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 adjacency matrix. Let $G$ be a graph which is cospectral with $Kite_{p,q}$ and the clique number of $G$ is denoted by $w(G)$. 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 the spectral characterization of pineapple graphs

The pineapple graph $K_p^q$ is obtained by appending $q$ pendant edges to a vertex of a complete graph $K_{p}$ ($q\geq 1,\ p\geq 3$). Zhang and Zhang ["Some graphs determined by their spectra", Linear Algebra and its Applications, 431 (2009) 1443-1454] claim that the pineapple graphs are determined by their adjacency spectrum. We show that their claim is false by constructing graphs which are cospectral and non-isomorphic with $K_p^q$ for every $p\geq 4$ and various values of $q$. In addition we prove that the claim is true if $q=2$, and refer to the literature for $q=1$, $p=3$, and $(p,q)=(4,3)$

Cospectrality Results for Signed Graphs with Two Eigenvalues Unequal to ±1\pm 1

Recently the collection $\cal G$ of all signed graphs for which the adjacency matrix has all but at most two eigenvalues equal to $\pm 1$ has been determined. Here we investigate $\cal G$ for cospectral pairs, and for signed graphs determined by their spectrum (up to switching). If the order is at most 20, the outcome is presented in a clear table. If the spectrum is symmetric we find all signed graphs in $\cal G$ determined by their spectrum, and we obtain all signed graphs cospectral with the bipartite double of the complete graph. In addition we determine all signed graphs cospectral with the Friendship graph $F_\ell$, and show that there is no connected signed graph cospectral but not switching equivalent with $F_\ell$

Cospectrality Results for Signed Graphs with Two Eigenvalues Unequal to ±1\pm 1

Recently the collection $\cal G$ of all signed graphs for which the adjacency matrix has all but at most two eigenvalues equal to $\pm 1$ has been determined. Here we investigate $\cal G$ for cospectral pairs, and for signed graphs determined by their spectrum (up to switching). If the order is at most 20, the outcome is presented in a clear table. If the spectrum is symmetric we find all signed graphs in $\cal G$ determined by their spectrum, and we obtain all signed graphs cospectral with the bipartite double of the complete graph. In addition we determine all signed graphs cospectral with the Friendship graph $F_\ell$, and show that there is no connected signed graph cospectral but not switching equivalent with $F_\ell$

On Signed Graphs With at Most Two Eigenvalues Unequal to ±1\pm 1

We present the first steps towards the determination of the signed graphs for which the adjacency matrix has all but at most two eigenvalues equal to 1 or -1. Here we deal with the disconnected, the bipartite and the complete signed graphs. In addition, we present many examples which cannot be obtained from an unsigned graph or its negative by switching

On the spectral characterization of mixed extensions of P₃

A mixed extension of a graph G is a graph H obtained from G by replacing each vertex of G by a clique or a coclique, whilst two vertices in H corresponding to distinct vertices x and y of G are adjacent whenever x and y are adjacent in G. If G is the path P3, then H has at most three adjacency eigenvalues unequal to 0 and -1. Recently, the first author classified the graphs with the mentioned eigenvalue property. Using this classification we investigate mixed extension of P3 on being determined by the adjacency spectrum. We present several cospectral families, and with the help of a computer we find all graphs on at most 25 vertices that are cospectral with a mixed extension of P_3

Informal carers’ experiences of caring for someone with Multiple Sclerosis: A photovoice investigation

ObjectivesThis study explores the lived experiences of carers of people with Multiple Sclerosis (MS), specifically in relation to their quality of life (QoL), through the use of images and narratives, with the aim of gaining a nuanced insight into the complex nature of QoL in the MS caregiving context.DesignReal‐time qualitative design using the photovoice method.MethodsTwelve MS carers (aged 30–73 years) took photographs of objects/places/events that represented enhancement or compromise to their QoL and composed written narratives for each photograph based on their experiences of caregiving. In total, 126 photographs and their corresponding narratives were analysed using content analysis.ResultsSeven inter‐related themes were identified. MS caregiving‐related challenges, sense of loss (e.g., loss of activities), emotional impact (e.g., feeling lonely), urge to escape, and sense of anxiety over the unpredictability of MS carer role were discussed in relation to the negative experiences that compromised their QoL. The themes precious moments (e.g., time spent with loved ones or hobbies) and helpful support (e.g., family and pets) encompassed participants’ positive experiences that enhanced their QoL.ConclusionsFindings demonstrated the multi‐faceted and complex nature of MS caregiver’s QoL and highlighted that although the experiences of MS carers were mostly negative, there were also some positive aspects to caregiving, that helped enhance carers’ QoL by ameliorating these negative experiences. These findings can be used to inform support programmes and enhance service provision for MS carers

Some relations among the largest eigenvalues of product matrix and graph matrices

( ) ve ( ) sırasıyla bir grafının komşuluk matrisi ve nokta derecelerinin bir köşegen matrisi olmak üzere ( ) ( ) ( ) matrisini tanımlarız. Bu makalede bu çarpım matrisinin spektral yarıçapı ile graf matrislerinin en büyük öz değerleri arasında bazı bağıntılar elde edilmiştir. Ayrıca nümerik sonuçlar da verilmiştir.We define product matrix as ( ) ( ) ( ) , where ( ) is an adjacency matrix and ( ) is a diagonal matrix of vertex degrees of a graph . In this paper, some relations among the spectral radius of product matrix and the largest eigenvalues of graph matrices are obtained. We also give numerical results for them