

The adjacency spectrum of a graph Γ, which is denoted by Spec(Γ), is the multiset of eigenvalues of its adjacency matrix. We say that two graphs Γ and Γ′ are cospectral if Spec(Γ)=Spec(Γ′). In this paper for each prime number p, p≥23, we construct a large family of cospectral non-isomorphic Cayley graphs over the dihedral group of order 2p.

On the distance spectra of coronas

In this article, we give a complete description of the eigenvalues and the corresponding eigenvectors of the distance matrix and the distance signless Laplacian matrix of corona of two graphs and (denoted by ), when is connected transmission regular and is regular. Further, when is connected transmission regular, we give a complete description of the eigenvalues and the eigenvectors of the distance Laplacian matrix of for any graph . As an application, we find families of distance, distance Laplacian and distance signless Laplacian cospectral nonisomorphic graphs.

The spectral characterization of butterfly-like graphs

Let a(k)=(a1,a2,…,ak) be a sequence of positive integers. A butterfly-like graphWp(s);a(k) is a graph consisting of s(≥1) cycle of lengths p+1, and k(≥1) paths Pa1+1, Pa2+1, …, Pak+1 intersecting in a single vertex. The girth of a graph G is the length of a shortest cycle in G. Two graphs are said to be A-cospectral if they have the same adjacency spectrum. For a graph G, if there does not exist another non-isomorphic graph H such that G and H share the same Laplacian (respectively, signless Laplacian) spectrum, then we say that G is L−DS (respectively, Q−DS). In this paper, we firstly prove that no two non-isomorphic butterfly-like graphs with the same girth are A-cospectral, and then present a new upper and lower bounds for the i-th largest eigenvalue of L(G) and Q(G), respectively. By applying these new results, we give a positive answer to an open problem in Wen et al. (2015) [17] by proving that all the butterfly-like graphs W2(s);a(k) are both Q−DS and L−DS.

Total edge irregularity strength of disjoint union of double wheel graphs

An edge irregular total k-labeling f : V ∪ E → {1, 2, 3,...,k} of a graph G = (V, E) is a labeling of vertices and edges of G in such a way thatfor any two different edges uv and u'v' their weights f (u) + f (uv) + f (v) and f (u') + f (u'v') + f (v') are distinct. The total edge irregularity strength tes(G) is defined as the minimum k for which the graph G has an edge irregular total k-labeling. In this paper, we determine the total edge irregularity strength of disjoint union of p isomorphic double wheel graphs and disjoint union of p consecutive non-isomorphic double wheel graphs.

Enumeration of symmetric (45,12,3) designs with nontrivial automorphisms

We show that there are exactly 4285 symmetric (45,12,3) designs that admit nontrivial automorphisms. Among them there are 1161 self-dual designs and 1562 pairs of mutually dual designs. We describe the full automorphism groups of these designs and analyze their ternary codes. R. Mathon and E. Spence have constructed 1136 symmetric (45,12,3) designs with trivial automorphism group, which means that there are at least 5421 symmetric (45,12,3) designs. Further, we discuss trigeodetic graphs obtained from the symmetric $(45,12,3)$ designs. We prove that $k$-geodetic graphs constructed from mutually non-isomorphic designs are mutually non-isomorphic, hence there are at least 5421 mutually non-isomorphic trigeodetic graphs obtained from symmetric $(45,12,3)$ designs.

On the spectral characterization of kite graphs

The \textit{Kite graph}, denoted by $Kite_{p,q}$ is obtained by appending a complete graph $K_{p}$ to a pendant vertex of a path $P_{q}$. In this paper, firstly we show that no two non-isomorphic kite graphs are cospectral w.r.t the adjacency matrix. Let $G$ be a graph which is cospectral with $Kite_{p,q}$ and let $w(G)$ be the clique number of $G$. Then, it is shown that $w(G)\geq p-2q+1$. Also, we prove that $Kite_{p,2}$ graphs are determined by their adjacency spectrum.

On strongly asymmetric and controllable primitive graphs

A strongly asymmetric graph is a graph all of whose overgraphs are non-isomorphic. A graph whose 0–1 adjacency matrix has distinct and main eigenvalues is said to be controllable. We show that all controllable graphs are strongly asymmetric, thus strengthening the well-known result that controllable graphs have a trivial automorphism group. Moreover, we define the subclass of controllable primitive graphs (CP-graphs), and show that every controllable graph is either primitive or is the union or join of two controllable graphs. With the aid of CP-graphs, we prove that controllable graphs on at least six vertices have an induced path on four vertices. Furthermore, we show that the number of non-isomorphic controllable graphs and the number of non-isomorphic controllable primitive graphs on n≥2 vertices are both even. Using a computer search, we show that most of the controllable graphs on up to ten vertices are primitive. Among the controllable graphs on n<12 vertices, the controllable non-primitive graphs are proved to be related to the singular controllable graphs on n−1 vertices.

Cospectrality of complete bipartite graphs

Richard Brualdi proposed in [Research problems from the Aveiro workshop on graph spectra, Linear Algebra Appl. 2007;423:172–181] the following problem: (Problem AWGS.4) Let and be two non-isomorphic graphs on n vertices with spectrarespectively. Define the distance between the spectra of and asDefine the cospectrality of byLetProblem AInvestigate for special classes of graphs. Problem BFind a good upper bound on . In this paper, we study Problem A and determine the cospectrality of all complete bipartite graphs by the Euclidian distance. More precisely, we show that for all positive integers m and n there are some positive integers r, s and a non-negative integer t such that , where is the complete bipartite graph with parts of sizes m and n.

On the domination polynomials of friendship graphs

Let G be a simple graph of order n. The domination polynomial of G is the polynomial D(G,x)= n?i=0 d(G,i)xi, where d(G,i) is the number of dominating sets of G of size i. Let n be any positive integer and Fn be the Friendship graph with 2n + 1 vertices and 3n edges, formed by the join of K1 with nK2. We study the domination polynomials of this family of graphs, and in particular examine the domination roots of the family, and find the limiting curve for the roots. We also show that for every n &gt; 2, Fn is not D-unique, that is, there is another non-isomorphic graph with the same domination polynomial. Also we construct some families of graphs whose real domination roots are only -2 and 0. Finally, we conclude by discussing the domination polynomials of a related family of graphs, the n-book graphs Bn, formed by joining n copies of the cycle graph C4 with a common edge.

Signless Laplacian spectral characterization of graphs with isolated vertices

A graph is said to be DQS if there is no other non-isomorphic graph with the same signless Laplacian spectrum. For a DQS graph G, we show that G ? rK1 is DQS under certain conditions. Applying these results, some DQS graphs with isolated vertices are obtained.

A Characterization of $K_{2,4}$-Minor-Free Graphs

We provide a complete structural characterization of $K_{2,4}$-minor-free graphs. The 3-connected $K_{2,4}$-minor-free graphs consist of nine small graphs on at most eight vertices, together with a family of planar graphs that contains $2n-8$ nonisomorphic graphs of order $n$ for each $n \geq 5$ as well as $K_4$. To describe the 2-connected $K_{2,4}$-minor-free graphs we use $xy$-outerplanar graphs, graphs embeddable in the plane with a Hamilton $xy$-path so that all other edges lie on one side of this path. We show that, subject to an appropriate connectivity condition, $xy$-outerplanar graphs are precisely the graphs that have no rooted $K_{2,2}$ minor where $x$ and $y$ correspond to the two vertices on one side of the bipartition of $K_{2,2}$. Each 2-connected $K_{2,4}$-minor-free graph is then (i) outerplanar, (ii) the union of three $xy$-outerplanar graphs and possibly the edge $xy$, or (iii) obtained from a 3-connected $K_{2,4}$-minor-free graph by replacing each edge $x_iy_i$ in a set $\{x_1 y_1, x...

Per-spectral and adjacency spectral characterizations of a complete graph removing six edges

Cámara and Haemers (2014) investigated when a complete graph with some edges deleted is determined by its adjacency spectrum (DAS for short). They claimed: for any m≥6 and every large enough n one can obtain graphs which are not DAS by removing m edges from a complete graph Kn. Let Gn denote the set of all graphs obtained from a complete graph Kn by deleting six edges. In this paper, we show that all graphs in Gn are uniquely determined by their permanental spectra. However, we show that for each n≥7 or n=5 there is just one pair of nonisomorphic cospectral graphs in Gn, and for n=4 or 6 all graphs in Gn are DAS.

Edit distance measure for graphs

In this paper, we investigate a measure of similarity of graphs similar to the Ramsey number. We present values and bounds for g(n, l), the biggest number k guaranteeing that there exist l graphs on n vertices, each two having edit distance at least k. By edit distance of two graphs G, F we mean the number of edges needed to be added to or deleted from graph G to obtain graph F. This new extremal number g(n, l) is closely linked to the edit distance of graphs. Using probabilistic methods we show that g(n, l) is close to \(\frac{1} {2}\left( {\begin{array}{*{20}c} n 2 \end{array} } \right)\) for small values of l > 2. We also present some exact values for small n and lower bounds for very large l close to the number of non-isomorphic graphs of n vertices.

Functional graphs of polynomials over finite fields

Given a function f in a finite field Fq of q elements, we define the functional graph of f as a directed graph on q nodes labelled by the elements of Fq where there is an edge from u to v if and only if f(u)=v. We obtain some theoretical estimates on the number of non-isomorphic graphs generated by all polynomials of a given degree. We then develop a simple and practical algorithm to test the isomorphism of quadratic polynomials that has linear memory and time complexities. Furthermore, we extend this isomorphism testing algorithm to the general case of functional graphs, and prove that, while its time complexity deviates from linear by a (usually small) multiplier dependent on graph parameters, its memory complexity remains linear. We exploit this algorithm to provide an upper bound on the number of functional graphs corresponding to polynomials of degree d over Fq. Finally, we present some numerical results and compare function graphs of quadratic polynomials with those generated by random maps and pose interesting new problems.

Total Edge Irregularity Strength of Disjoint Union of Wheel Graphs

In this paper, we determine the total edge irregularity strength of the disjoint union of p isomorphic wheel graphs, disjoint union of p consecutive non-isomorphic wheel graphs and the total edge irregularity strength of p(≥3) disjoint union of any wheel graphs.